The Cryosphere 7 1455ndash1471 2013wwwthe-cryospherenet714552013doi105194tc-7-1455-2013copy Author(s) 2013 CC Attribution 30 License

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Changes in glacier equilibrium-line altitude in the western Alpsfrom 1984 to 2010 evaluation by remote sensing and modeling ofthe morpho-topographic and climate controls

A Rabatel1 A Letreacuteguilly 1 J-P Dedieu2 and N Eckert3

1Univ Grenoble Alpes CNRS LGGE UMR5183 38000 Grenoble France2Univ Grenoble Alpes CNRS IRD LTHE UMR5564 38000 Grenoble France3IRSTEA UR ETGR 38000 Grenoble France

Correspondence toA Rabatel (antoinerabatelujf-grenoblefr)

Received 16 April 2013 ndash Published in The Cryosphere Discuss 3 June 2013Revised 2 August 2013 ndash Accepted 13 August 2013 ndash Published 24 September 2013

Abstract We present time series of equilibrium-line altitude(ELA) measured from the end-of-summer snow line altitudecomputed using satellite images for 43 glaciers in the west-ern Alps over the 1984ndash2010 period More than 120 satelliteimages acquired by Landsat SPOT and ASTER were usedIn parallel changes in climate variables summer cumulativepositive degree days (CPDD) and winter precipitation wereanalyzed over the same time period using 22 weather sta-tions located inside and around the study area Assuming acontinuous linear trend over the study period (1) the averageELA of the 43 glaciers increased by about 170 m (2) sum-mer CPDD increased by about 150 PDD at 3000 m asl and(3) winter precipitation remained rather stationary SummerCPDD showed homogeneous spatial and temporal variabil-ity winter precipitation showed homogeneous temporal vari-ability but some stations showed a slightly different spatialpattern Regarding ELAs temporal variability between the43 glaciers was also homogeneous but spatially glaciers inthe southern part of the study area differed from glaciersin the northern part mainly due to a different precipitationpattern A sensitivity analysis of the ELAs to climate andmorpho-topographic variables (elevation aspect latitude)highlighted the following (1) the average ELA over the studyperiod of each glacier is strongly controlled by morpho-topographic variables and (2) the interannual variability ofthe ELA is strongly controlled by climate variables with theobserved increasing trend mainly driven by increasing tem-peratures even if significant nonlinear low-frequency fluc-tuations appear to be driven by winter precipitation anoma-

lies Finally we used an expansion of Lliboutryrsquos approachto reconstruct fluctuations in the ELA of any glacier of thestudy area with respect to morpho-topographic and climatevariables by quantifying their respective weight and the re-lated uncertainties in a consistent manner within a hierarchi-cal Bayesian framework This method was tested and vali-dated using the ELA measured on the satellite images

1 Introduction

Historically the glacierrsquos annual surface mass balance andequilibrium-line altitude (ELA) have been computed from di-rect field measurements of snow accumulation and snowiceablation through point measurements using a network ofablation stakes and snow pits on individual glaciers Thelongest mass balance data series started in the late 1940s forStorglaciaumlren (Sweden) Taku Glacier (USA) Limmern andPlattalva glaciers (Switzerland) Sarennes Glacier (France)and Storbreen (Norway) The World Glacier Monitoring Ser-vice (WGMS) has mass balance data on around 260 glaciersworldwide (note that the World Glacier Inventory containsmore than 130 000 glaciers) but uninterrupted time seriesspanning more than 40 yr are available for only 37 glaciers(WGMS 2012) The small number of available data seriesis due to the cost in terms of money manpower and timeof this laborious method The small number of data seriesis an obstacle that needs to be overcome if we are to im-prove our knowledge of the relationship between climate and

Published by Copernicus Publications on behalf of the European Geosciences Union

1456 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

glacier changes from mountain-range up to regional scaleas well as our knowledge of the contribution of glaciers towater resources in the functioning of high-altitude water-sheds Remote sensing provides a unique opportunity to ad-dress the question of glacier changes at regional scale Sincethe availability of satellite data in the 1970s several meth-ods have been developed to compute changes in glacier vol-ume at multiannual to decadal scale using digital elevationmodels (DEMs) (eg Echelmeyer et al 1996 Baltsaviaset al 1999 Berthier et al 2004 Gardelle et al 2012)or the annual ELA and mass balance (eg Oslashstrem 1975Rabatel et al 2005 2008 2012a) However satellite de-rived DEMs are not accurate enough to compute variations inthe annual volume of mountain glaciers Another alternativeemerged from recent studies based on ELA and distributedmass balance modeling using meteorological input fields de-rived from local weather stations data reanalysis or regionalclimate models (eg Zemp et al 2007 Machguth et al2009 Marzeion et al 2012) However such modeling ap-proaches still have a number of limitations eg accumulationis underestimated in mountainous regions On the other handfor mid-latitude mountain glaciers the end-of-summer snowline altitude (SLA) is a good indicator of the ELA and thus ofthe annual mass balance (Lliboutry 1965 Braithwaite 1984Rabatel et al 2005) This enables ELA changes to be recon-structed for long time periods from remote-sensing data (De-muth and Pietroniro 1999 Rabatel et al 2002 2005 2008Barcaza et al 2009 Mathieu et al 2009) because the snowline is generally easy to identify using aerial photographs andsatellite images (Meier 1980 Rees 2005) Consequentlyit is possible to study the climatendashglacier relationship at amassif- or regional scale (Clare et al 2002 Chinn et al2005) which is particularly useful in remote areas where nodirect measurements are available

In the current study we rely on previous studies con-ducted in the French Alps (Dedieu and Reynaud 1990 Ra-batel et al 2002 2005 2008) to reconstruct ELA timeseries for more than 40 glaciers over the 1984ndash2010 pe-riod using the end-of-summer snow line detected on satel-lite images Our aim are (1) to quantify at a regional scalethe temporal and spatial changes of the ELA (2) to char-acterize the relationships between ELA and both morpho-topographic and climate variables and (3) to reconstruct thespatio-temporal variability of annual ELA time series by in-corporating the above-mentioned relationships in an expan-sion of Lliboutryrsquos variance decomposition model (1974)

2 Study area data and method

21 Selection of the study sites

A recent update of the glacier inventory of the French Alpslists 593 glaciers (Rabatel et al 2012b) The total glacialcoverage in the French Alps was about 340 km2 in the mid-

Fig 1 The 43 glaciers studied (red) among the 593 glaciers in theFrench Alpine glacier inventory (in blue) plus additional glaciers onthe Italian and Swiss sides of the Franco-Italian and Franco-Swissborders The three boxes represent the main glacierized rangesquoted in the text and figures ie Mont Blanc Vanoise and Ecrins

1980s and had decreased to about 275 km2 in the late 2000sIn the current study 43 glaciers located in the French Alpsor just next to the border with Switzerland and Italy wereselected (Table 1 Fig 1) The selection was based on the fol-lowing criteria (1) glaciers had to have a high enough max-imum elevation to allow observation of the snow line everyyear during the study period (2) glaciers with all aspects hadto be represented and (3) glaciers located in all the glacier-ized mountain ranges in the French Alps from the southern-most (4450prime N) to the northern-most (4600prime N) had to beincluded

Among the selected glaciers three belong to the GLACIO-CLIM observatory which runs a permanent mass balancemonitoring program These three glaciers are Saint SorlinGlacier monitored since 1957 ( 31 in Table 1 and Fig 1)Argentiegravere Glacier monitored since 1975 ( 4 in Table 1 andFig 1) and Geacutebroulaz Glacier monitored since 1983 ( 28in Table 1 and Fig 1) Mass balance data available on these

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1457

Table 1 List of the glaciers studied and the weather stations used in this study Numbers (for the glaciers) and codes (for the weatherstations) refer to Fig 1 All glaciers are located in the French Alps except when specified in brackets CH= Switzerland and IT= Italy Forthe weather stations letters in brackets identify the data producer MF= Meacuteteacuteo-France EDF= Electriciteacute de France MS= Meacuteteacuteo-SuisseThe coordinates are given in degrees minutes for latitude and longitude and in m asl for elevation

Glaciers Weather stations Name Name Name (data producer) ndash Code on Fig 1 Coord m asl

1 Trient (CH) 23 Grand Meacutean Chamonix-Mt-Blanc (MF) ndash Cham 4555prime N 0652prime E 10422 Tour 24 Pelve Contamine-sur-Arve (MF) ndashOff the map 4607prime N 0620prime E 4523 Saleina (CH) 25 Vallonnet Contamines-Montjoie (EDF) ndash Cont M 4549prime N 0643prime E 11804 Argentiegravere 26 Arpont Hauteluce (EDF) ndash Haut 4546prime N 0638prime E 12155 Talegravefre 27 Mahure Bourg-Saint-Maurice (MF) ndash Bg St M 4536prime N 0645prime E 8656 Preacute de Bar (IT) 28 Geacutebroulaz Peisey-Nancroix (EDF) ndash Pe Na 4532prime N 0645prime E 13507 Triolet (IT) 29 Baounet Pralognan-La-Vanoise (EDF) ndash Pra la V 4523prime N 0643prime E 14208 Leschaux 30 Rochemelon Bessans (EDF) ndash Bes 4519prime N 0659prime E 17159 Mont Blanc (IT) 31 Saint Sorlin Termignon (MF) ndash Ter 4516prime N 0648prime E 128010 Freiney (IT) 32 Quirlies Lyon ndash Bron (MF) ndashOff the map 4543prime N 0456prime E 19811 Brouillard (IT) 33 Girose St-Etienne-de-St-Geoirs (MF) ndashOff the map 4521prime N 0518prime E 38412 Treacute la Tecircte 34 Lautaret Allemond (EDF) ndashOff the map 4512prime N 0602prime E 127013 Leacutee Blanche (IT) 35 Mont de Lans Vaujany (EDF) ndash Vau 4509prime N 0602prime E 77214 des Glaciers 36 Selle Besse-en-Oisans (EDF) ndash B en O 4504prime N 0610prime E 152515 Rutor (IT) 37 Casset La Grave (EDF) ndash LaG 4503prime N 0617prime E 178016 Savinaz 38 Blanc Monetier-Les-Bains (EDF) ndash M les B 4458prime N 0630prime E 145917 Gurraz 39 Vallon Pilatte St-Christophe-en-Oisans (EDF) ndash StC 4456prime N 0611prime E 157018 Sassiegravere 40 Violettes Pelvoux (EDF) ndash Pel 4452prime N 0628prime E 126019 Tsantelaina (IT) 41 Rouies Chapelle-en-Valgaudemar (EDF) ndashOff the map 4448prime N 0611prime E 127020 Grande Motte 42 Seacuteleacute Embrun (MF) ndashOff the map 4433prime N 0630prime E 87121 Mulinet 43 Pilatte Genegraveve (MS) ndashOff the map 4614prime N 0605prime E 42022 Arcellin Grand St Bernard (MS) ndashOff the map 4552prime N 0710prime E 2472

three glaciers were used to assess the representativeness ofthe end-of-summer SLA computed from the satellite imagesas an indicator of the ELA and of the annual mass balancecomputed from field measurements (see Sect 31 below)

22 Snow line altitude retrieved from satellite images

221 Satellite images and snow line delineation

A total of 122 images of the 43 glaciers were used to coverthe 27 yr study period Unfortunately in some years usableimages were not available for all the glaciers because of(1) cloud cover hiding the underlying terrain and (2) snow-falls that can occur in late summer and completely cover theglaciers This was mainly the case in the 1980s and 1990swhen fewer satellites were in orbit than in the 2000s Theimages we used were acquired by the following satellitesLandsat 4TM 5TM 7ETM+ SPOT 1 to 5 and ASTER withspatial resolutions ranging from 25 to 30 m (see supplemen-tary material Table 1 for a detailed list of the images used)

For the snow line delineation on multispectral images atest of band combinations and band ratios to facilitate theidentification of the snow line on the images is described inRabatel et al (2012a see Fig 4) These authors concludedthat the combination of the green near-infrared and short-wave infrared bands (see Table 2 for details about the wave-

Table 2 Characteristics of the wavelength of the spectral bandsused to identify the snow line on the satellite images of the threesensors

Wavelength (microm)Green Near-infrared Shortwave

infrared

Landsat 052ndash060 077ndash090 155ndash175SPOT 050ndash059 079ndash089 160ndash170ASTER 052ndash060 076ndash086 158ndash175

lengths of each satellite) was the most appropriate to identifythe limit between snow and ice The snow line was delin-eated on the central part of the glaciers to avoid border ef-fects on the glacier banks shadows from surrounding slopesadditional snow input by avalanches overaccumulation dueto wind (Fig 2) which could generate equilibrium-line po-sition dependence on local conditions (Rabatel et al 2005)The delineation was performed manually because automaticmethods hardly succeed in distinguishing the snow line fromthe firn-line when both to be presented on the glacier Thisdistinction results to be also visually difficult when the pixelsize of the satellite images is too coarse (see the discussionsection)

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1458 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 2 Example of Landsat image used to delineate the snow line(red line) with a combination of spectral bands green near-infraredand shortwave infrared Argentiegravere Glacier (Mont Blanc Massif)Landsat TM5 image acquired on 9 September 1987

222 Digital elevation model computation of thealtitude of the snow line and of the glaciermorpho-topographic variables

The average altitude of each snow line was calculated byoverlaying the shapefiles containing the digitized snow lineson a DEM For each glacier the DEM was also used to com-pute the following morpho-topographic variables used in thisstudy surface area mean elevation aspect latitude and lon-gitude of the glacier centroid The mean elevation of eachglacier has been computed as the arithmetic mean of the ele-vation of each pixel of the DEM included within the glaciersoutline This mean elevation is rather close to the median el-evation of the glaciers Indeed the difference between thetwo variables is 10 m on average for the 43 studied glaciersThis shows that the studied glaciers have on average an al-most symmetrical area-altitude distribution (Braithwaite andRaper 2010)

Several DEMs are available for the study area Amongthese one comes from the French National Geographical In-stitute (IGN) with a resolution of 25 m and dates from theearly 1980s Another one is the ASTERGDEM with a resolu-tion of 30 m and dates from the mid-2000s The average dif-ference between the two DEMs at the mean elevation of theSLA of the 43 glaciers studied over the whole study periodie approx 3050 m asl is about 20 m This matches the ver-tical accuracy of the DEMs and is small in comparison to theinterannual variability of the SLA (the standard deviation ofthe measured SLA over the whole study period ranges be-tween 75 and 255 m depending on the glacier) However be-cause the glacier surface lowering can reach several tens ofmeters at lower elevation (up to 80 m at 1500 m asl) due to

the important glacier shrinkage over the last decades (Thib-ert et al 2005 Paul and Haeberli 2008 Vincent et al 2009Berthier and Vincent 2012) when the SLA is at lower eleva-tion as was the case in 2001 (2839plusmn 129 m asl) it is betterto use a DEM as close as possible to the date of the used im-ages Accordingly the DEM from the French IGN was usedfor the first half of the period (till the late 1990s) and theASTERGDEM was used for the second part of the periodChanging from one DEM to the other in the late 1990s hasno impact on the results because at that time the SLA waslocated between 3000 and 3200 m asl an altitudinal rangefor which the difference in elevation between the DEMs islower than their vertical accuracies

An uncertainty was estimated for each SLA Uncertaintyresults from different sources of error (Rabatel et al 20022012a) (1) the pixel size of the images which ranged be-tween 25 and 30 m depending on the sensor (2) the slope ofthe glacier in the vicinity of the SLA which ranged from 7

to 32 depending on the glacier and the zone where the SLAwas located in any given year (3) the vertical accuracy of theDEM which was about 20 m and (4) the standard deviationof the calculation of the average SLA along its delineationwhich ranged from 10 m to 170 m depending on the glacierand on the year concerned The last source of error (4) is themost important The resulting total uncertainty of the SLAs(root of the quadratic sum of the different independent er-rors) ranged fromplusmn15 to plusmn170 m depending on the yearand glacier concerned The uncertainty was greater when theSLA was located on a part of a glacier where the slope issteepest because in this case the standard deviation of thecomputed SLA is high

223 Method used to fill data gaps in the snow linealtitude time series

In spite of the large number of images used in this study therewere still some gaps in the SLA time series In the 27 yr studyperiod for the 43 glaciers studied the SLA was measured in85 of the 1081 cases Thus data were missing in 15 ofthe cases

To fill these gaps a bi-linear interpolation was appliedOriginally this approach was used by Lliboutry (1965 1974)to extract the spatial and temporal terms from a very incom-plete table of mass balance measurements (70 gaps) in theablation zone of Saint Sorlin Glacier This approach is basedon the fact that the glacier mass balance depends on both thesite and the time of measurement This approach was subse-quently extended to a set of glaciers in the Alps by Reynaud(1980) and later on to glaciers in other mountain ranges byLetreacuteguilly and Reynaud (1990) Here we applied it to ourSLA table that had far fewer gaps writing simply

SLAit = αi + βt + εit (1)

where SLAit is the snow line altitude of the glacieri forthe the yeart αi is long-term mean for each glacier over

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1459

the period of record βt is a term (depending on the yearonly) which is common to all the glaciers analyzed andεit = SLAit minusαi minusβt with εitsimN(0σ 2

ε ) a table of centeredresiduals assumed to be independent and Gaussian (N de-notes the normal distribution) The purpose is not simply toobtain theαi andβt terms but to estimate each missing valueSLAit on glacieri and yeart as

αi =1

T

Tsumt=1

SLAit (2)

βt =1

M

Msumi=1

(SLAit minus αi

) (3)

ˆSLAit = αi + βt (4)

whereT is the number of years of the study periodM is thenumber of glaciers studied and the circumflex is a classicalstatistical estimate

To maximize the consistency of the reconstructionαi wascalculated using only the years for which no data was miss-ing However only six years (out of the 27) had the SLA foreach of the 43 glaciers which is not quite enough to obtainan average SLA that is not too dependant on a specific yearConsequently the initial data set was divided into three ge-ographical subsets the Ecrins Massif the Vanoise area andthe Mont Blanc Massif and the computation was applied in-dependently to each one of these subsets In this way moreyears with SLAs for all the glaciers of each subset were avail-able (see Table 3 where the statistics of each subset are pre-sented) The year 1995 was the only exception For this spe-cific year because very few values were available the recon-struction was made considering the three subsets together

The RMSE ofεit representing the model errors was 73 mThis is less than half the standard deviation of the measuredSLAs (174 m) showing that the use of the bi-linear modelmakes sense on this data set

23 Meteorological data

Table 1 also lists the weather stations we used which arelocated in and around the study area (Fig 1) Among the105 weather stations available in the Meacuteteacuteo-France database(including stations operated by Meacuteteacuteo-France and Electriciteacutede France) in the French departments (Hautes-Alpes IsegravereSavoie and Haute-Savoie) in the study area 40 were se-lected on the basis of data availability and their proximityto the glacierized areas (except the Lyon-Bron weather sta-tion which was selected as a regional reference) Among the40 weather stations only 20 of which had a continuous se-ries of winter precipitation and summer temperatures overthe period 1984ndash2010 and were finally used Two weatherstations in Switzerland were also used The climate variables

Table 3 Statistics of the three subsets used to reconstruct missingvalues

Name of Number Missing Number of RMSE RMSEsubset of glaciers values complete SLA εit

years usedfor αi

Ecrins 13 11 14 148 m 62 mVanoise 16 20 12 160 m 66 mMont Blanc 14 14 14 201 m 90 m

used in the analysis were (1) cumulative positive degree days(CPDD) from 15 May to 15 September for each yeart ex-trapolated to the altitude of 3000 m asl using a standardgradient of 6C kmminus1 (3000 m asl being the approximatemean elevation of the SLA of the 43 glaciers studied overthe whole study period) and (2) cumulated winter precipi-tation from 15 September of the yeart-1 to 15 May of theyeart During this period liquid precipitation is negligible at3000 m asl

For the CPDD the date of 15 September was chosen be-cause it matches the average date of the satellite images usedto delineate the snow line Furthermore it should be notedthat at 3000 m asl in our study area after 15 SeptemberCPDD are negligible (Thibert et al 2013)

24 Modeling of the control of the morpho-topographicand climate variables

From our data set of SLA series glacier morpho-topographicvariables and climate variables we propose a simple expan-sion of Lliboutryrsquos approach (1974) to model and reconstructthe SLA fluctuations of any glacier in our study area over thestudy period This approach also enables us to quantify therespective control of each morpho-topographic and climatevariables on the SLA spatio-temporal variability

Following the ideas of Eckert et al (2011) the modelis implemented in a hierarchical Bayesian framework(eg Wikle 2003) The main advantage over more empir-ical approaches is treating the different sources of uncer-tainty in a consistent manner For instance the available in-formation (missing values and repeated observations amongcorrelated glacier series) is ldquorespectedrdquo when inferring thespatio-temporal patterns of interest more specifically the re-gression variables relating ELA fluctuations their morpho-topographic and climate drivers and their respective weight

In detail starting from the available SLA table with hol-low (ie without reconstructed values) and Eq (1) we furtherdecomposed the spatial term as

αi = a0 +

sumk

Xkiak + Vi (5)

where ao is the interannual regional averageXki are thespatial explanatory variables considered for each glacier iethe morpho-topographic variables latitude average altitude

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1460 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

surface area and aspect andVi sim N(0σ 2

v

)is the spatial

residual ie the local effect for the glacieri which is notexplained by the spatial explanatory variables considered

Similarly the temporal term is decomposed as

βt =

sumk

Xktbk + Zt (6)

where theXkt are the temporal explanatory variables con-sidered ie the climate variables summer CPDD and win-ter precipitation andZt sim N

(0σ 2

z

)is the temporal residual

ie the annual effect common to all glaciers for any yeart which is not explained by the temporal explanatory variablesconsidered

For the sake of simplicity the model is fed with reducedstandardized variables (by dividing by the standard devia-tion the difference between each value and the average of theseries) This avoids identifiability problems by granting thatsumt

βt = 0 and makes theak andbk coefficients more directly

interpretable at least for non-heavily correlated explanatoryvariables (see below)

Without the Vi and Zt terms the model would be aclassical spatio-temporal linear model with easy inferenceeg with likelihood maximization providing analytical so-lutions for point estimates and asymptotic standard errorsHowever this would not allow us to distinguish between ex-plained and unexplained variance in the spatial and temporalterms Hence theVi andZt terms make the model hierar-chical ie richer but also more tricky to estimate This chal-lenge was solved using Bayesian Markov chain Monte Carlomethods (MCMC Brooks 1998) As detailed in Eckert etal (2007 2010b) for similar problems the robustness of theinference was checked using tests based on starting differ-ent simulation runs at different points of the parameter space(Brooks and Gelman 1998) Poorly informativeprior prob-ability distributions were used for all parameters howeverthis standard choice allows us to obtainposteriorestimatesonly driven by information conveyed by the data From thejoint posteriordistribution of all parameters latent variablesand missing values we retained point estimates (theposte-rior mean)posteriorstandard deviations and 95 credibil-ity intervals the Bayesian counterpart of the classical confi-dence interval (Table 4) The construction of the model en-sures thatαi βt andεit are independent FurthermoreVi andsumk

Xkiak are independent so areZt andsumk

Xktbk Hence the

total variance of the data table is

VAR(SLAit ) = VAR

(sumk

Xkiak

)

+VAR

(sumk

Xktbk

)+ σ 2

v + σ 2z + σ 2 (7)

By analogy to the classicalR2 variance ratios can then becomputed to evaluate the respective weight of the different

Table 4Posterior characteristics of the proposed model applied onthe 43 ELA time series for the 1984ndash2010 periodak andbk denotethe regression parameters for each explanatory variable in bracketsOverall 25 and 975 denote the lower and upper bound of the95 credible interval

Mean 25 975

a1 (average altitude) 0381 0257 0502a2 (latitude) minus0123 minus0269 0023a3 (aspect) 0108 minus0022 0238a4 (surface area) minus0066 minus0199 0067b1 (summer CPDD) 0417 0268 0557b2 (winter precipitation) minus0311 minus0459 minus0155R2

clim 031 019 041R2

clim res 011 006 019R2

topo 024 016 032

R2topo res 009 005 014

R2res 025 019 030

R2tot 073 060 081

R2space 072 057 082

R2time 073 052 084

R2timespace 056 043 068

terms in ELA variability

R2topo = VAR

(sumk

Xkiak

)VAR(SLAit ) (8)

R2clim = VAR

(sumk

Xktbk

)VAR(SLAit ) (9)

R2topo res= σ 2

v VAR(SLAit ) (10)

R2clim res= σ 2

z VAR(SLAit ) (11)

R2res= σ 2VAR(SLAit ) (12)

Similarly ratios can be computed for the spatial and tem-poral terms solely

R2space= VAR

(sumk

Xkiak

)

(VAR

(sumk

Xkiak

)+ σ 2

v

) (13)

R2time = VAR

(sumk

Xktbk

)

(VAR

(sumk

Xktbk

)+ σ 2

z

) (14)

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1461

and getting rid of the random spatio temporal fluctuationsthe overall timespace ratio and determination coefficient inthe decomposition can be computed as

R2timespace=

(VAR

(sumk

Xktbk

)+ σ 2

z

)(

VAR (SLAit ) minus σ 2) (15)

R2tot =

(VAR

(sumk

Xkiak

)+ VAR

(sumk

Xktbk

))(

VAR (SLAit ) minus σ 2) (16)

All these statistics quantify the weight of mean spatial andtemporal effects in the total variability and the capacity ofthe chosen explanatory variables to model these mean ef-fects At the observation level of each yearglacier of themodel local and annual adjustment statistics can also becomputed as

δt =minus1

M

sumi

εit (17)

δi =minus1

T

sumt

εit (18)

δ2t = 1minus

radicradicradicradicradicradicsumi

ε2itsum

i

SLA2it

(19)

δ2i = 1minus

radicradicradicradicradicradicsumt

ε2itsum

t

SLA2it

(20)

respectively representing the annual and local bias and theannual and local determination coefficient withM the num-ber of glaciers considered andT the number of years studiedThey quantify the goodness of fit of each of the annuallocalseries

Variance ratios and adjustment statistics were evaluated bycomputing their values at each point of the MCMC iterativesimulation run giving access to point estimates and relateduncertainties taking into account the number of missing val-ues for each yearglacier (logically when nearly all observa-tions are missing for a given yearglacier the correspondingdetermination coefficientsδ2

tandδ2

iare unknown)

3 Results

Here we give the results of both the SLA and the climatevariables In Sect 31 we present the temporal and spatial

variability of the SLA measured on the 43 glaciers for the1984ndash2010 period In Sect 32 we present the variability ofclimate variables (ie summer CPDD and winter precipita-tion)

31 Changes in the equilibrium-line altitude

As mentioned above it has been demonstrated that on mid-latitude glaciers where superimposed ice is negligible theend-of-summer SLA is an accurate indicator of the ELA (seeFig 6 in Rabatel et al 2005 and Fig 3B in Rabatel et al2008) Because of this strong correlation and for the sake ofsimplicity we hereafter only use the term ELA

311 Temporal variability of the equilibrium-linealtitude

Figure 3a illustrates changes in the ELA over the 1984ndash2010period for the 43 glaciers studied (see also see SupplementTable II where all the data are presented) Considering allthe glaciers the average ELA for the whole period was lo-cated at 3035plusmn 120 m asl ie the interannual variabilityof the average ELA of all glaciers was high (120 m on av-erage) The difference between extreme years was as highas 460 m with the lowest average ELA measured in 1984(2790plusmn 180 m asl) and the highest average ELA measuredin 2003 (3250plusmn 135 m asl) In addition over the study pe-riod the ELA time series showed an average increasing trendof 64 m yrminus1 assuming a linear trend which results to be sta-tistically significant considering a risk of error of 5 Thisis the equivalent of an average increase of 170 m over the1984ndash2010 period ie higher than the interannual variabilityof the average ELA

The average annual ELA during the first five years of thestudy period was lower than the average ELA for the wholeperiod (except in 1986 when it was slightly higher than theaverage) These years match the end of a 15 yr period (fromthe mid-1970s to the late 1980s) during which alpine glaciersincreased in size due to higher winter accumulation and re-duced summer ablation (Vincent 2002 Thibert et al 2013)Since 2003 the average annual ELA has consistently beenabove the average ELA for the whole period Considering thetwo subsets ie before and after 2003 no significant trendappeared in either of the subsets The second subset is shorter(8 yr) than the first (19 yr) and trends computed on short pe-riods should be interpreted with caution However the aver-age ELA between the two subsets increased by about 140 mwhich is 82 of the total change Hence the year 2003 canalso be considered as a breakpoint in the time series It shouldbe noted that the 2003 extreme heat wave resulted in a reduc-tion or even complete loss on some glaciers of the firn areahence introducing a positive feedback on glacier mass bal-ances after 2003 consistent with higher ELAs

Considering each glacier individually Fig 4 shows therate of increase in the ELA over the whole period for each

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1462 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 3 Changes over the 1984ndash2010 period in(A) the ELA for the 43 glaciers studied (see Fig 1 and Table 1)(B) summer CPDD and(C)winter precipitation recorded by the weather stations used (see Fig 1 and Table 1) In each graph the horizontal black bar represents theannual average of the sample the gray box represents the median interval (Q3minusQ1) and the vertical black lines show the interval betweenthe first and the last decile (D1 and D9) On each graph (described in Sects 311 and 32) the dashed line represents the smooth underlyingtrend captured by a cubic smoothing spline regression (Lavigne et al 2012) The dotted line is the corresponding 95 confidence intervalThe yellow and purple boxes highlight the years in which the average ELA at the scale of the study region is linked with positive (yellow) ornegative (purple) winter precipitation anomalies (see Sect 42)

glacier plotted against its aspect In this study the aspectof the glacier matched the average aspect of the area of theglacier where the ELA fluctuated over the whole study pe-riod It is notable that (1) the increasing trend of the ELA var-ied between less than 1 m yrminus1 and more than 13 m yrminus1 de-pending on the glacier and (2) glaciers facing east displayeda more pronounced increasing trend of the ELA Howeverthis result should be interpreted with caution because the dis-tribution of the glaciers studied were not homogeneous withrespect to the aspect of the majority of glaciers in the FrenchAlps which are north facing Furthermore one of our se-lection criteria should also be recalled to measure the ELAfor each year the glaciers had to reach at least 3250 m asl(highest average ELA measured in 2003) and south-facingglaciers reaching this elevation are rare and can only be foundin the Mont Blanc Massif

Fig 4 Average increasing rate of the ELA (m yrminus1) over the1984ndash2010 period of each of the glaciers studied with respect to theaspect of the glacier The color of the diamond identifies the mas-sif in which the glacier is located blue= Ecrins green= Vanoisered= Mont Blanc

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

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1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

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1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

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1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

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1456 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

glacier changes from mountain-range up to regional scaleas well as our knowledge of the contribution of glaciers towater resources in the functioning of high-altitude water-sheds Remote sensing provides a unique opportunity to ad-dress the question of glacier changes at regional scale Sincethe availability of satellite data in the 1970s several meth-ods have been developed to compute changes in glacier vol-ume at multiannual to decadal scale using digital elevationmodels (DEMs) (eg Echelmeyer et al 1996 Baltsaviaset al 1999 Berthier et al 2004 Gardelle et al 2012)or the annual ELA and mass balance (eg Oslashstrem 1975Rabatel et al 2005 2008 2012a) However satellite de-rived DEMs are not accurate enough to compute variations inthe annual volume of mountain glaciers Another alternativeemerged from recent studies based on ELA and distributedmass balance modeling using meteorological input fields de-rived from local weather stations data reanalysis or regionalclimate models (eg Zemp et al 2007 Machguth et al2009 Marzeion et al 2012) However such modeling ap-proaches still have a number of limitations eg accumulationis underestimated in mountainous regions On the other handfor mid-latitude mountain glaciers the end-of-summer snowline altitude (SLA) is a good indicator of the ELA and thus ofthe annual mass balance (Lliboutry 1965 Braithwaite 1984Rabatel et al 2005) This enables ELA changes to be recon-structed for long time periods from remote-sensing data (De-muth and Pietroniro 1999 Rabatel et al 2002 2005 2008Barcaza et al 2009 Mathieu et al 2009) because the snowline is generally easy to identify using aerial photographs andsatellite images (Meier 1980 Rees 2005) Consequentlyit is possible to study the climatendashglacier relationship at amassif- or regional scale (Clare et al 2002 Chinn et al2005) which is particularly useful in remote areas where nodirect measurements are available

In the current study we rely on previous studies con-ducted in the French Alps (Dedieu and Reynaud 1990 Ra-batel et al 2002 2005 2008) to reconstruct ELA timeseries for more than 40 glaciers over the 1984ndash2010 pe-riod using the end-of-summer snow line detected on satel-lite images Our aim are (1) to quantify at a regional scalethe temporal and spatial changes of the ELA (2) to char-acterize the relationships between ELA and both morpho-topographic and climate variables and (3) to reconstruct thespatio-temporal variability of annual ELA time series by in-corporating the above-mentioned relationships in an expan-sion of Lliboutryrsquos variance decomposition model (1974)

2 Study area data and method

21 Selection of the study sites

A recent update of the glacier inventory of the French Alpslists 593 glaciers (Rabatel et al 2012b) The total glacialcoverage in the French Alps was about 340 km2 in the mid-

Fig 1 The 43 glaciers studied (red) among the 593 glaciers in theFrench Alpine glacier inventory (in blue) plus additional glaciers onthe Italian and Swiss sides of the Franco-Italian and Franco-Swissborders The three boxes represent the main glacierized rangesquoted in the text and figures ie Mont Blanc Vanoise and Ecrins

1980s and had decreased to about 275 km2 in the late 2000sIn the current study 43 glaciers located in the French Alpsor just next to the border with Switzerland and Italy wereselected (Table 1 Fig 1) The selection was based on the fol-lowing criteria (1) glaciers had to have a high enough max-imum elevation to allow observation of the snow line everyyear during the study period (2) glaciers with all aspects hadto be represented and (3) glaciers located in all the glacier-ized mountain ranges in the French Alps from the southern-most (4450prime N) to the northern-most (4600prime N) had to beincluded

Among the selected glaciers three belong to the GLACIO-CLIM observatory which runs a permanent mass balancemonitoring program These three glaciers are Saint SorlinGlacier monitored since 1957 ( 31 in Table 1 and Fig 1)Argentiegravere Glacier monitored since 1975 ( 4 in Table 1 andFig 1) and Geacutebroulaz Glacier monitored since 1983 ( 28in Table 1 and Fig 1) Mass balance data available on these

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1457

Table 1 List of the glaciers studied and the weather stations used in this study Numbers (for the glaciers) and codes (for the weatherstations) refer to Fig 1 All glaciers are located in the French Alps except when specified in brackets CH= Switzerland and IT= Italy Forthe weather stations letters in brackets identify the data producer MF= Meacuteteacuteo-France EDF= Electriciteacute de France MS= Meacuteteacuteo-SuisseThe coordinates are given in degrees minutes for latitude and longitude and in m asl for elevation

Glaciers Weather stations Name Name Name (data producer) ndash Code on Fig 1 Coord m asl

1 Trient (CH) 23 Grand Meacutean Chamonix-Mt-Blanc (MF) ndash Cham 4555prime N 0652prime E 10422 Tour 24 Pelve Contamine-sur-Arve (MF) ndashOff the map 4607prime N 0620prime E 4523 Saleina (CH) 25 Vallonnet Contamines-Montjoie (EDF) ndash Cont M 4549prime N 0643prime E 11804 Argentiegravere 26 Arpont Hauteluce (EDF) ndash Haut 4546prime N 0638prime E 12155 Talegravefre 27 Mahure Bourg-Saint-Maurice (MF) ndash Bg St M 4536prime N 0645prime E 8656 Preacute de Bar (IT) 28 Geacutebroulaz Peisey-Nancroix (EDF) ndash Pe Na 4532prime N 0645prime E 13507 Triolet (IT) 29 Baounet Pralognan-La-Vanoise (EDF) ndash Pra la V 4523prime N 0643prime E 14208 Leschaux 30 Rochemelon Bessans (EDF) ndash Bes 4519prime N 0659prime E 17159 Mont Blanc (IT) 31 Saint Sorlin Termignon (MF) ndash Ter 4516prime N 0648prime E 128010 Freiney (IT) 32 Quirlies Lyon ndash Bron (MF) ndashOff the map 4543prime N 0456prime E 19811 Brouillard (IT) 33 Girose St-Etienne-de-St-Geoirs (MF) ndashOff the map 4521prime N 0518prime E 38412 Treacute la Tecircte 34 Lautaret Allemond (EDF) ndashOff the map 4512prime N 0602prime E 127013 Leacutee Blanche (IT) 35 Mont de Lans Vaujany (EDF) ndash Vau 4509prime N 0602prime E 77214 des Glaciers 36 Selle Besse-en-Oisans (EDF) ndash B en O 4504prime N 0610prime E 152515 Rutor (IT) 37 Casset La Grave (EDF) ndash LaG 4503prime N 0617prime E 178016 Savinaz 38 Blanc Monetier-Les-Bains (EDF) ndash M les B 4458prime N 0630prime E 145917 Gurraz 39 Vallon Pilatte St-Christophe-en-Oisans (EDF) ndash StC 4456prime N 0611prime E 157018 Sassiegravere 40 Violettes Pelvoux (EDF) ndash Pel 4452prime N 0628prime E 126019 Tsantelaina (IT) 41 Rouies Chapelle-en-Valgaudemar (EDF) ndashOff the map 4448prime N 0611prime E 127020 Grande Motte 42 Seacuteleacute Embrun (MF) ndashOff the map 4433prime N 0630prime E 87121 Mulinet 43 Pilatte Genegraveve (MS) ndashOff the map 4614prime N 0605prime E 42022 Arcellin Grand St Bernard (MS) ndashOff the map 4552prime N 0710prime E 2472

three glaciers were used to assess the representativeness ofthe end-of-summer SLA computed from the satellite imagesas an indicator of the ELA and of the annual mass balancecomputed from field measurements (see Sect 31 below)

22 Snow line altitude retrieved from satellite images

221 Satellite images and snow line delineation

A total of 122 images of the 43 glaciers were used to coverthe 27 yr study period Unfortunately in some years usableimages were not available for all the glaciers because of(1) cloud cover hiding the underlying terrain and (2) snow-falls that can occur in late summer and completely cover theglaciers This was mainly the case in the 1980s and 1990swhen fewer satellites were in orbit than in the 2000s Theimages we used were acquired by the following satellitesLandsat 4TM 5TM 7ETM+ SPOT 1 to 5 and ASTER withspatial resolutions ranging from 25 to 30 m (see supplemen-tary material Table 1 for a detailed list of the images used)

For the snow line delineation on multispectral images atest of band combinations and band ratios to facilitate theidentification of the snow line on the images is described inRabatel et al (2012a see Fig 4) These authors concludedthat the combination of the green near-infrared and short-wave infrared bands (see Table 2 for details about the wave-

Table 2 Characteristics of the wavelength of the spectral bandsused to identify the snow line on the satellite images of the threesensors

Wavelength (microm)Green Near-infrared Shortwave

infrared

Landsat 052ndash060 077ndash090 155ndash175SPOT 050ndash059 079ndash089 160ndash170ASTER 052ndash060 076ndash086 158ndash175

lengths of each satellite) was the most appropriate to identifythe limit between snow and ice The snow line was delin-eated on the central part of the glaciers to avoid border ef-fects on the glacier banks shadows from surrounding slopesadditional snow input by avalanches overaccumulation dueto wind (Fig 2) which could generate equilibrium-line po-sition dependence on local conditions (Rabatel et al 2005)The delineation was performed manually because automaticmethods hardly succeed in distinguishing the snow line fromthe firn-line when both to be presented on the glacier Thisdistinction results to be also visually difficult when the pixelsize of the satellite images is too coarse (see the discussionsection)

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1458 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 2 Example of Landsat image used to delineate the snow line(red line) with a combination of spectral bands green near-infraredand shortwave infrared Argentiegravere Glacier (Mont Blanc Massif)Landsat TM5 image acquired on 9 September 1987

222 Digital elevation model computation of thealtitude of the snow line and of the glaciermorpho-topographic variables

The average altitude of each snow line was calculated byoverlaying the shapefiles containing the digitized snow lineson a DEM For each glacier the DEM was also used to com-pute the following morpho-topographic variables used in thisstudy surface area mean elevation aspect latitude and lon-gitude of the glacier centroid The mean elevation of eachglacier has been computed as the arithmetic mean of the ele-vation of each pixel of the DEM included within the glaciersoutline This mean elevation is rather close to the median el-evation of the glaciers Indeed the difference between thetwo variables is 10 m on average for the 43 studied glaciersThis shows that the studied glaciers have on average an al-most symmetrical area-altitude distribution (Braithwaite andRaper 2010)

Several DEMs are available for the study area Amongthese one comes from the French National Geographical In-stitute (IGN) with a resolution of 25 m and dates from theearly 1980s Another one is the ASTERGDEM with a resolu-tion of 30 m and dates from the mid-2000s The average dif-ference between the two DEMs at the mean elevation of theSLA of the 43 glaciers studied over the whole study periodie approx 3050 m asl is about 20 m This matches the ver-tical accuracy of the DEMs and is small in comparison to theinterannual variability of the SLA (the standard deviation ofthe measured SLA over the whole study period ranges be-tween 75 and 255 m depending on the glacier) However be-cause the glacier surface lowering can reach several tens ofmeters at lower elevation (up to 80 m at 1500 m asl) due to

the important glacier shrinkage over the last decades (Thib-ert et al 2005 Paul and Haeberli 2008 Vincent et al 2009Berthier and Vincent 2012) when the SLA is at lower eleva-tion as was the case in 2001 (2839plusmn 129 m asl) it is betterto use a DEM as close as possible to the date of the used im-ages Accordingly the DEM from the French IGN was usedfor the first half of the period (till the late 1990s) and theASTERGDEM was used for the second part of the periodChanging from one DEM to the other in the late 1990s hasno impact on the results because at that time the SLA waslocated between 3000 and 3200 m asl an altitudinal rangefor which the difference in elevation between the DEMs islower than their vertical accuracies

An uncertainty was estimated for each SLA Uncertaintyresults from different sources of error (Rabatel et al 20022012a) (1) the pixel size of the images which ranged be-tween 25 and 30 m depending on the sensor (2) the slope ofthe glacier in the vicinity of the SLA which ranged from 7

to 32 depending on the glacier and the zone where the SLAwas located in any given year (3) the vertical accuracy of theDEM which was about 20 m and (4) the standard deviationof the calculation of the average SLA along its delineationwhich ranged from 10 m to 170 m depending on the glacierand on the year concerned The last source of error (4) is themost important The resulting total uncertainty of the SLAs(root of the quadratic sum of the different independent er-rors) ranged fromplusmn15 to plusmn170 m depending on the yearand glacier concerned The uncertainty was greater when theSLA was located on a part of a glacier where the slope issteepest because in this case the standard deviation of thecomputed SLA is high

223 Method used to fill data gaps in the snow linealtitude time series

In spite of the large number of images used in this study therewere still some gaps in the SLA time series In the 27 yr studyperiod for the 43 glaciers studied the SLA was measured in85 of the 1081 cases Thus data were missing in 15 ofthe cases

To fill these gaps a bi-linear interpolation was appliedOriginally this approach was used by Lliboutry (1965 1974)to extract the spatial and temporal terms from a very incom-plete table of mass balance measurements (70 gaps) in theablation zone of Saint Sorlin Glacier This approach is basedon the fact that the glacier mass balance depends on both thesite and the time of measurement This approach was subse-quently extended to a set of glaciers in the Alps by Reynaud(1980) and later on to glaciers in other mountain ranges byLetreacuteguilly and Reynaud (1990) Here we applied it to ourSLA table that had far fewer gaps writing simply

SLAit = αi + βt + εit (1)

where SLAit is the snow line altitude of the glacieri forthe the yeart αi is long-term mean for each glacier over

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1459

the period of record βt is a term (depending on the yearonly) which is common to all the glaciers analyzed andεit = SLAit minusαi minusβt with εitsimN(0σ 2

ε ) a table of centeredresiduals assumed to be independent and Gaussian (N de-notes the normal distribution) The purpose is not simply toobtain theαi andβt terms but to estimate each missing valueSLAit on glacieri and yeart as

αi =1

T

Tsumt=1

SLAit (2)

βt =1

M

Msumi=1

(SLAit minus αi

) (3)

ˆSLAit = αi + βt (4)

whereT is the number of years of the study periodM is thenumber of glaciers studied and the circumflex is a classicalstatistical estimate

To maximize the consistency of the reconstructionαi wascalculated using only the years for which no data was miss-ing However only six years (out of the 27) had the SLA foreach of the 43 glaciers which is not quite enough to obtainan average SLA that is not too dependant on a specific yearConsequently the initial data set was divided into three ge-ographical subsets the Ecrins Massif the Vanoise area andthe Mont Blanc Massif and the computation was applied in-dependently to each one of these subsets In this way moreyears with SLAs for all the glaciers of each subset were avail-able (see Table 3 where the statistics of each subset are pre-sented) The year 1995 was the only exception For this spe-cific year because very few values were available the recon-struction was made considering the three subsets together

The RMSE ofεit representing the model errors was 73 mThis is less than half the standard deviation of the measuredSLAs (174 m) showing that the use of the bi-linear modelmakes sense on this data set

23 Meteorological data

Table 1 also lists the weather stations we used which arelocated in and around the study area (Fig 1) Among the105 weather stations available in the Meacuteteacuteo-France database(including stations operated by Meacuteteacuteo-France and Electriciteacutede France) in the French departments (Hautes-Alpes IsegravereSavoie and Haute-Savoie) in the study area 40 were se-lected on the basis of data availability and their proximityto the glacierized areas (except the Lyon-Bron weather sta-tion which was selected as a regional reference) Among the40 weather stations only 20 of which had a continuous se-ries of winter precipitation and summer temperatures overthe period 1984ndash2010 and were finally used Two weatherstations in Switzerland were also used The climate variables

Table 3 Statistics of the three subsets used to reconstruct missingvalues

Name of Number Missing Number of RMSE RMSEsubset of glaciers values complete SLA εit

years usedfor αi

Ecrins 13 11 14 148 m 62 mVanoise 16 20 12 160 m 66 mMont Blanc 14 14 14 201 m 90 m

used in the analysis were (1) cumulative positive degree days(CPDD) from 15 May to 15 September for each yeart ex-trapolated to the altitude of 3000 m asl using a standardgradient of 6C kmminus1 (3000 m asl being the approximatemean elevation of the SLA of the 43 glaciers studied overthe whole study period) and (2) cumulated winter precipi-tation from 15 September of the yeart-1 to 15 May of theyeart During this period liquid precipitation is negligible at3000 m asl

For the CPDD the date of 15 September was chosen be-cause it matches the average date of the satellite images usedto delineate the snow line Furthermore it should be notedthat at 3000 m asl in our study area after 15 SeptemberCPDD are negligible (Thibert et al 2013)

24 Modeling of the control of the morpho-topographicand climate variables

From our data set of SLA series glacier morpho-topographicvariables and climate variables we propose a simple expan-sion of Lliboutryrsquos approach (1974) to model and reconstructthe SLA fluctuations of any glacier in our study area over thestudy period This approach also enables us to quantify therespective control of each morpho-topographic and climatevariables on the SLA spatio-temporal variability

Following the ideas of Eckert et al (2011) the modelis implemented in a hierarchical Bayesian framework(eg Wikle 2003) The main advantage over more empir-ical approaches is treating the different sources of uncer-tainty in a consistent manner For instance the available in-formation (missing values and repeated observations amongcorrelated glacier series) is ldquorespectedrdquo when inferring thespatio-temporal patterns of interest more specifically the re-gression variables relating ELA fluctuations their morpho-topographic and climate drivers and their respective weight

In detail starting from the available SLA table with hol-low (ie without reconstructed values) and Eq (1) we furtherdecomposed the spatial term as

αi = a0 +

sumk

Xkiak + Vi (5)

where ao is the interannual regional averageXki are thespatial explanatory variables considered for each glacier iethe morpho-topographic variables latitude average altitude

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1460 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

surface area and aspect andVi sim N(0σ 2

v

)is the spatial

residual ie the local effect for the glacieri which is notexplained by the spatial explanatory variables considered

Similarly the temporal term is decomposed as

βt =

sumk

Xktbk + Zt (6)

where theXkt are the temporal explanatory variables con-sidered ie the climate variables summer CPDD and win-ter precipitation andZt sim N

(0σ 2

z

)is the temporal residual

ie the annual effect common to all glaciers for any yeart which is not explained by the temporal explanatory variablesconsidered

For the sake of simplicity the model is fed with reducedstandardized variables (by dividing by the standard devia-tion the difference between each value and the average of theseries) This avoids identifiability problems by granting thatsumt

βt = 0 and makes theak andbk coefficients more directly

interpretable at least for non-heavily correlated explanatoryvariables (see below)

Without the Vi and Zt terms the model would be aclassical spatio-temporal linear model with easy inferenceeg with likelihood maximization providing analytical so-lutions for point estimates and asymptotic standard errorsHowever this would not allow us to distinguish between ex-plained and unexplained variance in the spatial and temporalterms Hence theVi andZt terms make the model hierar-chical ie richer but also more tricky to estimate This chal-lenge was solved using Bayesian Markov chain Monte Carlomethods (MCMC Brooks 1998) As detailed in Eckert etal (2007 2010b) for similar problems the robustness of theinference was checked using tests based on starting differ-ent simulation runs at different points of the parameter space(Brooks and Gelman 1998) Poorly informativeprior prob-ability distributions were used for all parameters howeverthis standard choice allows us to obtainposteriorestimatesonly driven by information conveyed by the data From thejoint posteriordistribution of all parameters latent variablesand missing values we retained point estimates (theposte-rior mean)posteriorstandard deviations and 95 credibil-ity intervals the Bayesian counterpart of the classical confi-dence interval (Table 4) The construction of the model en-sures thatαi βt andεit are independent FurthermoreVi andsumk

Xkiak are independent so areZt andsumk

Xktbk Hence the

total variance of the data table is

VAR(SLAit ) = VAR

(sumk

Xkiak

)

+VAR

(sumk

Xktbk

)+ σ 2

v + σ 2z + σ 2 (7)

By analogy to the classicalR2 variance ratios can then becomputed to evaluate the respective weight of the different

Table 4Posterior characteristics of the proposed model applied onthe 43 ELA time series for the 1984ndash2010 periodak andbk denotethe regression parameters for each explanatory variable in bracketsOverall 25 and 975 denote the lower and upper bound of the95 credible interval

Mean 25 975

a1 (average altitude) 0381 0257 0502a2 (latitude) minus0123 minus0269 0023a3 (aspect) 0108 minus0022 0238a4 (surface area) minus0066 minus0199 0067b1 (summer CPDD) 0417 0268 0557b2 (winter precipitation) minus0311 minus0459 minus0155R2

clim 031 019 041R2

clim res 011 006 019R2

topo 024 016 032

R2topo res 009 005 014

R2res 025 019 030

R2tot 073 060 081

R2space 072 057 082

R2time 073 052 084

R2timespace 056 043 068

terms in ELA variability

R2topo = VAR

(sumk

Xkiak

)VAR(SLAit ) (8)

R2clim = VAR

(sumk

Xktbk

)VAR(SLAit ) (9)

R2topo res= σ 2

v VAR(SLAit ) (10)

R2clim res= σ 2

z VAR(SLAit ) (11)

R2res= σ 2VAR(SLAit ) (12)

Similarly ratios can be computed for the spatial and tem-poral terms solely

R2space= VAR

(sumk

Xkiak

)

(VAR

(sumk

Xkiak

)+ σ 2

v

) (13)

R2time = VAR

(sumk

Xktbk

)

(VAR

(sumk

Xktbk

)+ σ 2

z

) (14)

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1461

and getting rid of the random spatio temporal fluctuationsthe overall timespace ratio and determination coefficient inthe decomposition can be computed as

R2timespace=

(VAR

(sumk

Xktbk

)+ σ 2

z

)(

VAR (SLAit ) minus σ 2) (15)

R2tot =

(VAR

(sumk

Xkiak

)+ VAR

(sumk

Xktbk

))(

VAR (SLAit ) minus σ 2) (16)

All these statistics quantify the weight of mean spatial andtemporal effects in the total variability and the capacity ofthe chosen explanatory variables to model these mean ef-fects At the observation level of each yearglacier of themodel local and annual adjustment statistics can also becomputed as

δt =minus1

M

sumi

εit (17)

δi =minus1

T

sumt

εit (18)

δ2t = 1minus

radicradicradicradicradicradicsumi

ε2itsum

i

SLA2it

(19)

δ2i = 1minus

radicradicradicradicradicradicsumt

ε2itsum

t

SLA2it

(20)

respectively representing the annual and local bias and theannual and local determination coefficient withM the num-ber of glaciers considered andT the number of years studiedThey quantify the goodness of fit of each of the annuallocalseries

Variance ratios and adjustment statistics were evaluated bycomputing their values at each point of the MCMC iterativesimulation run giving access to point estimates and relateduncertainties taking into account the number of missing val-ues for each yearglacier (logically when nearly all observa-tions are missing for a given yearglacier the correspondingdetermination coefficientsδ2

tandδ2

iare unknown)

3 Results

Here we give the results of both the SLA and the climatevariables In Sect 31 we present the temporal and spatial

variability of the SLA measured on the 43 glaciers for the1984ndash2010 period In Sect 32 we present the variability ofclimate variables (ie summer CPDD and winter precipita-tion)

31 Changes in the equilibrium-line altitude

As mentioned above it has been demonstrated that on mid-latitude glaciers where superimposed ice is negligible theend-of-summer SLA is an accurate indicator of the ELA (seeFig 6 in Rabatel et al 2005 and Fig 3B in Rabatel et al2008) Because of this strong correlation and for the sake ofsimplicity we hereafter only use the term ELA

311 Temporal variability of the equilibrium-linealtitude

Figure 3a illustrates changes in the ELA over the 1984ndash2010period for the 43 glaciers studied (see also see SupplementTable II where all the data are presented) Considering allthe glaciers the average ELA for the whole period was lo-cated at 3035plusmn 120 m asl ie the interannual variabilityof the average ELA of all glaciers was high (120 m on av-erage) The difference between extreme years was as highas 460 m with the lowest average ELA measured in 1984(2790plusmn 180 m asl) and the highest average ELA measuredin 2003 (3250plusmn 135 m asl) In addition over the study pe-riod the ELA time series showed an average increasing trendof 64 m yrminus1 assuming a linear trend which results to be sta-tistically significant considering a risk of error of 5 Thisis the equivalent of an average increase of 170 m over the1984ndash2010 period ie higher than the interannual variabilityof the average ELA

The average annual ELA during the first five years of thestudy period was lower than the average ELA for the wholeperiod (except in 1986 when it was slightly higher than theaverage) These years match the end of a 15 yr period (fromthe mid-1970s to the late 1980s) during which alpine glaciersincreased in size due to higher winter accumulation and re-duced summer ablation (Vincent 2002 Thibert et al 2013)Since 2003 the average annual ELA has consistently beenabove the average ELA for the whole period Considering thetwo subsets ie before and after 2003 no significant trendappeared in either of the subsets The second subset is shorter(8 yr) than the first (19 yr) and trends computed on short pe-riods should be interpreted with caution However the aver-age ELA between the two subsets increased by about 140 mwhich is 82 of the total change Hence the year 2003 canalso be considered as a breakpoint in the time series It shouldbe noted that the 2003 extreme heat wave resulted in a reduc-tion or even complete loss on some glaciers of the firn areahence introducing a positive feedback on glacier mass bal-ances after 2003 consistent with higher ELAs

Considering each glacier individually Fig 4 shows therate of increase in the ELA over the whole period for each

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1462 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 3 Changes over the 1984ndash2010 period in(A) the ELA for the 43 glaciers studied (see Fig 1 and Table 1)(B) summer CPDD and(C)winter precipitation recorded by the weather stations used (see Fig 1 and Table 1) In each graph the horizontal black bar represents theannual average of the sample the gray box represents the median interval (Q3minusQ1) and the vertical black lines show the interval betweenthe first and the last decile (D1 and D9) On each graph (described in Sects 311 and 32) the dashed line represents the smooth underlyingtrend captured by a cubic smoothing spline regression (Lavigne et al 2012) The dotted line is the corresponding 95 confidence intervalThe yellow and purple boxes highlight the years in which the average ELA at the scale of the study region is linked with positive (yellow) ornegative (purple) winter precipitation anomalies (see Sect 42)

glacier plotted against its aspect In this study the aspectof the glacier matched the average aspect of the area of theglacier where the ELA fluctuated over the whole study pe-riod It is notable that (1) the increasing trend of the ELA var-ied between less than 1 m yrminus1 and more than 13 m yrminus1 de-pending on the glacier and (2) glaciers facing east displayeda more pronounced increasing trend of the ELA Howeverthis result should be interpreted with caution because the dis-tribution of the glaciers studied were not homogeneous withrespect to the aspect of the majority of glaciers in the FrenchAlps which are north facing Furthermore one of our se-lection criteria should also be recalled to measure the ELAfor each year the glaciers had to reach at least 3250 m asl(highest average ELA measured in 2003) and south-facingglaciers reaching this elevation are rare and can only be foundin the Mont Blanc Massif

Fig 4 Average increasing rate of the ELA (m yrminus1) over the1984ndash2010 period of each of the glaciers studied with respect to theaspect of the glacier The color of the diamond identifies the mas-sif in which the glacier is located blue= Ecrins green= Vanoisered= Mont Blanc

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

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1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

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1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1457

Table 1 List of the glaciers studied and the weather stations used in this study Numbers (for the glaciers) and codes (for the weatherstations) refer to Fig 1 All glaciers are located in the French Alps except when specified in brackets CH= Switzerland and IT= Italy Forthe weather stations letters in brackets identify the data producer MF= Meacuteteacuteo-France EDF= Electriciteacute de France MS= Meacuteteacuteo-SuisseThe coordinates are given in degrees minutes for latitude and longitude and in m asl for elevation

Glaciers Weather stations Name Name Name (data producer) ndash Code on Fig 1 Coord m asl

1 Trient (CH) 23 Grand Meacutean Chamonix-Mt-Blanc (MF) ndash Cham 4555prime N 0652prime E 10422 Tour 24 Pelve Contamine-sur-Arve (MF) ndashOff the map 4607prime N 0620prime E 4523 Saleina (CH) 25 Vallonnet Contamines-Montjoie (EDF) ndash Cont M 4549prime N 0643prime E 11804 Argentiegravere 26 Arpont Hauteluce (EDF) ndash Haut 4546prime N 0638prime E 12155 Talegravefre 27 Mahure Bourg-Saint-Maurice (MF) ndash Bg St M 4536prime N 0645prime E 8656 Preacute de Bar (IT) 28 Geacutebroulaz Peisey-Nancroix (EDF) ndash Pe Na 4532prime N 0645prime E 13507 Triolet (IT) 29 Baounet Pralognan-La-Vanoise (EDF) ndash Pra la V 4523prime N 0643prime E 14208 Leschaux 30 Rochemelon Bessans (EDF) ndash Bes 4519prime N 0659prime E 17159 Mont Blanc (IT) 31 Saint Sorlin Termignon (MF) ndash Ter 4516prime N 0648prime E 128010 Freiney (IT) 32 Quirlies Lyon ndash Bron (MF) ndashOff the map 4543prime N 0456prime E 19811 Brouillard (IT) 33 Girose St-Etienne-de-St-Geoirs (MF) ndashOff the map 4521prime N 0518prime E 38412 Treacute la Tecircte 34 Lautaret Allemond (EDF) ndashOff the map 4512prime N 0602prime E 127013 Leacutee Blanche (IT) 35 Mont de Lans Vaujany (EDF) ndash Vau 4509prime N 0602prime E 77214 des Glaciers 36 Selle Besse-en-Oisans (EDF) ndash B en O 4504prime N 0610prime E 152515 Rutor (IT) 37 Casset La Grave (EDF) ndash LaG 4503prime N 0617prime E 178016 Savinaz 38 Blanc Monetier-Les-Bains (EDF) ndash M les B 4458prime N 0630prime E 145917 Gurraz 39 Vallon Pilatte St-Christophe-en-Oisans (EDF) ndash StC 4456prime N 0611prime E 157018 Sassiegravere 40 Violettes Pelvoux (EDF) ndash Pel 4452prime N 0628prime E 126019 Tsantelaina (IT) 41 Rouies Chapelle-en-Valgaudemar (EDF) ndashOff the map 4448prime N 0611prime E 127020 Grande Motte 42 Seacuteleacute Embrun (MF) ndashOff the map 4433prime N 0630prime E 87121 Mulinet 43 Pilatte Genegraveve (MS) ndashOff the map 4614prime N 0605prime E 42022 Arcellin Grand St Bernard (MS) ndashOff the map 4552prime N 0710prime E 2472

three glaciers were used to assess the representativeness ofthe end-of-summer SLA computed from the satellite imagesas an indicator of the ELA and of the annual mass balancecomputed from field measurements (see Sect 31 below)

22 Snow line altitude retrieved from satellite images

221 Satellite images and snow line delineation

A total of 122 images of the 43 glaciers were used to coverthe 27 yr study period Unfortunately in some years usableimages were not available for all the glaciers because of(1) cloud cover hiding the underlying terrain and (2) snow-falls that can occur in late summer and completely cover theglaciers This was mainly the case in the 1980s and 1990swhen fewer satellites were in orbit than in the 2000s Theimages we used were acquired by the following satellitesLandsat 4TM 5TM 7ETM+ SPOT 1 to 5 and ASTER withspatial resolutions ranging from 25 to 30 m (see supplemen-tary material Table 1 for a detailed list of the images used)

For the snow line delineation on multispectral images atest of band combinations and band ratios to facilitate theidentification of the snow line on the images is described inRabatel et al (2012a see Fig 4) These authors concludedthat the combination of the green near-infrared and short-wave infrared bands (see Table 2 for details about the wave-

Table 2 Characteristics of the wavelength of the spectral bandsused to identify the snow line on the satellite images of the threesensors

Wavelength (microm)Green Near-infrared Shortwave

infrared

Landsat 052ndash060 077ndash090 155ndash175SPOT 050ndash059 079ndash089 160ndash170ASTER 052ndash060 076ndash086 158ndash175

lengths of each satellite) was the most appropriate to identifythe limit between snow and ice The snow line was delin-eated on the central part of the glaciers to avoid border ef-fects on the glacier banks shadows from surrounding slopesadditional snow input by avalanches overaccumulation dueto wind (Fig 2) which could generate equilibrium-line po-sition dependence on local conditions (Rabatel et al 2005)The delineation was performed manually because automaticmethods hardly succeed in distinguishing the snow line fromthe firn-line when both to be presented on the glacier Thisdistinction results to be also visually difficult when the pixelsize of the satellite images is too coarse (see the discussionsection)

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1458 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 2 Example of Landsat image used to delineate the snow line(red line) with a combination of spectral bands green near-infraredand shortwave infrared Argentiegravere Glacier (Mont Blanc Massif)Landsat TM5 image acquired on 9 September 1987

222 Digital elevation model computation of thealtitude of the snow line and of the glaciermorpho-topographic variables

The average altitude of each snow line was calculated byoverlaying the shapefiles containing the digitized snow lineson a DEM For each glacier the DEM was also used to com-pute the following morpho-topographic variables used in thisstudy surface area mean elevation aspect latitude and lon-gitude of the glacier centroid The mean elevation of eachglacier has been computed as the arithmetic mean of the ele-vation of each pixel of the DEM included within the glaciersoutline This mean elevation is rather close to the median el-evation of the glaciers Indeed the difference between thetwo variables is 10 m on average for the 43 studied glaciersThis shows that the studied glaciers have on average an al-most symmetrical area-altitude distribution (Braithwaite andRaper 2010)

Several DEMs are available for the study area Amongthese one comes from the French National Geographical In-stitute (IGN) with a resolution of 25 m and dates from theearly 1980s Another one is the ASTERGDEM with a resolu-tion of 30 m and dates from the mid-2000s The average dif-ference between the two DEMs at the mean elevation of theSLA of the 43 glaciers studied over the whole study periodie approx 3050 m asl is about 20 m This matches the ver-tical accuracy of the DEMs and is small in comparison to theinterannual variability of the SLA (the standard deviation ofthe measured SLA over the whole study period ranges be-tween 75 and 255 m depending on the glacier) However be-cause the glacier surface lowering can reach several tens ofmeters at lower elevation (up to 80 m at 1500 m asl) due to

the important glacier shrinkage over the last decades (Thib-ert et al 2005 Paul and Haeberli 2008 Vincent et al 2009Berthier and Vincent 2012) when the SLA is at lower eleva-tion as was the case in 2001 (2839plusmn 129 m asl) it is betterto use a DEM as close as possible to the date of the used im-ages Accordingly the DEM from the French IGN was usedfor the first half of the period (till the late 1990s) and theASTERGDEM was used for the second part of the periodChanging from one DEM to the other in the late 1990s hasno impact on the results because at that time the SLA waslocated between 3000 and 3200 m asl an altitudinal rangefor which the difference in elevation between the DEMs islower than their vertical accuracies

An uncertainty was estimated for each SLA Uncertaintyresults from different sources of error (Rabatel et al 20022012a) (1) the pixel size of the images which ranged be-tween 25 and 30 m depending on the sensor (2) the slope ofthe glacier in the vicinity of the SLA which ranged from 7

to 32 depending on the glacier and the zone where the SLAwas located in any given year (3) the vertical accuracy of theDEM which was about 20 m and (4) the standard deviationof the calculation of the average SLA along its delineationwhich ranged from 10 m to 170 m depending on the glacierand on the year concerned The last source of error (4) is themost important The resulting total uncertainty of the SLAs(root of the quadratic sum of the different independent er-rors) ranged fromplusmn15 to plusmn170 m depending on the yearand glacier concerned The uncertainty was greater when theSLA was located on a part of a glacier where the slope issteepest because in this case the standard deviation of thecomputed SLA is high

223 Method used to fill data gaps in the snow linealtitude time series

In spite of the large number of images used in this study therewere still some gaps in the SLA time series In the 27 yr studyperiod for the 43 glaciers studied the SLA was measured in85 of the 1081 cases Thus data were missing in 15 ofthe cases

To fill these gaps a bi-linear interpolation was appliedOriginally this approach was used by Lliboutry (1965 1974)to extract the spatial and temporal terms from a very incom-plete table of mass balance measurements (70 gaps) in theablation zone of Saint Sorlin Glacier This approach is basedon the fact that the glacier mass balance depends on both thesite and the time of measurement This approach was subse-quently extended to a set of glaciers in the Alps by Reynaud(1980) and later on to glaciers in other mountain ranges byLetreacuteguilly and Reynaud (1990) Here we applied it to ourSLA table that had far fewer gaps writing simply

SLAit = αi + βt + εit (1)

where SLAit is the snow line altitude of the glacieri forthe the yeart αi is long-term mean for each glacier over

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1459

the period of record βt is a term (depending on the yearonly) which is common to all the glaciers analyzed andεit = SLAit minusαi minusβt with εitsimN(0σ 2

ε ) a table of centeredresiduals assumed to be independent and Gaussian (N de-notes the normal distribution) The purpose is not simply toobtain theαi andβt terms but to estimate each missing valueSLAit on glacieri and yeart as

αi =1

T

Tsumt=1

SLAit (2)

βt =1

M

Msumi=1

(SLAit minus αi

) (3)

ˆSLAit = αi + βt (4)

whereT is the number of years of the study periodM is thenumber of glaciers studied and the circumflex is a classicalstatistical estimate

To maximize the consistency of the reconstructionαi wascalculated using only the years for which no data was miss-ing However only six years (out of the 27) had the SLA foreach of the 43 glaciers which is not quite enough to obtainan average SLA that is not too dependant on a specific yearConsequently the initial data set was divided into three ge-ographical subsets the Ecrins Massif the Vanoise area andthe Mont Blanc Massif and the computation was applied in-dependently to each one of these subsets In this way moreyears with SLAs for all the glaciers of each subset were avail-able (see Table 3 where the statistics of each subset are pre-sented) The year 1995 was the only exception For this spe-cific year because very few values were available the recon-struction was made considering the three subsets together

The RMSE ofεit representing the model errors was 73 mThis is less than half the standard deviation of the measuredSLAs (174 m) showing that the use of the bi-linear modelmakes sense on this data set

23 Meteorological data

Table 1 also lists the weather stations we used which arelocated in and around the study area (Fig 1) Among the105 weather stations available in the Meacuteteacuteo-France database(including stations operated by Meacuteteacuteo-France and Electriciteacutede France) in the French departments (Hautes-Alpes IsegravereSavoie and Haute-Savoie) in the study area 40 were se-lected on the basis of data availability and their proximityto the glacierized areas (except the Lyon-Bron weather sta-tion which was selected as a regional reference) Among the40 weather stations only 20 of which had a continuous se-ries of winter precipitation and summer temperatures overthe period 1984ndash2010 and were finally used Two weatherstations in Switzerland were also used The climate variables

Table 3 Statistics of the three subsets used to reconstruct missingvalues

Name of Number Missing Number of RMSE RMSEsubset of glaciers values complete SLA εit

years usedfor αi

Ecrins 13 11 14 148 m 62 mVanoise 16 20 12 160 m 66 mMont Blanc 14 14 14 201 m 90 m

used in the analysis were (1) cumulative positive degree days(CPDD) from 15 May to 15 September for each yeart ex-trapolated to the altitude of 3000 m asl using a standardgradient of 6C kmminus1 (3000 m asl being the approximatemean elevation of the SLA of the 43 glaciers studied overthe whole study period) and (2) cumulated winter precipi-tation from 15 September of the yeart-1 to 15 May of theyeart During this period liquid precipitation is negligible at3000 m asl

For the CPDD the date of 15 September was chosen be-cause it matches the average date of the satellite images usedto delineate the snow line Furthermore it should be notedthat at 3000 m asl in our study area after 15 SeptemberCPDD are negligible (Thibert et al 2013)

24 Modeling of the control of the morpho-topographicand climate variables

From our data set of SLA series glacier morpho-topographicvariables and climate variables we propose a simple expan-sion of Lliboutryrsquos approach (1974) to model and reconstructthe SLA fluctuations of any glacier in our study area over thestudy period This approach also enables us to quantify therespective control of each morpho-topographic and climatevariables on the SLA spatio-temporal variability

Following the ideas of Eckert et al (2011) the modelis implemented in a hierarchical Bayesian framework(eg Wikle 2003) The main advantage over more empir-ical approaches is treating the different sources of uncer-tainty in a consistent manner For instance the available in-formation (missing values and repeated observations amongcorrelated glacier series) is ldquorespectedrdquo when inferring thespatio-temporal patterns of interest more specifically the re-gression variables relating ELA fluctuations their morpho-topographic and climate drivers and their respective weight

In detail starting from the available SLA table with hol-low (ie without reconstructed values) and Eq (1) we furtherdecomposed the spatial term as

αi = a0 +

sumk

Xkiak + Vi (5)

where ao is the interannual regional averageXki are thespatial explanatory variables considered for each glacier iethe morpho-topographic variables latitude average altitude

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1460 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

surface area and aspect andVi sim N(0σ 2

v

)is the spatial

residual ie the local effect for the glacieri which is notexplained by the spatial explanatory variables considered

Similarly the temporal term is decomposed as

βt =

sumk

Xktbk + Zt (6)

where theXkt are the temporal explanatory variables con-sidered ie the climate variables summer CPDD and win-ter precipitation andZt sim N

(0σ 2

z

)is the temporal residual

ie the annual effect common to all glaciers for any yeart which is not explained by the temporal explanatory variablesconsidered

For the sake of simplicity the model is fed with reducedstandardized variables (by dividing by the standard devia-tion the difference between each value and the average of theseries) This avoids identifiability problems by granting thatsumt

βt = 0 and makes theak andbk coefficients more directly

interpretable at least for non-heavily correlated explanatoryvariables (see below)

Without the Vi and Zt terms the model would be aclassical spatio-temporal linear model with easy inferenceeg with likelihood maximization providing analytical so-lutions for point estimates and asymptotic standard errorsHowever this would not allow us to distinguish between ex-plained and unexplained variance in the spatial and temporalterms Hence theVi andZt terms make the model hierar-chical ie richer but also more tricky to estimate This chal-lenge was solved using Bayesian Markov chain Monte Carlomethods (MCMC Brooks 1998) As detailed in Eckert etal (2007 2010b) for similar problems the robustness of theinference was checked using tests based on starting differ-ent simulation runs at different points of the parameter space(Brooks and Gelman 1998) Poorly informativeprior prob-ability distributions were used for all parameters howeverthis standard choice allows us to obtainposteriorestimatesonly driven by information conveyed by the data From thejoint posteriordistribution of all parameters latent variablesand missing values we retained point estimates (theposte-rior mean)posteriorstandard deviations and 95 credibil-ity intervals the Bayesian counterpart of the classical confi-dence interval (Table 4) The construction of the model en-sures thatαi βt andεit are independent FurthermoreVi andsumk

Xkiak are independent so areZt andsumk

Xktbk Hence the

total variance of the data table is

VAR(SLAit ) = VAR

(sumk

Xkiak

)

+VAR

(sumk

Xktbk

)+ σ 2

v + σ 2z + σ 2 (7)

By analogy to the classicalR2 variance ratios can then becomputed to evaluate the respective weight of the different

Table 4Posterior characteristics of the proposed model applied onthe 43 ELA time series for the 1984ndash2010 periodak andbk denotethe regression parameters for each explanatory variable in bracketsOverall 25 and 975 denote the lower and upper bound of the95 credible interval

Mean 25 975

a1 (average altitude) 0381 0257 0502a2 (latitude) minus0123 minus0269 0023a3 (aspect) 0108 minus0022 0238a4 (surface area) minus0066 minus0199 0067b1 (summer CPDD) 0417 0268 0557b2 (winter precipitation) minus0311 minus0459 minus0155R2

clim 031 019 041R2

clim res 011 006 019R2

topo 024 016 032

R2topo res 009 005 014

R2res 025 019 030

R2tot 073 060 081

R2space 072 057 082

R2time 073 052 084

R2timespace 056 043 068

terms in ELA variability

R2topo = VAR

(sumk

Xkiak

)VAR(SLAit ) (8)

R2clim = VAR

(sumk

Xktbk

)VAR(SLAit ) (9)

R2topo res= σ 2

v VAR(SLAit ) (10)

R2clim res= σ 2

z VAR(SLAit ) (11)

R2res= σ 2VAR(SLAit ) (12)

Similarly ratios can be computed for the spatial and tem-poral terms solely

R2space= VAR

(sumk

Xkiak

)

(VAR

(sumk

Xkiak

)+ σ 2

v

) (13)

R2time = VAR

(sumk

Xktbk

)

(VAR

(sumk

Xktbk

)+ σ 2

z

) (14)

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1461

and getting rid of the random spatio temporal fluctuationsthe overall timespace ratio and determination coefficient inthe decomposition can be computed as

R2timespace=

(VAR

(sumk

Xktbk

)+ σ 2

z

)(

VAR (SLAit ) minus σ 2) (15)

R2tot =

(VAR

(sumk

Xkiak

)+ VAR

(sumk

Xktbk

))(

VAR (SLAit ) minus σ 2) (16)

All these statistics quantify the weight of mean spatial andtemporal effects in the total variability and the capacity ofthe chosen explanatory variables to model these mean ef-fects At the observation level of each yearglacier of themodel local and annual adjustment statistics can also becomputed as

δt =minus1

M

sumi

εit (17)

δi =minus1

T

sumt

εit (18)

δ2t = 1minus

radicradicradicradicradicradicsumi

ε2itsum

i

SLA2it

(19)

δ2i = 1minus

radicradicradicradicradicradicsumt

ε2itsum

t

SLA2it

(20)

respectively representing the annual and local bias and theannual and local determination coefficient withM the num-ber of glaciers considered andT the number of years studiedThey quantify the goodness of fit of each of the annuallocalseries

Variance ratios and adjustment statistics were evaluated bycomputing their values at each point of the MCMC iterativesimulation run giving access to point estimates and relateduncertainties taking into account the number of missing val-ues for each yearglacier (logically when nearly all observa-tions are missing for a given yearglacier the correspondingdetermination coefficientsδ2

tandδ2

iare unknown)

3 Results

Here we give the results of both the SLA and the climatevariables In Sect 31 we present the temporal and spatial

variability of the SLA measured on the 43 glaciers for the1984ndash2010 period In Sect 32 we present the variability ofclimate variables (ie summer CPDD and winter precipita-tion)

31 Changes in the equilibrium-line altitude

As mentioned above it has been demonstrated that on mid-latitude glaciers where superimposed ice is negligible theend-of-summer SLA is an accurate indicator of the ELA (seeFig 6 in Rabatel et al 2005 and Fig 3B in Rabatel et al2008) Because of this strong correlation and for the sake ofsimplicity we hereafter only use the term ELA

311 Temporal variability of the equilibrium-linealtitude

Figure 3a illustrates changes in the ELA over the 1984ndash2010period for the 43 glaciers studied (see also see SupplementTable II where all the data are presented) Considering allthe glaciers the average ELA for the whole period was lo-cated at 3035plusmn 120 m asl ie the interannual variabilityof the average ELA of all glaciers was high (120 m on av-erage) The difference between extreme years was as highas 460 m with the lowest average ELA measured in 1984(2790plusmn 180 m asl) and the highest average ELA measuredin 2003 (3250plusmn 135 m asl) In addition over the study pe-riod the ELA time series showed an average increasing trendof 64 m yrminus1 assuming a linear trend which results to be sta-tistically significant considering a risk of error of 5 Thisis the equivalent of an average increase of 170 m over the1984ndash2010 period ie higher than the interannual variabilityof the average ELA

The average annual ELA during the first five years of thestudy period was lower than the average ELA for the wholeperiod (except in 1986 when it was slightly higher than theaverage) These years match the end of a 15 yr period (fromthe mid-1970s to the late 1980s) during which alpine glaciersincreased in size due to higher winter accumulation and re-duced summer ablation (Vincent 2002 Thibert et al 2013)Since 2003 the average annual ELA has consistently beenabove the average ELA for the whole period Considering thetwo subsets ie before and after 2003 no significant trendappeared in either of the subsets The second subset is shorter(8 yr) than the first (19 yr) and trends computed on short pe-riods should be interpreted with caution However the aver-age ELA between the two subsets increased by about 140 mwhich is 82 of the total change Hence the year 2003 canalso be considered as a breakpoint in the time series It shouldbe noted that the 2003 extreme heat wave resulted in a reduc-tion or even complete loss on some glaciers of the firn areahence introducing a positive feedback on glacier mass bal-ances after 2003 consistent with higher ELAs

Considering each glacier individually Fig 4 shows therate of increase in the ELA over the whole period for each

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1462 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 3 Changes over the 1984ndash2010 period in(A) the ELA for the 43 glaciers studied (see Fig 1 and Table 1)(B) summer CPDD and(C)winter precipitation recorded by the weather stations used (see Fig 1 and Table 1) In each graph the horizontal black bar represents theannual average of the sample the gray box represents the median interval (Q3minusQ1) and the vertical black lines show the interval betweenthe first and the last decile (D1 and D9) On each graph (described in Sects 311 and 32) the dashed line represents the smooth underlyingtrend captured by a cubic smoothing spline regression (Lavigne et al 2012) The dotted line is the corresponding 95 confidence intervalThe yellow and purple boxes highlight the years in which the average ELA at the scale of the study region is linked with positive (yellow) ornegative (purple) winter precipitation anomalies (see Sect 42)

glacier plotted against its aspect In this study the aspectof the glacier matched the average aspect of the area of theglacier where the ELA fluctuated over the whole study pe-riod It is notable that (1) the increasing trend of the ELA var-ied between less than 1 m yrminus1 and more than 13 m yrminus1 de-pending on the glacier and (2) glaciers facing east displayeda more pronounced increasing trend of the ELA Howeverthis result should be interpreted with caution because the dis-tribution of the glaciers studied were not homogeneous withrespect to the aspect of the majority of glaciers in the FrenchAlps which are north facing Furthermore one of our se-lection criteria should also be recalled to measure the ELAfor each year the glaciers had to reach at least 3250 m asl(highest average ELA measured in 2003) and south-facingglaciers reaching this elevation are rare and can only be foundin the Mont Blanc Massif

Fig 4 Average increasing rate of the ELA (m yrminus1) over the1984ndash2010 period of each of the glaciers studied with respect to theaspect of the glacier The color of the diamond identifies the mas-sif in which the glacier is located blue= Ecrins green= Vanoisered= Mont Blanc

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

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1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

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1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1458 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 2 Example of Landsat image used to delineate the snow line(red line) with a combination of spectral bands green near-infraredand shortwave infrared Argentiegravere Glacier (Mont Blanc Massif)Landsat TM5 image acquired on 9 September 1987

222 Digital elevation model computation of thealtitude of the snow line and of the glaciermorpho-topographic variables

The average altitude of each snow line was calculated byoverlaying the shapefiles containing the digitized snow lineson a DEM For each glacier the DEM was also used to com-pute the following morpho-topographic variables used in thisstudy surface area mean elevation aspect latitude and lon-gitude of the glacier centroid The mean elevation of eachglacier has been computed as the arithmetic mean of the ele-vation of each pixel of the DEM included within the glaciersoutline This mean elevation is rather close to the median el-evation of the glaciers Indeed the difference between thetwo variables is 10 m on average for the 43 studied glaciersThis shows that the studied glaciers have on average an al-most symmetrical area-altitude distribution (Braithwaite andRaper 2010)

Several DEMs are available for the study area Amongthese one comes from the French National Geographical In-stitute (IGN) with a resolution of 25 m and dates from theearly 1980s Another one is the ASTERGDEM with a resolu-tion of 30 m and dates from the mid-2000s The average dif-ference between the two DEMs at the mean elevation of theSLA of the 43 glaciers studied over the whole study periodie approx 3050 m asl is about 20 m This matches the ver-tical accuracy of the DEMs and is small in comparison to theinterannual variability of the SLA (the standard deviation ofthe measured SLA over the whole study period ranges be-tween 75 and 255 m depending on the glacier) However be-cause the glacier surface lowering can reach several tens ofmeters at lower elevation (up to 80 m at 1500 m asl) due to

the important glacier shrinkage over the last decades (Thib-ert et al 2005 Paul and Haeberli 2008 Vincent et al 2009Berthier and Vincent 2012) when the SLA is at lower eleva-tion as was the case in 2001 (2839plusmn 129 m asl) it is betterto use a DEM as close as possible to the date of the used im-ages Accordingly the DEM from the French IGN was usedfor the first half of the period (till the late 1990s) and theASTERGDEM was used for the second part of the periodChanging from one DEM to the other in the late 1990s hasno impact on the results because at that time the SLA waslocated between 3000 and 3200 m asl an altitudinal rangefor which the difference in elevation between the DEMs islower than their vertical accuracies

An uncertainty was estimated for each SLA Uncertaintyresults from different sources of error (Rabatel et al 20022012a) (1) the pixel size of the images which ranged be-tween 25 and 30 m depending on the sensor (2) the slope ofthe glacier in the vicinity of the SLA which ranged from 7

to 32 depending on the glacier and the zone where the SLAwas located in any given year (3) the vertical accuracy of theDEM which was about 20 m and (4) the standard deviationof the calculation of the average SLA along its delineationwhich ranged from 10 m to 170 m depending on the glacierand on the year concerned The last source of error (4) is themost important The resulting total uncertainty of the SLAs(root of the quadratic sum of the different independent er-rors) ranged fromplusmn15 to plusmn170 m depending on the yearand glacier concerned The uncertainty was greater when theSLA was located on a part of a glacier where the slope issteepest because in this case the standard deviation of thecomputed SLA is high

223 Method used to fill data gaps in the snow linealtitude time series

In spite of the large number of images used in this study therewere still some gaps in the SLA time series In the 27 yr studyperiod for the 43 glaciers studied the SLA was measured in85 of the 1081 cases Thus data were missing in 15 ofthe cases

To fill these gaps a bi-linear interpolation was appliedOriginally this approach was used by Lliboutry (1965 1974)to extract the spatial and temporal terms from a very incom-plete table of mass balance measurements (70 gaps) in theablation zone of Saint Sorlin Glacier This approach is basedon the fact that the glacier mass balance depends on both thesite and the time of measurement This approach was subse-quently extended to a set of glaciers in the Alps by Reynaud(1980) and later on to glaciers in other mountain ranges byLetreacuteguilly and Reynaud (1990) Here we applied it to ourSLA table that had far fewer gaps writing simply

SLAit = αi + βt + εit (1)

where SLAit is the snow line altitude of the glacieri forthe the yeart αi is long-term mean for each glacier over

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1459

the period of record βt is a term (depending on the yearonly) which is common to all the glaciers analyzed andεit = SLAit minusαi minusβt with εitsimN(0σ 2

ε ) a table of centeredresiduals assumed to be independent and Gaussian (N de-notes the normal distribution) The purpose is not simply toobtain theαi andβt terms but to estimate each missing valueSLAit on glacieri and yeart as

αi =1

T

Tsumt=1

SLAit (2)

βt =1

M

Msumi=1

(SLAit minus αi

) (3)

ˆSLAit = αi + βt (4)

whereT is the number of years of the study periodM is thenumber of glaciers studied and the circumflex is a classicalstatistical estimate

To maximize the consistency of the reconstructionαi wascalculated using only the years for which no data was miss-ing However only six years (out of the 27) had the SLA foreach of the 43 glaciers which is not quite enough to obtainan average SLA that is not too dependant on a specific yearConsequently the initial data set was divided into three ge-ographical subsets the Ecrins Massif the Vanoise area andthe Mont Blanc Massif and the computation was applied in-dependently to each one of these subsets In this way moreyears with SLAs for all the glaciers of each subset were avail-able (see Table 3 where the statistics of each subset are pre-sented) The year 1995 was the only exception For this spe-cific year because very few values were available the recon-struction was made considering the three subsets together

The RMSE ofεit representing the model errors was 73 mThis is less than half the standard deviation of the measuredSLAs (174 m) showing that the use of the bi-linear modelmakes sense on this data set

23 Meteorological data

Table 1 also lists the weather stations we used which arelocated in and around the study area (Fig 1) Among the105 weather stations available in the Meacuteteacuteo-France database(including stations operated by Meacuteteacuteo-France and Electriciteacutede France) in the French departments (Hautes-Alpes IsegravereSavoie and Haute-Savoie) in the study area 40 were se-lected on the basis of data availability and their proximityto the glacierized areas (except the Lyon-Bron weather sta-tion which was selected as a regional reference) Among the40 weather stations only 20 of which had a continuous se-ries of winter precipitation and summer temperatures overthe period 1984ndash2010 and were finally used Two weatherstations in Switzerland were also used The climate variables

Table 3 Statistics of the three subsets used to reconstruct missingvalues

Name of Number Missing Number of RMSE RMSEsubset of glaciers values complete SLA εit

years usedfor αi

Ecrins 13 11 14 148 m 62 mVanoise 16 20 12 160 m 66 mMont Blanc 14 14 14 201 m 90 m

used in the analysis were (1) cumulative positive degree days(CPDD) from 15 May to 15 September for each yeart ex-trapolated to the altitude of 3000 m asl using a standardgradient of 6C kmminus1 (3000 m asl being the approximatemean elevation of the SLA of the 43 glaciers studied overthe whole study period) and (2) cumulated winter precipi-tation from 15 September of the yeart-1 to 15 May of theyeart During this period liquid precipitation is negligible at3000 m asl

For the CPDD the date of 15 September was chosen be-cause it matches the average date of the satellite images usedto delineate the snow line Furthermore it should be notedthat at 3000 m asl in our study area after 15 SeptemberCPDD are negligible (Thibert et al 2013)

24 Modeling of the control of the morpho-topographicand climate variables

From our data set of SLA series glacier morpho-topographicvariables and climate variables we propose a simple expan-sion of Lliboutryrsquos approach (1974) to model and reconstructthe SLA fluctuations of any glacier in our study area over thestudy period This approach also enables us to quantify therespective control of each morpho-topographic and climatevariables on the SLA spatio-temporal variability

Following the ideas of Eckert et al (2011) the modelis implemented in a hierarchical Bayesian framework(eg Wikle 2003) The main advantage over more empir-ical approaches is treating the different sources of uncer-tainty in a consistent manner For instance the available in-formation (missing values and repeated observations amongcorrelated glacier series) is ldquorespectedrdquo when inferring thespatio-temporal patterns of interest more specifically the re-gression variables relating ELA fluctuations their morpho-topographic and climate drivers and their respective weight

In detail starting from the available SLA table with hol-low (ie without reconstructed values) and Eq (1) we furtherdecomposed the spatial term as

αi = a0 +

sumk

Xkiak + Vi (5)

where ao is the interannual regional averageXki are thespatial explanatory variables considered for each glacier iethe morpho-topographic variables latitude average altitude

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1460 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

surface area and aspect andVi sim N(0σ 2

v

)is the spatial

residual ie the local effect for the glacieri which is notexplained by the spatial explanatory variables considered

Similarly the temporal term is decomposed as

βt =

sumk

Xktbk + Zt (6)

where theXkt are the temporal explanatory variables con-sidered ie the climate variables summer CPDD and win-ter precipitation andZt sim N

(0σ 2

z

)is the temporal residual

ie the annual effect common to all glaciers for any yeart which is not explained by the temporal explanatory variablesconsidered

For the sake of simplicity the model is fed with reducedstandardized variables (by dividing by the standard devia-tion the difference between each value and the average of theseries) This avoids identifiability problems by granting thatsumt

βt = 0 and makes theak andbk coefficients more directly

interpretable at least for non-heavily correlated explanatoryvariables (see below)

Without the Vi and Zt terms the model would be aclassical spatio-temporal linear model with easy inferenceeg with likelihood maximization providing analytical so-lutions for point estimates and asymptotic standard errorsHowever this would not allow us to distinguish between ex-plained and unexplained variance in the spatial and temporalterms Hence theVi andZt terms make the model hierar-chical ie richer but also more tricky to estimate This chal-lenge was solved using Bayesian Markov chain Monte Carlomethods (MCMC Brooks 1998) As detailed in Eckert etal (2007 2010b) for similar problems the robustness of theinference was checked using tests based on starting differ-ent simulation runs at different points of the parameter space(Brooks and Gelman 1998) Poorly informativeprior prob-ability distributions were used for all parameters howeverthis standard choice allows us to obtainposteriorestimatesonly driven by information conveyed by the data From thejoint posteriordistribution of all parameters latent variablesand missing values we retained point estimates (theposte-rior mean)posteriorstandard deviations and 95 credibil-ity intervals the Bayesian counterpart of the classical confi-dence interval (Table 4) The construction of the model en-sures thatαi βt andεit are independent FurthermoreVi andsumk

Xkiak are independent so areZt andsumk

Xktbk Hence the

total variance of the data table is

VAR(SLAit ) = VAR

(sumk

Xkiak

)

+VAR

(sumk

Xktbk

)+ σ 2

v + σ 2z + σ 2 (7)

By analogy to the classicalR2 variance ratios can then becomputed to evaluate the respective weight of the different

Table 4Posterior characteristics of the proposed model applied onthe 43 ELA time series for the 1984ndash2010 periodak andbk denotethe regression parameters for each explanatory variable in bracketsOverall 25 and 975 denote the lower and upper bound of the95 credible interval

Mean 25 975

a1 (average altitude) 0381 0257 0502a2 (latitude) minus0123 minus0269 0023a3 (aspect) 0108 minus0022 0238a4 (surface area) minus0066 minus0199 0067b1 (summer CPDD) 0417 0268 0557b2 (winter precipitation) minus0311 minus0459 minus0155R2

clim 031 019 041R2

clim res 011 006 019R2

topo 024 016 032

R2topo res 009 005 014

R2res 025 019 030

R2tot 073 060 081

R2space 072 057 082

R2time 073 052 084

R2timespace 056 043 068

terms in ELA variability

R2topo = VAR

(sumk

Xkiak

)VAR(SLAit ) (8)

R2clim = VAR

(sumk

Xktbk

)VAR(SLAit ) (9)

R2topo res= σ 2

v VAR(SLAit ) (10)

R2clim res= σ 2

z VAR(SLAit ) (11)

R2res= σ 2VAR(SLAit ) (12)

Similarly ratios can be computed for the spatial and tem-poral terms solely

R2space= VAR

(sumk

Xkiak

)

(VAR

(sumk

Xkiak

)+ σ 2

v

) (13)

R2time = VAR

(sumk

Xktbk

)

(VAR

(sumk

Xktbk

)+ σ 2

z

) (14)

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1461

and getting rid of the random spatio temporal fluctuationsthe overall timespace ratio and determination coefficient inthe decomposition can be computed as

R2timespace=

(VAR

(sumk

Xktbk

)+ σ 2

z

)(

VAR (SLAit ) minus σ 2) (15)

R2tot =

(VAR

(sumk

Xkiak

)+ VAR

(sumk

Xktbk

))(

VAR (SLAit ) minus σ 2) (16)

All these statistics quantify the weight of mean spatial andtemporal effects in the total variability and the capacity ofthe chosen explanatory variables to model these mean ef-fects At the observation level of each yearglacier of themodel local and annual adjustment statistics can also becomputed as

δt =minus1

M

sumi

εit (17)

δi =minus1

T

sumt

εit (18)

δ2t = 1minus

radicradicradicradicradicradicsumi

ε2itsum

i

SLA2it

(19)

δ2i = 1minus

radicradicradicradicradicradicsumt

ε2itsum

t

SLA2it

(20)

respectively representing the annual and local bias and theannual and local determination coefficient withM the num-ber of glaciers considered andT the number of years studiedThey quantify the goodness of fit of each of the annuallocalseries

Variance ratios and adjustment statistics were evaluated bycomputing their values at each point of the MCMC iterativesimulation run giving access to point estimates and relateduncertainties taking into account the number of missing val-ues for each yearglacier (logically when nearly all observa-tions are missing for a given yearglacier the correspondingdetermination coefficientsδ2

tandδ2

iare unknown)

3 Results

Here we give the results of both the SLA and the climatevariables In Sect 31 we present the temporal and spatial

variability of the SLA measured on the 43 glaciers for the1984ndash2010 period In Sect 32 we present the variability ofclimate variables (ie summer CPDD and winter precipita-tion)

31 Changes in the equilibrium-line altitude

As mentioned above it has been demonstrated that on mid-latitude glaciers where superimposed ice is negligible theend-of-summer SLA is an accurate indicator of the ELA (seeFig 6 in Rabatel et al 2005 and Fig 3B in Rabatel et al2008) Because of this strong correlation and for the sake ofsimplicity we hereafter only use the term ELA

311 Temporal variability of the equilibrium-linealtitude

Figure 3a illustrates changes in the ELA over the 1984ndash2010period for the 43 glaciers studied (see also see SupplementTable II where all the data are presented) Considering allthe glaciers the average ELA for the whole period was lo-cated at 3035plusmn 120 m asl ie the interannual variabilityof the average ELA of all glaciers was high (120 m on av-erage) The difference between extreme years was as highas 460 m with the lowest average ELA measured in 1984(2790plusmn 180 m asl) and the highest average ELA measuredin 2003 (3250plusmn 135 m asl) In addition over the study pe-riod the ELA time series showed an average increasing trendof 64 m yrminus1 assuming a linear trend which results to be sta-tistically significant considering a risk of error of 5 Thisis the equivalent of an average increase of 170 m over the1984ndash2010 period ie higher than the interannual variabilityof the average ELA

The average annual ELA during the first five years of thestudy period was lower than the average ELA for the wholeperiod (except in 1986 when it was slightly higher than theaverage) These years match the end of a 15 yr period (fromthe mid-1970s to the late 1980s) during which alpine glaciersincreased in size due to higher winter accumulation and re-duced summer ablation (Vincent 2002 Thibert et al 2013)Since 2003 the average annual ELA has consistently beenabove the average ELA for the whole period Considering thetwo subsets ie before and after 2003 no significant trendappeared in either of the subsets The second subset is shorter(8 yr) than the first (19 yr) and trends computed on short pe-riods should be interpreted with caution However the aver-age ELA between the two subsets increased by about 140 mwhich is 82 of the total change Hence the year 2003 canalso be considered as a breakpoint in the time series It shouldbe noted that the 2003 extreme heat wave resulted in a reduc-tion or even complete loss on some glaciers of the firn areahence introducing a positive feedback on glacier mass bal-ances after 2003 consistent with higher ELAs

Considering each glacier individually Fig 4 shows therate of increase in the ELA over the whole period for each

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1462 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 3 Changes over the 1984ndash2010 period in(A) the ELA for the 43 glaciers studied (see Fig 1 and Table 1)(B) summer CPDD and(C)winter precipitation recorded by the weather stations used (see Fig 1 and Table 1) In each graph the horizontal black bar represents theannual average of the sample the gray box represents the median interval (Q3minusQ1) and the vertical black lines show the interval betweenthe first and the last decile (D1 and D9) On each graph (described in Sects 311 and 32) the dashed line represents the smooth underlyingtrend captured by a cubic smoothing spline regression (Lavigne et al 2012) The dotted line is the corresponding 95 confidence intervalThe yellow and purple boxes highlight the years in which the average ELA at the scale of the study region is linked with positive (yellow) ornegative (purple) winter precipitation anomalies (see Sect 42)

glacier plotted against its aspect In this study the aspectof the glacier matched the average aspect of the area of theglacier where the ELA fluctuated over the whole study pe-riod It is notable that (1) the increasing trend of the ELA var-ied between less than 1 m yrminus1 and more than 13 m yrminus1 de-pending on the glacier and (2) glaciers facing east displayeda more pronounced increasing trend of the ELA Howeverthis result should be interpreted with caution because the dis-tribution of the glaciers studied were not homogeneous withrespect to the aspect of the majority of glaciers in the FrenchAlps which are north facing Furthermore one of our se-lection criteria should also be recalled to measure the ELAfor each year the glaciers had to reach at least 3250 m asl(highest average ELA measured in 2003) and south-facingglaciers reaching this elevation are rare and can only be foundin the Mont Blanc Massif

Fig 4 Average increasing rate of the ELA (m yrminus1) over the1984ndash2010 period of each of the glaciers studied with respect to theaspect of the glacier The color of the diamond identifies the mas-sif in which the glacier is located blue= Ecrins green= Vanoisered= Mont Blanc

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

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1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

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1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

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1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1459

the period of record βt is a term (depending on the yearonly) which is common to all the glaciers analyzed andεit = SLAit minusαi minusβt with εitsimN(0σ 2

ε ) a table of centeredresiduals assumed to be independent and Gaussian (N de-notes the normal distribution) The purpose is not simply toobtain theαi andβt terms but to estimate each missing valueSLAit on glacieri and yeart as

αi =1

T

Tsumt=1

SLAit (2)

βt =1

M

Msumi=1

(SLAit minus αi

) (3)

ˆSLAit = αi + βt (4)

whereT is the number of years of the study periodM is thenumber of glaciers studied and the circumflex is a classicalstatistical estimate

To maximize the consistency of the reconstructionαi wascalculated using only the years for which no data was miss-ing However only six years (out of the 27) had the SLA foreach of the 43 glaciers which is not quite enough to obtainan average SLA that is not too dependant on a specific yearConsequently the initial data set was divided into three ge-ographical subsets the Ecrins Massif the Vanoise area andthe Mont Blanc Massif and the computation was applied in-dependently to each one of these subsets In this way moreyears with SLAs for all the glaciers of each subset were avail-able (see Table 3 where the statistics of each subset are pre-sented) The year 1995 was the only exception For this spe-cific year because very few values were available the recon-struction was made considering the three subsets together

The RMSE ofεit representing the model errors was 73 mThis is less than half the standard deviation of the measuredSLAs (174 m) showing that the use of the bi-linear modelmakes sense on this data set

23 Meteorological data

Table 1 also lists the weather stations we used which arelocated in and around the study area (Fig 1) Among the105 weather stations available in the Meacuteteacuteo-France database(including stations operated by Meacuteteacuteo-France and Electriciteacutede France) in the French departments (Hautes-Alpes IsegravereSavoie and Haute-Savoie) in the study area 40 were se-lected on the basis of data availability and their proximityto the glacierized areas (except the Lyon-Bron weather sta-tion which was selected as a regional reference) Among the40 weather stations only 20 of which had a continuous se-ries of winter precipitation and summer temperatures overthe period 1984ndash2010 and were finally used Two weatherstations in Switzerland were also used The climate variables

Table 3 Statistics of the three subsets used to reconstruct missingvalues

Name of Number Missing Number of RMSE RMSEsubset of glaciers values complete SLA εit

years usedfor αi

Ecrins 13 11 14 148 m 62 mVanoise 16 20 12 160 m 66 mMont Blanc 14 14 14 201 m 90 m

used in the analysis were (1) cumulative positive degree days(CPDD) from 15 May to 15 September for each yeart ex-trapolated to the altitude of 3000 m asl using a standardgradient of 6C kmminus1 (3000 m asl being the approximatemean elevation of the SLA of the 43 glaciers studied overthe whole study period) and (2) cumulated winter precipi-tation from 15 September of the yeart-1 to 15 May of theyeart During this period liquid precipitation is negligible at3000 m asl

For the CPDD the date of 15 September was chosen be-cause it matches the average date of the satellite images usedto delineate the snow line Furthermore it should be notedthat at 3000 m asl in our study area after 15 SeptemberCPDD are negligible (Thibert et al 2013)

24 Modeling of the control of the morpho-topographicand climate variables

From our data set of SLA series glacier morpho-topographicvariables and climate variables we propose a simple expan-sion of Lliboutryrsquos approach (1974) to model and reconstructthe SLA fluctuations of any glacier in our study area over thestudy period This approach also enables us to quantify therespective control of each morpho-topographic and climatevariables on the SLA spatio-temporal variability

Following the ideas of Eckert et al (2011) the modelis implemented in a hierarchical Bayesian framework(eg Wikle 2003) The main advantage over more empir-ical approaches is treating the different sources of uncer-tainty in a consistent manner For instance the available in-formation (missing values and repeated observations amongcorrelated glacier series) is ldquorespectedrdquo when inferring thespatio-temporal patterns of interest more specifically the re-gression variables relating ELA fluctuations their morpho-topographic and climate drivers and their respective weight

In detail starting from the available SLA table with hol-low (ie without reconstructed values) and Eq (1) we furtherdecomposed the spatial term as

αi = a0 +

sumk

Xkiak + Vi (5)

where ao is the interannual regional averageXki are thespatial explanatory variables considered for each glacier iethe morpho-topographic variables latitude average altitude

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1460 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

surface area and aspect andVi sim N(0σ 2

v

)is the spatial

residual ie the local effect for the glacieri which is notexplained by the spatial explanatory variables considered

Similarly the temporal term is decomposed as

βt =

sumk

Xktbk + Zt (6)

where theXkt are the temporal explanatory variables con-sidered ie the climate variables summer CPDD and win-ter precipitation andZt sim N

(0σ 2

z

)is the temporal residual

ie the annual effect common to all glaciers for any yeart which is not explained by the temporal explanatory variablesconsidered

For the sake of simplicity the model is fed with reducedstandardized variables (by dividing by the standard devia-tion the difference between each value and the average of theseries) This avoids identifiability problems by granting thatsumt

βt = 0 and makes theak andbk coefficients more directly

interpretable at least for non-heavily correlated explanatoryvariables (see below)

Without the Vi and Zt terms the model would be aclassical spatio-temporal linear model with easy inferenceeg with likelihood maximization providing analytical so-lutions for point estimates and asymptotic standard errorsHowever this would not allow us to distinguish between ex-plained and unexplained variance in the spatial and temporalterms Hence theVi andZt terms make the model hierar-chical ie richer but also more tricky to estimate This chal-lenge was solved using Bayesian Markov chain Monte Carlomethods (MCMC Brooks 1998) As detailed in Eckert etal (2007 2010b) for similar problems the robustness of theinference was checked using tests based on starting differ-ent simulation runs at different points of the parameter space(Brooks and Gelman 1998) Poorly informativeprior prob-ability distributions were used for all parameters howeverthis standard choice allows us to obtainposteriorestimatesonly driven by information conveyed by the data From thejoint posteriordistribution of all parameters latent variablesand missing values we retained point estimates (theposte-rior mean)posteriorstandard deviations and 95 credibil-ity intervals the Bayesian counterpart of the classical confi-dence interval (Table 4) The construction of the model en-sures thatαi βt andεit are independent FurthermoreVi andsumk

Xkiak are independent so areZt andsumk

Xktbk Hence the

total variance of the data table is

VAR(SLAit ) = VAR

(sumk

Xkiak

)

+VAR

(sumk

Xktbk

)+ σ 2

v + σ 2z + σ 2 (7)

By analogy to the classicalR2 variance ratios can then becomputed to evaluate the respective weight of the different

Table 4Posterior characteristics of the proposed model applied onthe 43 ELA time series for the 1984ndash2010 periodak andbk denotethe regression parameters for each explanatory variable in bracketsOverall 25 and 975 denote the lower and upper bound of the95 credible interval

Mean 25 975

a1 (average altitude) 0381 0257 0502a2 (latitude) minus0123 minus0269 0023a3 (aspect) 0108 minus0022 0238a4 (surface area) minus0066 minus0199 0067b1 (summer CPDD) 0417 0268 0557b2 (winter precipitation) minus0311 minus0459 minus0155R2

clim 031 019 041R2

clim res 011 006 019R2

topo 024 016 032

R2topo res 009 005 014

R2res 025 019 030

R2tot 073 060 081

R2space 072 057 082

R2time 073 052 084

R2timespace 056 043 068

terms in ELA variability

R2topo = VAR

(sumk

Xkiak

)VAR(SLAit ) (8)

R2clim = VAR

(sumk

Xktbk

)VAR(SLAit ) (9)

R2topo res= σ 2

v VAR(SLAit ) (10)

R2clim res= σ 2

z VAR(SLAit ) (11)

R2res= σ 2VAR(SLAit ) (12)

Similarly ratios can be computed for the spatial and tem-poral terms solely

R2space= VAR

(sumk

Xkiak

)

(VAR

(sumk

Xkiak

)+ σ 2

v

) (13)

R2time = VAR

(sumk

Xktbk

)

(VAR

(sumk

Xktbk

)+ σ 2

z

) (14)

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1461

and getting rid of the random spatio temporal fluctuationsthe overall timespace ratio and determination coefficient inthe decomposition can be computed as

R2timespace=

(VAR

(sumk

Xktbk

)+ σ 2

z

)(

VAR (SLAit ) minus σ 2) (15)

R2tot =

(VAR

(sumk

Xkiak

)+ VAR

(sumk

Xktbk

))(

VAR (SLAit ) minus σ 2) (16)

All these statistics quantify the weight of mean spatial andtemporal effects in the total variability and the capacity ofthe chosen explanatory variables to model these mean ef-fects At the observation level of each yearglacier of themodel local and annual adjustment statistics can also becomputed as

δt =minus1

M

sumi

εit (17)

δi =minus1

T

sumt

εit (18)

δ2t = 1minus

radicradicradicradicradicradicsumi

ε2itsum

i

SLA2it

(19)

δ2i = 1minus

radicradicradicradicradicradicsumt

ε2itsum

t

SLA2it

(20)

respectively representing the annual and local bias and theannual and local determination coefficient withM the num-ber of glaciers considered andT the number of years studiedThey quantify the goodness of fit of each of the annuallocalseries

Variance ratios and adjustment statistics were evaluated bycomputing their values at each point of the MCMC iterativesimulation run giving access to point estimates and relateduncertainties taking into account the number of missing val-ues for each yearglacier (logically when nearly all observa-tions are missing for a given yearglacier the correspondingdetermination coefficientsδ2

tandδ2

iare unknown)

3 Results

Here we give the results of both the SLA and the climatevariables In Sect 31 we present the temporal and spatial

variability of the SLA measured on the 43 glaciers for the1984ndash2010 period In Sect 32 we present the variability ofclimate variables (ie summer CPDD and winter precipita-tion)

31 Changes in the equilibrium-line altitude

As mentioned above it has been demonstrated that on mid-latitude glaciers where superimposed ice is negligible theend-of-summer SLA is an accurate indicator of the ELA (seeFig 6 in Rabatel et al 2005 and Fig 3B in Rabatel et al2008) Because of this strong correlation and for the sake ofsimplicity we hereafter only use the term ELA

311 Temporal variability of the equilibrium-linealtitude

Figure 3a illustrates changes in the ELA over the 1984ndash2010period for the 43 glaciers studied (see also see SupplementTable II where all the data are presented) Considering allthe glaciers the average ELA for the whole period was lo-cated at 3035plusmn 120 m asl ie the interannual variabilityof the average ELA of all glaciers was high (120 m on av-erage) The difference between extreme years was as highas 460 m with the lowest average ELA measured in 1984(2790plusmn 180 m asl) and the highest average ELA measuredin 2003 (3250plusmn 135 m asl) In addition over the study pe-riod the ELA time series showed an average increasing trendof 64 m yrminus1 assuming a linear trend which results to be sta-tistically significant considering a risk of error of 5 Thisis the equivalent of an average increase of 170 m over the1984ndash2010 period ie higher than the interannual variabilityof the average ELA

The average annual ELA during the first five years of thestudy period was lower than the average ELA for the wholeperiod (except in 1986 when it was slightly higher than theaverage) These years match the end of a 15 yr period (fromthe mid-1970s to the late 1980s) during which alpine glaciersincreased in size due to higher winter accumulation and re-duced summer ablation (Vincent 2002 Thibert et al 2013)Since 2003 the average annual ELA has consistently beenabove the average ELA for the whole period Considering thetwo subsets ie before and after 2003 no significant trendappeared in either of the subsets The second subset is shorter(8 yr) than the first (19 yr) and trends computed on short pe-riods should be interpreted with caution However the aver-age ELA between the two subsets increased by about 140 mwhich is 82 of the total change Hence the year 2003 canalso be considered as a breakpoint in the time series It shouldbe noted that the 2003 extreme heat wave resulted in a reduc-tion or even complete loss on some glaciers of the firn areahence introducing a positive feedback on glacier mass bal-ances after 2003 consistent with higher ELAs

Considering each glacier individually Fig 4 shows therate of increase in the ELA over the whole period for each

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1462 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 3 Changes over the 1984ndash2010 period in(A) the ELA for the 43 glaciers studied (see Fig 1 and Table 1)(B) summer CPDD and(C)winter precipitation recorded by the weather stations used (see Fig 1 and Table 1) In each graph the horizontal black bar represents theannual average of the sample the gray box represents the median interval (Q3minusQ1) and the vertical black lines show the interval betweenthe first and the last decile (D1 and D9) On each graph (described in Sects 311 and 32) the dashed line represents the smooth underlyingtrend captured by a cubic smoothing spline regression (Lavigne et al 2012) The dotted line is the corresponding 95 confidence intervalThe yellow and purple boxes highlight the years in which the average ELA at the scale of the study region is linked with positive (yellow) ornegative (purple) winter precipitation anomalies (see Sect 42)

glacier plotted against its aspect In this study the aspectof the glacier matched the average aspect of the area of theglacier where the ELA fluctuated over the whole study pe-riod It is notable that (1) the increasing trend of the ELA var-ied between less than 1 m yrminus1 and more than 13 m yrminus1 de-pending on the glacier and (2) glaciers facing east displayeda more pronounced increasing trend of the ELA Howeverthis result should be interpreted with caution because the dis-tribution of the glaciers studied were not homogeneous withrespect to the aspect of the majority of glaciers in the FrenchAlps which are north facing Furthermore one of our se-lection criteria should also be recalled to measure the ELAfor each year the glaciers had to reach at least 3250 m asl(highest average ELA measured in 2003) and south-facingglaciers reaching this elevation are rare and can only be foundin the Mont Blanc Massif

Fig 4 Average increasing rate of the ELA (m yrminus1) over the1984ndash2010 period of each of the glaciers studied with respect to theaspect of the glacier The color of the diamond identifies the mas-sif in which the glacier is located blue= Ecrins green= Vanoisered= Mont Blanc

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

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1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

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1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

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1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1460 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

surface area and aspect andVi sim N(0σ 2

v

)is the spatial

residual ie the local effect for the glacieri which is notexplained by the spatial explanatory variables considered

Similarly the temporal term is decomposed as

βt =

sumk

Xktbk + Zt (6)

where theXkt are the temporal explanatory variables con-sidered ie the climate variables summer CPDD and win-ter precipitation andZt sim N

(0σ 2

z

)is the temporal residual

ie the annual effect common to all glaciers for any yeart which is not explained by the temporal explanatory variablesconsidered

For the sake of simplicity the model is fed with reducedstandardized variables (by dividing by the standard devia-tion the difference between each value and the average of theseries) This avoids identifiability problems by granting thatsumt

βt = 0 and makes theak andbk coefficients more directly

interpretable at least for non-heavily correlated explanatoryvariables (see below)

Without the Vi and Zt terms the model would be aclassical spatio-temporal linear model with easy inferenceeg with likelihood maximization providing analytical so-lutions for point estimates and asymptotic standard errorsHowever this would not allow us to distinguish between ex-plained and unexplained variance in the spatial and temporalterms Hence theVi andZt terms make the model hierar-chical ie richer but also more tricky to estimate This chal-lenge was solved using Bayesian Markov chain Monte Carlomethods (MCMC Brooks 1998) As detailed in Eckert etal (2007 2010b) for similar problems the robustness of theinference was checked using tests based on starting differ-ent simulation runs at different points of the parameter space(Brooks and Gelman 1998) Poorly informativeprior prob-ability distributions were used for all parameters howeverthis standard choice allows us to obtainposteriorestimatesonly driven by information conveyed by the data From thejoint posteriordistribution of all parameters latent variablesand missing values we retained point estimates (theposte-rior mean)posteriorstandard deviations and 95 credibil-ity intervals the Bayesian counterpart of the classical confi-dence interval (Table 4) The construction of the model en-sures thatαi βt andεit are independent FurthermoreVi andsumk

Xkiak are independent so areZt andsumk

Xktbk Hence the

total variance of the data table is

VAR(SLAit ) = VAR

(sumk

Xkiak

)

+VAR

(sumk

Xktbk

)+ σ 2

v + σ 2z + σ 2 (7)

By analogy to the classicalR2 variance ratios can then becomputed to evaluate the respective weight of the different

Table 4Posterior characteristics of the proposed model applied onthe 43 ELA time series for the 1984ndash2010 periodak andbk denotethe regression parameters for each explanatory variable in bracketsOverall 25 and 975 denote the lower and upper bound of the95 credible interval

Mean 25 975

a1 (average altitude) 0381 0257 0502a2 (latitude) minus0123 minus0269 0023a3 (aspect) 0108 minus0022 0238a4 (surface area) minus0066 minus0199 0067b1 (summer CPDD) 0417 0268 0557b2 (winter precipitation) minus0311 minus0459 minus0155R2

clim 031 019 041R2

clim res 011 006 019R2

topo 024 016 032

R2topo res 009 005 014

R2res 025 019 030

R2tot 073 060 081

R2space 072 057 082

R2time 073 052 084

R2timespace 056 043 068

terms in ELA variability

R2topo = VAR

(sumk

Xkiak

)VAR(SLAit ) (8)

R2clim = VAR

(sumk

Xktbk

)VAR(SLAit ) (9)

R2topo res= σ 2

v VAR(SLAit ) (10)

R2clim res= σ 2

z VAR(SLAit ) (11)

R2res= σ 2VAR(SLAit ) (12)

Similarly ratios can be computed for the spatial and tem-poral terms solely

R2space= VAR

(sumk

Xkiak

)

(VAR

(sumk

Xkiak

)+ σ 2

v

) (13)

R2time = VAR

(sumk

Xktbk

)

(VAR

(sumk

Xktbk

)+ σ 2

z

) (14)

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1461

and getting rid of the random spatio temporal fluctuationsthe overall timespace ratio and determination coefficient inthe decomposition can be computed as

R2timespace=

(VAR

(sumk

Xktbk

)+ σ 2

z

)(

VAR (SLAit ) minus σ 2) (15)

R2tot =

(VAR

(sumk

Xkiak

)+ VAR

(sumk

Xktbk

))(

VAR (SLAit ) minus σ 2) (16)

All these statistics quantify the weight of mean spatial andtemporal effects in the total variability and the capacity ofthe chosen explanatory variables to model these mean ef-fects At the observation level of each yearglacier of themodel local and annual adjustment statistics can also becomputed as

δt =minus1

M

sumi

εit (17)

δi =minus1

T

sumt

εit (18)

δ2t = 1minus

radicradicradicradicradicradicsumi

ε2itsum

i

SLA2it

(19)

δ2i = 1minus

radicradicradicradicradicradicsumt

ε2itsum

t

SLA2it

(20)

respectively representing the annual and local bias and theannual and local determination coefficient withM the num-ber of glaciers considered andT the number of years studiedThey quantify the goodness of fit of each of the annuallocalseries

Variance ratios and adjustment statistics were evaluated bycomputing their values at each point of the MCMC iterativesimulation run giving access to point estimates and relateduncertainties taking into account the number of missing val-ues for each yearglacier (logically when nearly all observa-tions are missing for a given yearglacier the correspondingdetermination coefficientsδ2

tandδ2

iare unknown)

3 Results

Here we give the results of both the SLA and the climatevariables In Sect 31 we present the temporal and spatial

variability of the SLA measured on the 43 glaciers for the1984ndash2010 period In Sect 32 we present the variability ofclimate variables (ie summer CPDD and winter precipita-tion)

31 Changes in the equilibrium-line altitude

As mentioned above it has been demonstrated that on mid-latitude glaciers where superimposed ice is negligible theend-of-summer SLA is an accurate indicator of the ELA (seeFig 6 in Rabatel et al 2005 and Fig 3B in Rabatel et al2008) Because of this strong correlation and for the sake ofsimplicity we hereafter only use the term ELA

311 Temporal variability of the equilibrium-linealtitude

Figure 3a illustrates changes in the ELA over the 1984ndash2010period for the 43 glaciers studied (see also see SupplementTable II where all the data are presented) Considering allthe glaciers the average ELA for the whole period was lo-cated at 3035plusmn 120 m asl ie the interannual variabilityof the average ELA of all glaciers was high (120 m on av-erage) The difference between extreme years was as highas 460 m with the lowest average ELA measured in 1984(2790plusmn 180 m asl) and the highest average ELA measuredin 2003 (3250plusmn 135 m asl) In addition over the study pe-riod the ELA time series showed an average increasing trendof 64 m yrminus1 assuming a linear trend which results to be sta-tistically significant considering a risk of error of 5 Thisis the equivalent of an average increase of 170 m over the1984ndash2010 period ie higher than the interannual variabilityof the average ELA

The average annual ELA during the first five years of thestudy period was lower than the average ELA for the wholeperiod (except in 1986 when it was slightly higher than theaverage) These years match the end of a 15 yr period (fromthe mid-1970s to the late 1980s) during which alpine glaciersincreased in size due to higher winter accumulation and re-duced summer ablation (Vincent 2002 Thibert et al 2013)Since 2003 the average annual ELA has consistently beenabove the average ELA for the whole period Considering thetwo subsets ie before and after 2003 no significant trendappeared in either of the subsets The second subset is shorter(8 yr) than the first (19 yr) and trends computed on short pe-riods should be interpreted with caution However the aver-age ELA between the two subsets increased by about 140 mwhich is 82 of the total change Hence the year 2003 canalso be considered as a breakpoint in the time series It shouldbe noted that the 2003 extreme heat wave resulted in a reduc-tion or even complete loss on some glaciers of the firn areahence introducing a positive feedback on glacier mass bal-ances after 2003 consistent with higher ELAs

Considering each glacier individually Fig 4 shows therate of increase in the ELA over the whole period for each

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1462 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 3 Changes over the 1984ndash2010 period in(A) the ELA for the 43 glaciers studied (see Fig 1 and Table 1)(B) summer CPDD and(C)winter precipitation recorded by the weather stations used (see Fig 1 and Table 1) In each graph the horizontal black bar represents theannual average of the sample the gray box represents the median interval (Q3minusQ1) and the vertical black lines show the interval betweenthe first and the last decile (D1 and D9) On each graph (described in Sects 311 and 32) the dashed line represents the smooth underlyingtrend captured by a cubic smoothing spline regression (Lavigne et al 2012) The dotted line is the corresponding 95 confidence intervalThe yellow and purple boxes highlight the years in which the average ELA at the scale of the study region is linked with positive (yellow) ornegative (purple) winter precipitation anomalies (see Sect 42)

glacier plotted against its aspect In this study the aspectof the glacier matched the average aspect of the area of theglacier where the ELA fluctuated over the whole study pe-riod It is notable that (1) the increasing trend of the ELA var-ied between less than 1 m yrminus1 and more than 13 m yrminus1 de-pending on the glacier and (2) glaciers facing east displayeda more pronounced increasing trend of the ELA Howeverthis result should be interpreted with caution because the dis-tribution of the glaciers studied were not homogeneous withrespect to the aspect of the majority of glaciers in the FrenchAlps which are north facing Furthermore one of our se-lection criteria should also be recalled to measure the ELAfor each year the glaciers had to reach at least 3250 m asl(highest average ELA measured in 2003) and south-facingglaciers reaching this elevation are rare and can only be foundin the Mont Blanc Massif

Fig 4 Average increasing rate of the ELA (m yrminus1) over the1984ndash2010 period of each of the glaciers studied with respect to theaspect of the glacier The color of the diamond identifies the mas-sif in which the glacier is located blue= Ecrins green= Vanoisered= Mont Blanc

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

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1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1461

and getting rid of the random spatio temporal fluctuationsthe overall timespace ratio and determination coefficient inthe decomposition can be computed as

R2timespace=

(VAR

(sumk

Xktbk

)+ σ 2

z

)(

VAR (SLAit ) minus σ 2) (15)

R2tot =

(VAR

(sumk

Xkiak

)+ VAR

(sumk

Xktbk

))(

VAR (SLAit ) minus σ 2) (16)

All these statistics quantify the weight of mean spatial andtemporal effects in the total variability and the capacity ofthe chosen explanatory variables to model these mean ef-fects At the observation level of each yearglacier of themodel local and annual adjustment statistics can also becomputed as

δt =minus1

M

sumi

εit (17)

δi =minus1

T

sumt

εit (18)

δ2t = 1minus

radicradicradicradicradicradicsumi

ε2itsum

i

SLA2it

(19)

δ2i = 1minus

radicradicradicradicradicradicsumt

ε2itsum

t

SLA2it

(20)

respectively representing the annual and local bias and theannual and local determination coefficient withM the num-ber of glaciers considered andT the number of years studiedThey quantify the goodness of fit of each of the annuallocalseries

Variance ratios and adjustment statistics were evaluated bycomputing their values at each point of the MCMC iterativesimulation run giving access to point estimates and relateduncertainties taking into account the number of missing val-ues for each yearglacier (logically when nearly all observa-tions are missing for a given yearglacier the correspondingdetermination coefficientsδ2

tandδ2

iare unknown)

3 Results

Here we give the results of both the SLA and the climatevariables In Sect 31 we present the temporal and spatial

variability of the SLA measured on the 43 glaciers for the1984ndash2010 period In Sect 32 we present the variability ofclimate variables (ie summer CPDD and winter precipita-tion)

31 Changes in the equilibrium-line altitude

As mentioned above it has been demonstrated that on mid-latitude glaciers where superimposed ice is negligible theend-of-summer SLA is an accurate indicator of the ELA (seeFig 6 in Rabatel et al 2005 and Fig 3B in Rabatel et al2008) Because of this strong correlation and for the sake ofsimplicity we hereafter only use the term ELA

311 Temporal variability of the equilibrium-linealtitude

Figure 3a illustrates changes in the ELA over the 1984ndash2010period for the 43 glaciers studied (see also see SupplementTable II where all the data are presented) Considering allthe glaciers the average ELA for the whole period was lo-cated at 3035plusmn 120 m asl ie the interannual variabilityof the average ELA of all glaciers was high (120 m on av-erage) The difference between extreme years was as highas 460 m with the lowest average ELA measured in 1984(2790plusmn 180 m asl) and the highest average ELA measuredin 2003 (3250plusmn 135 m asl) In addition over the study pe-riod the ELA time series showed an average increasing trendof 64 m yrminus1 assuming a linear trend which results to be sta-tistically significant considering a risk of error of 5 Thisis the equivalent of an average increase of 170 m over the1984ndash2010 period ie higher than the interannual variabilityof the average ELA

The average annual ELA during the first five years of thestudy period was lower than the average ELA for the wholeperiod (except in 1986 when it was slightly higher than theaverage) These years match the end of a 15 yr period (fromthe mid-1970s to the late 1980s) during which alpine glaciersincreased in size due to higher winter accumulation and re-duced summer ablation (Vincent 2002 Thibert et al 2013)Since 2003 the average annual ELA has consistently beenabove the average ELA for the whole period Considering thetwo subsets ie before and after 2003 no significant trendappeared in either of the subsets The second subset is shorter(8 yr) than the first (19 yr) and trends computed on short pe-riods should be interpreted with caution However the aver-age ELA between the two subsets increased by about 140 mwhich is 82 of the total change Hence the year 2003 canalso be considered as a breakpoint in the time series It shouldbe noted that the 2003 extreme heat wave resulted in a reduc-tion or even complete loss on some glaciers of the firn areahence introducing a positive feedback on glacier mass bal-ances after 2003 consistent with higher ELAs

Considering each glacier individually Fig 4 shows therate of increase in the ELA over the whole period for each

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1462 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 3 Changes over the 1984ndash2010 period in(A) the ELA for the 43 glaciers studied (see Fig 1 and Table 1)(B) summer CPDD and(C)winter precipitation recorded by the weather stations used (see Fig 1 and Table 1) In each graph the horizontal black bar represents theannual average of the sample the gray box represents the median interval (Q3minusQ1) and the vertical black lines show the interval betweenthe first and the last decile (D1 and D9) On each graph (described in Sects 311 and 32) the dashed line represents the smooth underlyingtrend captured by a cubic smoothing spline regression (Lavigne et al 2012) The dotted line is the corresponding 95 confidence intervalThe yellow and purple boxes highlight the years in which the average ELA at the scale of the study region is linked with positive (yellow) ornegative (purple) winter precipitation anomalies (see Sect 42)

glacier plotted against its aspect In this study the aspectof the glacier matched the average aspect of the area of theglacier where the ELA fluctuated over the whole study pe-riod It is notable that (1) the increasing trend of the ELA var-ied between less than 1 m yrminus1 and more than 13 m yrminus1 de-pending on the glacier and (2) glaciers facing east displayeda more pronounced increasing trend of the ELA Howeverthis result should be interpreted with caution because the dis-tribution of the glaciers studied were not homogeneous withrespect to the aspect of the majority of glaciers in the FrenchAlps which are north facing Furthermore one of our se-lection criteria should also be recalled to measure the ELAfor each year the glaciers had to reach at least 3250 m asl(highest average ELA measured in 2003) and south-facingglaciers reaching this elevation are rare and can only be foundin the Mont Blanc Massif

Fig 4 Average increasing rate of the ELA (m yrminus1) over the1984ndash2010 period of each of the glaciers studied with respect to theaspect of the glacier The color of the diamond identifies the mas-sif in which the glacier is located blue= Ecrins green= Vanoisered= Mont Blanc

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

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1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

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1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

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1462 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 3 Changes over the 1984ndash2010 period in(A) the ELA for the 43 glaciers studied (see Fig 1 and Table 1)(B) summer CPDD and(C)winter precipitation recorded by the weather stations used (see Fig 1 and Table 1) In each graph the horizontal black bar represents theannual average of the sample the gray box represents the median interval (Q3minusQ1) and the vertical black lines show the interval betweenthe first and the last decile (D1 and D9) On each graph (described in Sects 311 and 32) the dashed line represents the smooth underlyingtrend captured by a cubic smoothing spline regression (Lavigne et al 2012) The dotted line is the corresponding 95 confidence intervalThe yellow and purple boxes highlight the years in which the average ELA at the scale of the study region is linked with positive (yellow) ornegative (purple) winter precipitation anomalies (see Sect 42)

glacier plotted against its aspect In this study the aspectof the glacier matched the average aspect of the area of theglacier where the ELA fluctuated over the whole study pe-riod It is notable that (1) the increasing trend of the ELA var-ied between less than 1 m yrminus1 and more than 13 m yrminus1 de-pending on the glacier and (2) glaciers facing east displayeda more pronounced increasing trend of the ELA Howeverthis result should be interpreted with caution because the dis-tribution of the glaciers studied were not homogeneous withrespect to the aspect of the majority of glaciers in the FrenchAlps which are north facing Furthermore one of our se-lection criteria should also be recalled to measure the ELAfor each year the glaciers had to reach at least 3250 m asl(highest average ELA measured in 2003) and south-facingglaciers reaching this elevation are rare and can only be foundin the Mont Blanc Massif

Fig 4 Average increasing rate of the ELA (m yrminus1) over the1984ndash2010 period of each of the glaciers studied with respect to theaspect of the glacier The color of the diamond identifies the mas-sif in which the glacier is located blue= Ecrins green= Vanoisered= Mont Blanc

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

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1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

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A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

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1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

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1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1463

312 Spatial variability of the equilibrium-line altitude

Figure 5a shows the results of a factor analysis of the ELAtime series including all the glaciers and all the years inthe study period The factor analysis is a statistical methodallowing to describe the variability differences or simili-tudes among observed and correlated variables according toa lower number of factors ie unobserved variables for in-stance the space and time For the ELA time series the firsttwo factors explained 80 of the common variance with71 on the first factor The high percentage of common vari-ance on factor 1 represents the common temporal signal inthe interannual variability of the ELAs that is related to cli-matic forcing Factor 2 explained 9 of the common vari-ance of the glaciers studied and is related to different spa-tial patterns Hence with respect to the second factor twogroups can be distinguished corresponding to the northernAlps (Mont Blanc Vanoise) and the southern Alps (Ecrins)respectively One glacier in the Vanoise Massif GeacutebroulazGlacier is closer to the glacier group of the southern AlpsIts geographic location ie nearest to the southern sectoramong the Vanoise glaciers could partly explain the positionof this glacier in the graph The distinction between the twogroups could be related to a higher elevation of the ELA forthe glaciers of the southern sector (Fig 6a) which could beassociated with differences in winter accumulation betweenthe southern and northern Alps (see below) and warmer sum-mer temperature in the southern sector which would increasethe amount of CPDD and thus the ablation (note that the ef-fect of higher summer temperature on a shift of precipitationfrom snow to rain is not considered in this study becausewe do not use summer precipitation in our analysis) IndeedFig 6a shows the average ELA over the 1984ndash2010 periodfor each glacier plotted as a function of its latitude a cleardecreasing trend is apparent from south to north with an av-erage ELA located on average 150 m higher for the glaciersin the southern sector

32 Changes in climate conditions

Figure 3b shows the changes in summer CPDD overthe 1984ndash2010 period for the 22 weather stations usedSummer CPDD present an increasing trend averaging53plusmn 19 CPDD yrminus1 at 3000 m asl assuming a linear trendover the period 1984ndash2010 This trend is statistically signif-icant considering a risk of error of 5 Such an increase(150plusmn 19) in the CPDD over the study period is the equiva-lent of an additional energy supply of 14plusmn 6 W mminus2 whichcan result in an additional ablation of about 05 m we (as-suming that all of this energy is used to melt the snow andso that sublimation is negligible)

The year 2003 had the maximum summer CPDD in the1984ndash2010 period and 2003 is an outlier in the time seriesIn this particular year the ELA was about 300 meters higherthan the average for the whole period (in fact above many

glacierized summits) and is clearly associated with an un-usually warm summer (Beniston and Diaz 2004) that causedintense ablation because winter precipitation in 2003 wasaverage for the study period

Temporal variations in the summer CPDD at 3000 m aslwere highly homogeneous across the study area Indeed thefactor analysis of these data (Fig 5b) showed a common vari-ance of 85 on the first axis indicating similar temporal vari-ations between the stations located up to 150 km apart

Regarding winter precipitation no significant trend wasobserved over the study period (Fig 3c) However one cannote that on average interannual variability was lower af-ter 2001 (standard deviation divided by 2 after 2001 fallingfrom 140 mm to 62 mm) but it should be kept in mind thatthe time series after that date is short and this point thusneeds to be confirmed with longer data series Over the studyperiod maximum winter precipitation occurred in 2001 andthe same value was almost reached in 1995 These two yearscorrespond to low ELAs but not to the lowest of the wholetime series which occurred in 1984 and 1985 and appears tobe associated more with cooler summer temperatures

Variations in winter precipitation at the different stationsover time were not as homogeneous as the summer CPDDeven if the first factor of the factor analysis explained 71 of total variance (Fig 5c) As was the case for the ELAs twogroups can be distinguished on the second axis explaining9 of the variance This factor analysis showed that most ofthe variance was common to all the weather stations and wasdue to large-scale precipitation patterns but that local effectsalso exist This is particularly true for the weather stationslocated in the southern part of the study area which is in-fluenced by a Mediterranean climate in which some winterprecipitation is not only due to typical western disturbancesbut also to eastern events coming from the Gulf of Genoasuch as the one in December 2008 (Eckert et al 2010a)

4 Discussion

In this section we first discuss the bivariate relationshipsbetween each of the morpho-topographic variables consid-ered and the average ELA of each glacier for the study pe-riod Then in the second part we present the bivariate rela-tionships between the ELA interannual variability and eachclimate variable Finally we exploit the expansion of Lli-boutryrsquos approach (1974) that explicitly incorporates bothclimate and morpho-topographic variables to reconstruct an-nual ELAs

41 The role of morpho-topographic variables

Figure 6 shows the relationship between the average ELAof each glacier computed over the whole study period andthe latitude of the glacier (Fig 6a) the mean altitude of the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1464 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 5Factor analysis of(A) the ELAs measured on satellite images(B) the summer CPDD computed from weather stations and(C) winterprecipitation computed from weather stations Detailed information about each graph is provided in the text

Fig 6Average ELA of each one of the 43 glaciers studied over the 1984ndash2010 period as a function of(A) glacier latitude(B) glacier meanaltitude(C) glacier surface area and(D) glacier aspect The color of the diamonds in(A) and(D) identifies the massif in which the glacieris located blue= Ecrins green= Vanoise red= Mont Blanc

glacier (Fig 6b) the glacier surface area (Fig 6c) and theglacier aspect (Fig 6d)

We already mentioned the relationship between the ELAand the latitude (Fig 6a) in Sect 312 where we noted thatthe average ELA of the glaciers located in the southern partof the study area was about 150 m higher in altitude thanthe ELA of the glaciers located in the northern part of thestudy area This meridional effect is consistent with the drierand warmer conditions associated with the Mediterraneanclimate that prevails in the southern part of the study area

Figure 6b and c show the relationship between the averageELA and the average altitude of the glacier and the glaciersurface area respectively The correlation between the aver-age ELA and the glacier average altitude was positive mean-

ing that the lower the average altitude of the glacier the lowerthe average ELA Conversely the correlation between the av-erage ELA and the glacier surface area was negative mean-ing that the smaller the glacier the higher the ELA To un-derstand these correlations one has to keep in mind the re-lationship between the ELA and the accumulation area ra-tio (AAR) Indeed the ELA constitutes the lower limit ofthe accumulation zone which representssim 23 of the totalglacier surface area in a steady-state glacier Accordinglythe wider the accumulation zone the bigger the glacier thelower its snout the lower the mean altitude and consequentlythe lower the ELA of the glacier Current ELAs certainlycan not represent steady-state conditions nevertheless overa long period of time (almost 30 yr in our case) assuming

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1465

Fig 7 (A) Highest correlations among all the possible ldquoglacier vsstationrdquo pairs between each glacier (each symbol represents oneglacier) and CPDD plotted as a function of latitude(B) The samebut with the winter precipitation

pseudo-stationary conditions over this period the averagesurface area of a glacier as well as its average altitude canbe deemed to be representative of its average ELA This isin good agreement with the results presented by Braithwaiteand Raper (2010) showing on the basis of 94 glaciers thatthe glacier area median and mid-range altitudes are accurateproxies of the ELA for a period of time eg several decades

Figure 6d shows the average ELA of each glacier plot-ted as a function of the glacier aspect For this morpho-topographic variable we distinguished between the differentmassifs in the study area because when treated all togetherthe effects of aspect and latitude tended to cancel each otherout Indeed the average ELA of a glacier facing north in theEcrins Massif is almost the same as the average ELA of aglacier facing south in the Mont Blanc Massif As a resultthe meridional effect related to the latitude is stronger thanthe aspect effect However considering each massif sepa-rately and even if our samples are not homogeneous withrespect to aspect glaciers facing north and east have a loweraverage ELA than glaciers facing south and west as wouldbe expected for mid-latitude glaciers in the Northern Hemi-sphere

42 Climatic control of equilibrium-line altitudeinterannual variability

Analysis of the variances and correlations between the ELAof each glacier and the climate variables (summer CPDDand winter precipitation) was conducted for the study pe-riod Considering our 27 yr data set the correlation coeffi-cient r is statistically significant with a risk of error lowerthan 1 whenr gt 049 orr ltminus049 and lower than 5 whenr gt 038 orr ltminus038

Figure 7 shows the correlation coefficients for the two cli-mate variables obtained by selecting the weather station withthe highest value for each glacier The correlation coefficientswere plotted against the latitude of the glaciers in order toevaluate spatial variations in these coefficients For summerCPDD Fig 7a shows (1) a generally high and always sig-

nificant correlation coefficient regardless of the glacier andits location in the study region (2) a slight but not statisti-cally significant decrease in the correlation coefficient withincreasing latitude This means that the proportion of vari-ance of the ELAs explained by the summer CPDD is firstimportant and secondly of the same order regardless of theglacier concerned

This also appears clearly in Fig 3 which shows strongsimilarities between annual variability of ELAs (Fig 3a) andCPDD (Fig 3b) mostly an increasing trend over the wholestudy period although stronger before 1990 which appar-ently stopped after the 2003 maximum Hence the annualmeans of both variables are strongly correlated (r = 073)To further highlight their similarities a spline regression wasperformed which fit the data variability well apart from theexceptional CPDD in 2003 The correlation between theselow frequency signals was enhanced with regards to annualmeans reachingr = 084 confirming the very similar long-term variations in ELA and CPDD over the study area

Regarding winter precipitation Fig 7b shows the particu-lar behavior of the Ecrins Massif (blue triangles in Fig 7b)which could be related to a winter precipitation regime in-fluenced by the eastern disturbances coming from the Gulfof Genoa and presenting higher temporal variability than thetypical regime in which disturbances come from the westHowever except for three glaciers in the Ecrins Massif thecorrelation between the ELA and cumulative winter precip-itation was statistically significant considering a risk of er-ror lower than 5 and even 1 in most cases These threeglaciers are Glacier des Violettes (40) Glacier de la Girose(33) and Glacier du Mont de Lans (35) the low correlationwith precipitation for these glaciers could be due to topo-graphic effects influencing redistribution of the snow by thewind which can affect accumulation processes in one wayor the opposite ie overaccumulation or erosion (Girose andMont de Lans are located on a dome and Violettes in a nar-row and cached mountainside)

Hence the annual average ELAs (Fig 3a) and win-ter precipitation (Fig 3c) were rather strongly negativelycorrelated (r = minus063) Like for CPDDs a well-supportedspline regression enhanced their similarities (r = minus080)This showed that the irregularities in the increasing trendof the average ELA at the scale of the study region werelinked to winter precipitation anomalies excesses in 1995and 2001 and deficits in 1990 and 1998 which respectivelydecreasedincreased the annual average ELA

Finally concerning Fig 7 one has to keep in mind that(1) the highest correlation coefficient was found for summerCPDD and (2) except for few glaciers in the Ecrins Massifsignificant correlations were also found with winter precipi-tation but to a lesser extent than with summer CPDD Thesepoints are in good agreement with results found at a glacierscale based on direct field measurements showing that themass balance variability (and hence the ELA) is primarilycontrolled by summer ablation variability which is in turn

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1466 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 8 ELA anomalies (in meters) plotted versus CPDD anoma-lies (in C d left graph) and precipitation anomalies (in mm rightgraph) Each symbol represents one year in the 1984ndash2010 period

closely linked with summer CPDD (eg Martin 1974 Vallonet al 1998 Vincent 2002) However if Fig 3 confirms thatthe observed increasing trend in the average regional ELA isin fact mainly driven by increasingly warm summer tempera-tures it also shows that nonlinear low frequency fluctuationsin the average regional ELA are significantly related to win-ter precipitation anomalies

In terms of sensitivity of the ELA to climate variablesFig 8 shows the scatter plots of ELA anomalies vs summerCPDD anomalies (Fig 8a) and winter precipitation anoma-lies (Fig 8b) For this figure both the ELA anomalies andthe summer CPDD and winter precipitation anomalies werecomputed from the annual average values of the 43 glaciersand the six weather stations with the highest correlation co-efficients These two graphs show that the sensitivity of theELA to summer temperature was 115 mCminus1 (note that thelength of the period used to compute the summer CPDDwas 120 days in our case and that the temperature gradientused to compute the CPDD at 3000 m asl was 6C kmminus1)and the sensitivity of the ELA to winter precipitation was48 m100 mm This means that an additional amount of win-ter precipitation of about 240 mm is needed to compensatefor a daily average increase in summer temperature of 1CNote that in our estimate we do not consider that the in-crease in summer temperature would lead to a lengtheningof the ablation period

Our estimate of ELA sensitivity to summer temperature115 mCminus1 is in the middle range of values reported in theliterature which range from 60ndash70 mCminus1 (Vincent 2002)to 160 mCminus1 (Gerbaux et al 2005) Other values from theEuropean Alps and other mid-latitude temperate regions thatcan be found in the literature are 90ndash115 mCminus1 (Braith-waite and Zhang 2000) 100 mCminus1 (Zemp et al 2007)125 mCminus1 (Kuhn 1981) or 130 mCminus1 (Oerlemans andHoogendoorn 1989) Such a difference in values mainly re-sults from the approaches used that vary from empirical re-gressions from ablation stake measurements on a monitoredglacier and temperature data from one weather station lo-cated close to the glacier to modeling approaches based onphysical processes and providing information on the sensi-

tivity of the ELA to all meteorological variables Here ourapproach is empirical but relies on large number of glaciersand weather stations and can be considered as relevant at thescale of French Alpine glaciers

43 About the efficiency of our modeling approach toreconstruct equilibrium-line altitude time seriesfrom morpho-topographic and climate variables

Figures 9 and 10 summarize the results of the modeling ap-proach from the symmetric temporal and spatial point ofviews Hence in each figure A and B subplots illustratethe capacity of our method to model the common pattern oftime (respectively the interannual local effect) getting rid ofthe interannual effect of each glacier (respectively the annualcommon effect) C and D subplots quantify the goodness offit at the lower level of each annuallocal series in terms ofbias and determination coefficient

From Figs 9 and 10 and Table 4 one can first see that thetemporalspatial decomposition is very good with a lowR2

resequal to 025 This suggests a very clear distinction betweenthe effects of space and time in the total data variability Fur-thermore the average temporal effect for each year and theinterannual spatial effect for each glacier are well captured bythe symmetric temporal and spatial regression models withhigh determination coefficients in time (R2

time = 073) and inspace (R2

space= 072) leading to a total determination coeffi-

cient (R2tot = 073) This confirms the capacity of the chosen

explanatory variables to model the average spatial and tem-poral effects

This is particularly true for the temporal explanatory vari-ables with only two climate variables the regression ex-plains more than 30 of total variance (main term) lead-ing to the predominance of interannual fluctuations with re-gards to interglacier variability (R2

timespace= 056) As a con-sequence the average annual ELAs modeled from temporalexplanatory variables follow annual fluctuations of the meanELA (βt series) very well and roughly match the spline ad-justment presented in Fig 3a A Spearman test showed thatCPDD and winter precipitations are independent so thatsince all the explanatory variables are standardized anoma-lies the regression parameters given in Table 4 are directlythe marginal correlation coefficient with theβt series 042andminus031 respectively

For the spatial term the four morpho-topographic vari-ables explain aboutR2

topo= 25 of total variance and

R2space= 72 of the variability of the interannual average

ELA for each glacier (αi series) For the spatial explanatoryvariables a slight correlation exists between the surface areaand the average elevation of the glaciers However the valuesof ak (Eq 5) for each explanatory variable remain close tothe correlation values of the bivariate regressions (Sect 41)suggesting that the model is not overparameterized so that the

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1467

Fig 9 Results of temporal modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the annualaverage ELA modeled using the temporal explanatory variables only thus getting rid of the interannual mean specificity of each glacierao +βt (Eq 6) is the same plus ldquowhite noiserdquo not explained by the temporal explanatory variables(C) and(D) quantify the goodness of thefit for each annual series in terms of determination coefficient and bias respectively

full inferred multivariate relationship can be interpreted withreasonable confidence

Figures 9c and 10c (Figs 9d and 10d) present the annualand local determination coefficients (the annual and localbias) Since the bias is generally close to zero and the de-termination coefficient acceptable this confirms the overallcapacity of the model to reproduce the data set variability inaccordance with the inferred lowR2

res= 025 value for eachyeareach glacier In other words the assumption of completeseparability of the effects of space and time appears to becorrect for most of the yearsglaciers whose ELA time se-ries can be reasonably approximated by the regional annualmean seriesinterannual local mean series weighted by theinterannual mean effect for each glacierthe regional annualeffect for each year respectively However for some years orglaciers the high biaslow determination coefficients showthat this is not true ie that the spatio-temporal decomposi-

tion is insufficient to fully represent the variability of ELAvalues measured using the remote sensing images For thesepeculiar years or glaciers certain points are worth emphasiz-ing to explain the non-negligible spatio-temporal residuals

ndash In Fig 9c the years 1990 and 2008 showed low de-termination coefficients This means that for these twoyears the ELA variability between the glaciers con-cerned differed from that of the whole period Indeedwe mentioned in Sect 41 that over the whole studyperiod the average ELA of the glaciers of the southernsector of the study region was about 150 m higher inaltitude than that of the glaciers located in the northernsector In the years 1990 and 2008 the situation wasnot the same the ELA was homogeneous in 2008 andin 1990 the ELA of the glaciers located in the southernsector was lower than the ELA of the ones in the north-ern sector This situation is not clearly explained by the

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1468 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

Fig 10Results of spatial modeling(A) and(B) show the comparison between the ELA measured on the satellite images and the averageELA modeled using the spatial explanatory variables only thus getting rid of the annual effect common to all glaciersai is the same plusldquowhite noiserdquo not explained by the spatial explanatory variables(C) and(D) quantify the goodness of fit for each local series in terms ofdetermination coefficient and bias respectively

climate variables and further investigations are neededto explain what happened during these two years

ndash In Figure 9d the only year with an annual bias higherthanplusmn5 m was the year 2002 It should be noted thatthe year 2002 followed a year with a very high win-ter accumulation leading to a low ELA During thefield campaigns at the end of summer 2002 a firn line(firn from the year 2001) and the snow line were ob-served However the difference between the firn lineand the snow line is difficult to distinguish on imageswith 30 m resolution (most of the ELA in 2002 weremeasured on Landsat images) As a consequence it isquite possible that for some of the glaciers studied thefirn line was delineated instead of the snow line Thisused to be the limitation of the use of satellite imageshowever with the new sensors launched in the recent

years that have a resolution of few meters or even tensof centimeters this limitation is now reduced

ndash In Fig 10c some glaciers (eg Geacutebroulaz or Vio-lettes glaciers) show low determination coefficientThis means that for these glaciers the ELA interan-nual variability differed from the one computed forall the glaciers studied For Geacutebroulaz Glacier Raba-tel et al (2005) had already reported specific behaviorThese authors assumed that the peculiar accumulationconditions on this glacier could lead to a low mass-balance gradient This could be one of the causes ofthe different interannual variability of the ELA on thisglacier For Violettes Glaciers (the only glacier with alocal bias above 15 m Fig 10d) as already mentionedin Sect 42 the peculiar morpho-topographic condi-tions could be responsible for the different interannualvariability of its ELA

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1469

5 Conclusion

This study presents time series of ELA for 43 glaciers lo-cated in the western Alps over the 1984ndash2010 time periodELAs were derived from the end-of-summer snow line alti-tude an accurate proxy of the ELA for mid-latitude mountainglaciers More than 120 satellite images were used to obtainan annual coverage of the 43 glaciers during the 27 yr periodIn parallel this study analyzed changes in climate variablesin terms of summer CPDD and winter precipitation Thesetwo variables were obtained for the same time period from 22weather stations located in and around the study area Fromthese time series several points are worth emphasizing

ndash Assuming a linear trend over the study period theaverage ELA increased by about 170 m the summerCPDD increased by about 150 PDD at 3000 m aslwhile winter precipitation remained rather stationary

ndash Temporal variability of the ELAs summer CPDD andwinter precipitation were very homogeneous over thestudy region Regarding spatial variability homogene-ity was high for summer CPDD but glaciers in thesouthern part of the study area differed from the others

Bivariate and relationships between the ELAs and morpho-topographic and climate variables were also establishedThese relationships allow us to highlight the fact that

ndash The average ELA over the study period of each glacieris strongly controlled by its morpho-topographic vari-ables namely its average altitude surface area latitudeand aspect

ndash The interannual variability of the average ELA of theglaciers studied is strongly controlled by climate vari-ables with the observed increasing trend in the averageELA truly mainly driven by increasingly warm tem-peratures even if nonlinear low frequency fluctuationsin the average ELA time series appear to be signifi-cantly related to winter precipitation anomalies

From these relationships we propose an expansion ofLliboutryrsquos approach implemented within a hierarchicalBayesian framework to reconstruct the ELA fluctuations ofany glacier in the study area Considering two climate vari-ables and four morpho-topographics variables we show thatour approach is well suited to (1) isolate an average temporaleffect for each year and an interannual mean effect for eachglacier (2) model and explain their fluctuations from a re-duced number of climatic and topographic explanatory vari-ables whose respective weight and related uncertainties arequantified in a consistent manner and (3) well approximatethe full variability of our ELA data set for each year and foreach glacier and identify specific spatio-temporal behaviorsthat do not match the assumption of complete separability ofspace and time effects

Finally both methods presented in this study based on re-mote sensing and modeling are powerful tools to provide an-nual series of ELA that will be useful in hydrological model-ing of high mountain watersheds where there is a need to in-tegrate the glacier component in such an approach and whenno field data are available

Supplementary material related to this article isavailable online athttpwwwthe-cryospherenet714552013tc-7-1455-2013-supplementpdf

AcknowledgementsThis study was conducted in the con-text of the French glacier observatory GLACIOCLIM(httpwww-lggeujf-grenoblefrServiceObsindexhtm) Weare grateful to Imanol Goni for his help with image processingThis work was possible thanks to the contributions of Meacuteteacuteo-France Electriciteacute de France and Meacuteteacuteo-Suisse who provided themeteorological data USGS-EDC and GLIMS program who grantedfree access to Landsat images and ASTER images respectivelySPOT images were provided through the CNESSPOT-Image ISISprogram contracts 0103-157 0412-725 2010-405 and 2010-435We thank G Hilmar Gudmundsson (handling editor) M Zempand an anonymous referee for their constructive comments used toimprove the paper

Edited by G H Gudmundsson

The publication of this articleis financed by CNRS-INSU

References

Barcaza G Aniya M Matsumoto T and Aoki T Satellite-derived equilibrium lines in Northern Patagonia Icefield Chileand their implications to glacier variations Arct Antarct AlpRes 41 174ndash182 doi1016571938-4246-412174 2009

Baltsavias E P Favey E Bauder A Bosch H and Pateraki MDigital surface modelling by airborne laser scanning and digitalphotogrammetry for glacier monitoring Photogramm Rec 17243ndash273 doi1011110031-868X00182 1999

Beniston M and Diaz H F The 2003 heat wave as an exampleof summers in a greenhouse climate Observations and climatemodel simulations for Basel Switzerland Glob Planet Change44 73ndash81 doi101016jgloplacha200406006 2004

Berthier E and Vincent C Relative contribution of surface massbalance and ice flux changes to the accelerated thinning of theMer de Glace (Alps) over 1979ndash2008 J Glaciol 58 501-512doi1031892012JoG11J083 2012

Berthier E Arnaud Y Baratoux D Vincent C and Reacutemy FRecent rapid thinning of the ldquoMer de Glacerdquo glacier derived

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013

1470 A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps

from satellite optical images Geophys Res Lett 31 L17401doi1010292004GL020706 2004

Braithwaite R J Can the mass balance of a glacier be esti-mated from its equilibrium-line altitude J Glaciol 30 364ndash368 1984

Braithwaite R J and Raper S C B Estimating equilibrium-linealtitude (ELA) from glacier inventory data Ann Glaciol 50127ndash132 doi103189172756410790595930 2010

Braithwaite R J and Zhang Y Sensitivity of mass balances offive Swiss glaciers to temperature changes assessed by tuning adegreeday model J Glaciol 46 7ndash14 2000

Brooks S P Markov Chain Monte Carlo method and itsapplication The Statistician 47 69ndash100 doi1011111467-988400117 1998Brooks S P and Gelman A General methods for monitoringconvergence of iterative simulations J Comput Graph Stat 7434ndash455 1998

Clare G R Fitzharris B Chinn T and Salinger M Interannualvariation in end-of-summer snowlines of the Southern Alps ofNew Zealand and relationships with southern hemisphere atmo-spheric circulation and sea surface temperature patterns Int JClimatol 22 107ndash120 doi101002joc722 2002

Chinn T Heydenrych C and Salinger M Use of the ELAas a practical method of monitoring glacier response to cli-mate in New Zealandrsquos Southern Alps J Glaciol 51 85ndash95doi103189172756505781829593 2005

Dedieu J-P and Reynaud L Teacuteleacutedeacutetection appliqueacutee au suivides glaciers des Alpes franccedilaises Houille Blanche 5 355ndash3581990

Demuth M N and Pietroniro A Inferring glacier mass balanceusing RADARSAT results from Peyto Glacier Canada GeogrAnn 81 521ndash540 doi1011111468-045900081 1999

Eckert N Parent E Belanger L and Garcia S Hierarchicalmodelling for spatial analysis of the number of avalanche occur-rences at the scale of the township Cold Reg Sci Technol 5097ndash112 doi101016jcoldregions200701008 2007

Eckert N Coleou C Castebrunet H Giraud G Descha-tres M and Gaume J Cross-comparison of meteorolog-ical and avalanche data for characterising avalanche cy-cles the example of December 2008 in the eastern partof the French Alps Cold Reg Sci Technol 64 119ndash136doi101016jcoldregions201008009 2010a

Eckert N Naaim M and Parent E Long-term avalanche hazardassessment with a Bayesian depth-averaged propagation modelJ Glaciol 56 563ndash586 doi1031890022143107931463312010b

Eckert N Baya H Thibert E and Vincent C Ex-tracting the temporal signal from a winter and summermass-balance series application to a six-decade record atGlacier de Sarennes French Alps J Glaciol 57 134ndash150doi103189002214311795306673 2011

Echelmeyer K Harrison W Larsen C Sapiano J Mitchell JEDemallie J Rabus B Adalgeirsdottir G and Sombardier LAirborne surface profiling of glaciers A case-study in Alaska JGlaciol 42 538ndash547 1996

Gardelle J Berthier E and Arnaud Y Impact of resolu-tion and radar penetration on glacier elevation changes com-puted from DEM differencing J Glaciol 58 419ndash422doi1031892012JoG11J175 2012

Gerbaux M Genthon C Etchevers P Vincent C and DedieuJ-P Surface mass balance of glaciers in the French Alps dis-tributed modelling and sensitivity to climate change J Glaciol51 561ndash572 doi103189172756505781829133 2005

Kuhn M Climate and glaciers IAHS 131 3ndash20 1981Lavigne A Bel L Parent E and Eckert N A model for spatio-

temporal clustering using multinomial probit regression applica-tion to avalanche counts in the French Alps Envirometrics 23522ndash534 2012

Letreacuteguilly A and Reynaud L Space and time distribution ofglacier mass-balance in northern hemisphere Arct Alp Res 2243ndash50 1990

Lliboutry L Traiteacute de glaciologie Tome II Glaciers variations duclimat sols geleacutes Paris Masson et Cie 1965

Lliboutry L Multivariate statistical analysis of glacier annual bal-ances J Glaciol 13 371ndash392 1974

Machguth H Paul F Kotlarski S and Hoelzle M Calculat-ing distributed glacier mass balance for the Swiss Alps fromregional climate model output A methodical description andinterpretation of the results J Geophys Res 114 D19106doi1010292009JD011775 2009

Mathieu R Chinn T and Fitzharris B Detecting theequilibrium-line altitudes of New Zealand glaciers using ASTERsatellite images New Zeal J Geol Geop 52 209ndash222doi10108000288300909509887 2009

Martin S Correacutelation bilans de masse annuels-facteursmeacuteteacuteorologiques dans les Grandes Rousses Z GletscherkGlazialgeol BD X 89ndash100 1974

Marzeion B Hofer M Jarosch A H Kaser G and Moumllg TA minimal model for reconstructing interannual mass balancevariability of glaciers in the European Alps The Cryosphere 671ndash84 doi105194tc-6-71-2012 2012

Meier M F Remote sensing of snow and ice Hydrol Sci Bul25 307ndash330 1980

Oerlemans J and Hoogendoorn N C Mass-balance gradients andclimate change J Glaciol 35 399ndash405 1989

Oslashstrem G ERTS data in glaciology ndash an effort to monitor glaciermass balance from satellite imagery J Glaciol 15 403ndash4151975

Paul F and Haeberli W Spatial variability of glacier eleva-tion changes in the Swiss Alps obtained from two digital ele-vation models Geophys Res Lett 35 L21502 doi1010292008GL034718 2008

Rabatel A Dedieu J-P and Reynaud L Reconstitution des fluc-tuations du bilan de masse du Glacier Blanc (Massif des EcrinsFrance) par teacuteleacutedeacutetection optique (imagerie Spot et Landsat)Houille Blanche 67 64ndash71 2002

Rabatel A Dedieu J-P and Vincent C Using remote-sensingdata to determine equilibrium-line altitude and mass-balancetime series validation on three French glaciers 1994-2002J Glaciol 51 539ndash546 doi1031891727565057818291062005

Rabatel A Dedieu J-P Thibert E Letreguilly A and Vin-cent C 25 years (1981ndash2005) of equilibrium-line altitude andmass-balance reconstruction on Glacier Blanc French Alps us-ing remote-sensing methods and meteorological data J Glaciol54 307ndash314 doi103189002214308784886063 2008

Rabatel A Bermejo A Loarte E Soruco A Gomez J Leonar-dini G Vincent C and Sicart J E Can the snowline be

The Cryosphere 7 1455ndash1471 2013 wwwthe-cryospherenet714552013

A Rabatel et al Changes in glacier equilibrium-line altitude in the western Alps 1471

used as an indicator of the equilibrium line and mass balancefor glaciers in the outer tropics J Glaciol 58 1027ndash1036doi1031892012JoG12J027 2012a

Rabatel A Dedieu J-P Letreacuteguilly A and Six D Remote-sesing monitoring of the evolution of glacier surface area andequilibrium-line altitude in the French Alps Proceedings of the25th symposium of the International Association of Climatology5ndash8 September 2012 Grenoble France 2012b

Rees W G Remote sensing of snow and ice CRC Press BocaRaton Florida 285 pp 2005

Reynaud L Can the linear balance model be extended to the wholeAlps IAHS AISH 126 273ndash284 1980

Thibert E Faure J and Vincent C Mass balances of the GlacierBlanc between 1952 1981 and 2002 calculated from digital ele-vation models Houille Blanche 2 72ndash78 2005

Thibert E Eckert N and Vincent C Climatic drivers ofseasonal glacier mass balances an analysis of 6 decades atGlacier de Sarennes (French Alps) The Cryosphere 7 47ndash66doi105194tc-7-47-2013 2013

Vallon M Vincent C and Reynaud L Altitudinal gradient ofmass balance sensitivity to climatic change from 18 years of ob-servations on glacier drsquoArgentiere France J Glaciol 44 93ndash961998

Vincent C Influence of climate change over the 20th Century onfour French glacier mass balances J Geophys Res 107 4375doi1010292001JD000832 2002

Vincent C Soruco A Six D and Le Meur E Glacierthickening and decay analysis from 50 years of glacio-logical observations performed on Glacier drsquoArgentiegravereMont Blanc area France Ann Glaciol 50 73ndash79doi103189172756409787769500 2009

Wikle C Hierarchical Bayesian models for predicting thespread of ecological processes Ecology 84 1382ndash1394doi1018900012-9658(2003)084[1382HBMFPT]20CO22003

WGMS Fluctuations of Glaciers 2005ndash2010 Volume Xedited by Zemp M Frey H Gaumlrtner-Roer I Nuss-baumer S U Hoelzle M Paul F and Haeberli WICSU(WDS)IUGG(IACS)UNEPUNESCOWMO WorldGlacier Monitoring Service Zurich Switzerland 336 pppublication based on database version doi105904wgms-fog-2012-11 2012

Zemp M Hoelzle M and Haeberli W Distributed modellingof the regional climatic equilibrium line altitude of glaciersin the European Alps Glob Planet Change 56 83ndash100doi101016jgloplacha200607002 2007

wwwthe-cryospherenet714552013 The Cryosphere 7 1455ndash1471 2013