TCThe CryosphereTCThe Cryosphere1994-0424Copernicus GmbHGöttingen, Germany10.5194/tc-9-2135-2015From Doktor Kurowski's Schneegrenze to our modern glacier equilibrium line
altitude (ELA)BraithwaiteR. J.r.braithwaite@manchester.ac.ukSchool of Environment, Education and Development (SEED), University of
Manchester, Manchester, UKR. J. Braithwaite (r.braithwaite@manchester.ac.uk)18November2015962135214815May201517June201521September20152October2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/9/2135/2015/tc-9-2135-2015.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/9/2135/2015/tc-9-2135-2015.pdf
Translated into modern terminology, Kurowski suggested in 1891 that the
equilibrium line altitude (ELA) of a glacier is equal to the mean altitude
of the glacier when the whole glacier is in balance between accumulation and
ablation. Kurowski's method has been widely misunderstood, partly due to
inappropriate use of statistical terminology by later workers, and has only
been tested by Braithwaite and Müller in a 1980 paper (for 32 glaciers).
I now compare Kurowski's mean altitude with balanced-budget ELA
calculated for 103 present-day glaciers with measured surface mass-balance
data. Kurowski's mean altitude is significantly higher (at 95 % level)
than balanced-budget ELA for 19 outlet and 42 valley glaciers, but not
significantly higher for 34 mountain glaciers. The error in Kurowski mean
altitude as a predictor of balanced-budget ELA might be due to generally
lower balance gradients in accumulation areas compared with ablation areas
for many glaciers, as suggested by several workers, but some glaciers have
higher gradients, presumably due to precipitation increase with altitude.
The relatively close agreement between balanced-budget ELA and mean altitude
for mountain glaciers (mean error – 8 m with standard deviation 59 m) may
reflect smaller altitude ranges for these glaciers such that there is less
room for effects of different balance gradients to manifest themselves.
Introduction
Ludwig Kurowski was born in 1866 in Napajedla, Moravia (then in the Austrian
Empire and now in the Czech Republic), and died in 1912 in Vienna (http://mahren.germanistika.cz). For his doctoral-thesis research at the
University of Vienna, Kurowski (1891) studied the snow line (German:
Schneegrenze) in the Finsteraarhorn region of the Swiss Alps. He suggested the altitude
of the snow line on a glacier is equal to the mean altitude of the glacier
when snow accumulation and melt are in balance for the whole glacier. A
relatively recent definition of snow line (Armstrong et al., 1973) is “the
line or zone on land that separates areas in which fallen snow disappears in
summer from areas in which snow remains throughout the year. The altitude of
the snow line is controlled by temperature and the amount of snowfall (cf.
equilibrium line and firn line)”. Students of snow line in the 19th century would have broadly
agreed with this definition before Ratzel (1886) introduced extra terms like
climatic and orographic to qualify snow line. Ratzel (1886) also argued that the material left
at the end of the melt season is firn rather than snow but Kurowski (1891) does not
use Ratzel's preferred term Firngrenze.
The snow line definition above explicitly refers to the landscape at the end
of summer as being snow-covered or snow-free, but a mass-balance concept
is also implicit in the definition, i.e. snowmelt equals snow accumulation
at the snow line, and this is the aspect of snow line studied by Kurowski (1891).
In early modern mass-balance studies in the 1940s and 1950s, the altitude on
the glacier where mass balance is zero for a particular year was termed
altitude of “firn line”, corresponding to the German Firngrenze. However, firn line
implies that “firn” is visible on the glacier surface above the
zero-balance line while we now know that the lower accumulation zone of
some glaciers can consist of ice (“superimposed ice”) formed by refreezing
of water from melting snow; see Fig. 2.1 in Paterson (1994). Baird (1952)
seems to have been the first to use the term “equilibrium line altitude”
(ELA) for the zero-balance altitude, and this usage became accepted as
standard by the late 1960s (Anonymous, 1969). The distinction between firn
line and equilibrium line is marked for high latitude glaciers, e.g. in
Greenland or on Arctic islands, but is quite unimportant for the Alpine
glaciers studied by Kurowski and other pioneers. We can therefore translate
Kurowski's Schneegrenze (where snowmelt equals snow accumulation) as equilibrium line
(where mass balance is zero) and regard most late 19th century snow
line (a.k.a. firn line) methods as being equally applicable to equilibrium
line. The best review in English of models for indirect estimation of firn
lines (a.k.a. equilibrium lines) is in an obscure book-chapter by
Osmaston (1975), which I only discovered when preparing a late draft of the
present paper.
The simple theory of Kurowski (1891) depends on the assumption that mass-balance gradient is constant across the whole altitude range of the glacier.
This was criticized by Hess (1904) and Reid (1908), and several modern
authors have attempted to account for variations in mass-balance gradients
by defining a ratio (“balance ratio”) between balance gradients in the
ablation and accumulation zones (Furbish and Andrews, 1984; Osmaston, 2005;
Rea, 2009). Kurowski himself argued that non-linearity in the
balance–altitude equation need not cause a large error as low and high
altitudes on a glacier usually coincide with small areas, and are not
weighted heavily in calculating mean altitude. It is surprising that nobody
has verified the basic Kurowski theory with observed mass-balance data
except for Braithwaite and Müller (1980). The main purpose of the
present paper is to critically test the original Kurowski (1891) theory with
observed mass-balance data from more glaciers, and then to discuss the
results together with balance ratio data from Rea (2009).
Readers need not share my wish to honour Kurowski's pioneering work,
involving one of the earliest quantitative models in glaciology, but they
should agree that the estimation of glacier ELA from topographic proxies is
still an active and legitimate area of research in glaciology and quaternary
science. Recent ELA-related work includes Benn and Lehmkuhl (2000), Kaser
and Osmaston (2002), Cogley and McIntyre (2003), Leonard and Fountain (2003), Carrivick and Brewer (2004), Benn et al. (2005), Osmaston (2005),
Dyurgerov et al. (2009), Braithwaite and Raper (2009), Rea (2009), Kern and
László (2010), Bakke and Nesje (2011), Rabatel et al. (2013),
Ignéczi and Nagy (2013) and Heymann (2014), to cite only a few. The
possibilities of monitoring year-to-year variations in the
end-of-summer snow line (EOSS) from aircraft (Chinn, 1995) or from
satellite images (Rabatel et al., 2005, 2012; Mathieu et al., 2014) raise
similar needs to estimate proxy ELAs for present-day glaciers for which
there are no observed mass-balance data.
Tutorial on glacier altitudes
Kurowski's work has often been ignored or misquoted, his name is sometimes
wrongly spelled following Hess (1904), and Sissons (1974) and Sutherland (1984) rediscovered his method without citing him. Because of many
misquotes the reader may not understand Kurowski's method unless he/she has
him/herself read the original article. A PDF of the original article (kindly
provided by Hans-Dieter Schwartz of the Bavarian Academy of Sciences) is
available in the online Supplement. One underlying problem is the
widespread misuse of statistical terms like mean and median when applied to glacier
altitudes (Cox, 2004). This issue is so central to a discussion of Kurowski (1891) that I give a worked example, using the area–altitude distribution
of Hintereisferner in the Austrian Alps in the year 2001, to illustrate
concepts; see Sect. 5 for sources of data.
The graph of the area–altitude distribution in Fig. 1a looks like a
histogram (probability distribution function) of altitudes on
Hintereisferner and could have been obtained from a digital elevation model,
with area representing the number of pixels of equal area in each altitude
interval. The mean altitude for such a distribution is
h¯=Hmean=∑i=1i=Nhi×ai∑i=1i=Nai,
where ai is the area of the ith altitude band and hi is its
altitude and N is the number of altitude bands. For the given altitude–area
distribution (Fig. 1a) for Hintereisferner, the mean altitude Hmean is
3038 m a.s.l. This is the mean altitude of the glacier according to the
Kurowski (1891) method and it is obvious from his Table III that he
calculates his “Mittlere Höhe des Gletschers” from the altitude–area distribution of each glacier
according to Eq. (1).
Area–altitude distribution for Hintereisferner, Austrian Alps, for
the year 2001: (a) shows areas of altitude bands vs. their mean
altitudes; (b) shows percentage of glacier area above any particular
altitude.
Some authors incorrectly assert that Kurowski (1891) used an
accumulation–area ratio (AAR) of 50 % to locate the snow line (Müller,
1980; Kotlyakov and Krenke, 1982), and the guidelines of the World Glacier
Inventory (TTS, 1977) incorrectly refer to this altitude as “mean altitude”.
Figure 1b shows the percentage of the area lying above any particular altitude
(cumulative distribution function). The median altitude is that altitude
dividing the glacier area into equal halves, i.e. it is the altitude
(x-coordinate) corresponding to a y-coordinate of 50 %. For the given
altitude–area distribution (Fig. 1b), the median altitude H50 is 3056 m a.s.l. This is the altitude giving AAR = 50 %. In a similar way, the
altitude H60 above which 60 % of the glacier area lies is 2989 m a.s.l. Kurowski (1891) quotes Brückner (1886) as saying that 75 % of
the glacier lies above the snow line (which nobody would believe today), and
H75= 2878 m a.s.l. in the present case.
Some authors incorrectly assert that Kurowski (1891) used an average of
maximum and minimum glacier altitude to locate the snow line (Cogley and
McIntyre, 2003; Leonard and Fountain, 2003). The minimum and maximum
altitudes for the glacier are 2400 and 3727 m a.s.l respectively, and the
mid-range altitude of the glacier is
Hmid=Hmax+Hmin2
In the present case, the mid-range altitude (Hmid) is 3064 m a.s.l.
Manley (1959) estimated ELA (or snow line or firn line) as mid-range
altitude according to (2) but many authors incorrectly assert that he used
the “median” altitude although Manley does not even mention the word.
Authors incorrectly using “median” for this mid-range altitude include
Porter (1975), Meierding (1982), Hawkins (1985), Benn and Lehmkuhl (2000),
Carrivick and Brewer (2004), Benn et al. (2005), Osmaston (2005), Rea (2009), Dobhal (2011), and Bakke and Nesje (2011) to mention only a few.
Incorrect use of terminology can be inferred in any book or paper that
refers to both “median altitude” and to “AAR” without noting that the
correctly defined median altitude is identical to the altitude with
AAR = 50 %, e.g. Nesje and Dahl (2000), and Benn and Evans (2010).
Kurowski's theory was purely in terms of mean altitude, correctly defined in
(1), but median and mid-range altitudes for glaciers are generally close
to the mean altitude and would be identical to it if the area–altitude
distribution were symmetric. The area–altitude distribution of
Hintereisferner (Fig. 1a) is only slightly asymmetric, being somewhat skewed
to higher altitudes, but a wide variety can be found for other glaciers and
it is important not to conflate the various altitudes.
Snow line before Kurowski
The scientific concept of snow line was discovered by the French
geophysicist Pierre Bouguer (1698–1758) on an expedition to tropical South
America (Klengel, 1889). Up to the early 19th century, the snow line
had been observed in many areas so that Alexander von Humboldt
(1769–1859) could start to compile a global picture of snow line
variations. A version of von Humboldt's snow line table is given in English
by Kaemtz (1845, pp. 228–229) with snow line altitudes for 34 regions from
all over the world. Heim (1885, pp. 18–21) gives a greatly extended table,
and Hess (1904, map 1) plots a world map of glacier cover and snow line.
Paschinger (1912) makes the first climatological analysis of snow line in
various climatic regions.
Most of these snow line data were based on observations of an apparently sharp
delineation between snow-covered and snow-free areas as seen from a distance
of a few kilometres, typically by an observer in a valley or on a mountain
pass, looking upwards into the high mountains. It was known very early that
the snow line fluctuates with season, and from one year to the next, with
large local spatial variations due to topography and aspect, and that the
apparent sharp delineation between snow-covered and snow-free landscape
disappears on closer examination to be replaced by a broad zone of snow
patches, slowly morphing into a continuous snow cover (Mousson, 1854, p. 3;
Heim, 1885, pp. 9–21; Ratzel, 1886; Klengel, 1889; Kurowski, 1891, p. 120). To overcome these problems, snow line has sometimes been defined as
the boundary between > 50 % snow cover and < 50 %
snow cover on a flat surface (Escher, 1970). All of these problems can be
overcome with modern technology of regular remote sensing and image
processing (Tang et al., 2014; Gafurov et al., 2015) but would have been
nearly impossible with 19th century methods. In this sense, much of the
early work on the snow line as a measure of snow-covered landscape was
premature.
Ratzel (1886) was very critical of snow line observations based on
“traveller's tales” (this was obviously a poke at Alexander von Humboldt's
table) and introduced much of our modern armoury of regional, climatic,
temporary, and orographic snow line although these were not easy to measure
at the time. More fruitfully, a number of 19th century workers
recognized that glacier accumulation areas occupy most of the region above
the snow line so that the year-on-year accumulation of snow is offset by ice
flow to lower elevations. More attention was then focussed on glaciers which
were then being mapped in some detail for the first time in the Alps. One of
the resulting map-based methods to determine glacier snow line was by
Kurowski (1891).
Kurowski's work
Kurowski (1891) developed a simple theory for the altitude of snow line on a
glacier, which may be one of the first theories in glaciology. I translate
his theory into modern mass-balance terminology (Anonymous, 1969; Cogley
et al., 2011) in the present paper although we must remember that glacier
mass balance in its modern sense was not measured in the 19th century.
In essence, Kurowski (1891) assumed that specific mass balance bi at
any altitude is proportional to the height above or below the ELA0 for
which the whole glacier is in balance:
bi=k×hi-ELA0,
where k is balance gradient on the glacier (assumed constant for the whole
elevation range of the glacier) and ELA0 is the balanced-budget ELA. Some
people use the term steady-state to qualify this ELA but this implies zero change in a
multitude of factors rather than just the mass balance; see comments by M. F. Meier in the discussion following the papers by Braithwaite and
Müller (1980) and Radok (1980), and see also Cogley et al. (2011). I have
a similar objection to the term steady-state AAR used by some authors (Kern and László,
2010; Ignéczi and Nagy, 2013) and would prefer the term equilibrium AAR of
Dyurgerov et al. (2009) if not balanced-budget AAR.
Using modern terminology (Anonymous, 1969; Cogley et al., 2011), the mean
specific balance b¯ of the whole glacier is the area-weighted
sum of specific balances:
b¯=∑i=1i=Nai×bi∑i=1i=Nai.
Area-weighted averaging of both sides of (3) gives
b¯=k×h¯-k×ELA0,
where h¯ is the mean altitude of the glacier, defined by Equation
(1). Rearranging (5) and noting that b¯= zero (by assumption)
gives
ELA0=h¯=Hmean.
Equation (6) expresses the identity between balanced-budget ELA and the mean
altitude of the glacier. Kurowski himself did not assume constant balance
gradient casually but discussed available evidence (Kurowski, 1891, pp. 126–130), including application of an early version of the degree-day model,
to justify a nearly constant balance gradient. Remarkably, Kurowski (1891,
p. 127) suggested a value of 0.0056 m w.e. m-1 for vertical balance
gradient, which is not greatly out of line with modern results for Alpine
glaciers (Rabatel et al., 2005). He also tested a balance gradient
proportional to the square root of altitude (p. 130) and suggested that it
does not greatly affect the calculated ELA because of the relatively small
proportions of glacier area at the lowest and highest elevations. Osmaston (2005) appears
to misunderstand this as he says the “AA method” (his name for Kurowski's method) is based “on the principle of weighting the mass
balance in areas far above or below the ELA by more than in those close to
it”.
Kurowski (1891) presents his main results in Table III (pp. 142–147) of
his paper. The data consist of measured areas for altitude bands of 150 m
height from 1050 to 4200 m a.s.l. for 72 glaciers and 27 snow patches
(German: Schneefleck) in the Finsteraarhorn Group, Switzerland. The work involved
planimetric measurements of 744 individual area-elements, covering a
total glacierized area of 461.19 km2. The smallest snow patch was 0.04 km2 and the
largest glacier was 115.1 km2 (Gr. Aletschgletscher). Unfortunately,
there is no map showing delineations of separate glacial elements, and we
would have to guess which areas were included for which glaciers if we
wanted to replicate Kurowski's work (this is beyond the scope of the present
paper). According to the WGMS website (http://www.wgms.ch/products_fog/), the area of the presently delineated Gr. Aletschgletscher is
much smaller than given by Kurowski, i.e. only 83.02 km2. This smaller
area reflects: (1) a real reduction in glacier area since Kurowski's time;
(2) possible separation of the object seen by Kurowski into two or more
objects on modern maps, either due to glacier shrinkage or to better map
resolution; (3) possible overestimation of glacier-covered areas at higher
altitudes due to the oblique angle of observation by the 19th century
surveyors. Effects (1) and (2) are well documented for the Alps (Abermann et
al., 2009; Fischer, et al., 2014).
After so much tedious work with the planimeter, Kurowski must have been
frustrated that he had no easy way of verifying/validating his snow line
results. From Kurowski's Table III, I can calculate the average altitude for
all 99 glaciers and snow patches as 2867 m a.s.l. with a standard deviation
of ±181 m a.s.l., and there is a large range between minimum and
maximum altitudes of 2470 and 3211 m a.s.l. for individual glaciers/snow
patches. This variability within a single mountain group is in contrast to
Heim (1885, pp. 18–21), where the snow line in the Central Alps of
Switzerland is represented by the narrow range 2750–2800 m a.s.l., but is
consistent with modern results (Rabatel et al., 2013).
Kurowski (1891, pp. 152–155) discussed the influence of aspect on snow line.
According to him, glaciers with E and NE aspect have low snow line altitude,
glaciers with NW, N, and SW aspect have intermediate altitudes, and glaciers
with SE, S, and W aspect have higher altitudes. Modern studies of the effect
of aspect on glacier altitudes (Evans, 1977, 2006; Rabatel et al., 2013)
broadly confirm the importance of aspect claimed by Kurowski (1891).
The late 19th century work on glacier snow line by Kurowski and other
workers appeared to be so successful that Hess (1904, p. 68) stated simply
that snow line can be determined from maps of glacier regions rather than by
direct observation of snow line in nature.
Mass balance and equilibrium line altitude
For present purposes, the most important development in 20th century
glaciology was the systematic measurement of surface mass balance on
selected glaciers. This involves measuring the mass balance at many points
on the glacier surface using stakes and snow pits, and then averaging the
results over the whole glacier area. The first continuing, multi-year
series was started in 1946 on Storglaciären in northern Sweden (Schytt,
1981) and surface mass-balance measurements have gradually extended to
several hundred glaciers in all parts of the world (Haeberli et al., 2007).
The bulk of these surface mass-balance data, including ELA and AAR data and
various metadata, have been published in the five-yearly series
“Fluctuations of Glaciers” (http://wgms.ch/products_fog) and the less detailed two-yearly
series “Mass Balance Bulletin” (http://wgms.ch/products_gmbb) from the World Glacier
Monitoring Service. Jania and Hagen (1996), Dyurgerov (2002), and Dyurgerov
and Meier (2005) have published some additional data to those reported in
WGMS publications.
I have maintained my own database for surface mass balance since the
mid-1990s, consisting of a large data file compiled from the above sources
and a FORTRAN program to calculate statistics for the longer series
(Braithwaite, 2002, 2009). Updating and correction of data in 2012–2013
involved checking the database against the latest version of the WGMS data
(http://wgms.ch/products_gmbb). I now have surface mass-balance data for
371 glaciers, i.e. with ≥ 1 year of mass-balance data, for the period
1946–2010. This figure is volatile as new data can be expected, and the
database will be updated as necessary. Of these 371 glaciers, there are some
glaciers that do not appear to be in the WGMS database. This includes data
published in the first two volumes of Fluctuations and Glaciers (in hard copy) that were never
transferred to WGMS's digital database.
As mass-balance data became available from an increasing number of glaciers,
several workers (Liestøl, 1967; Hoinkes, 1970; Østrem, 1975;
Braithwaite and Müller, 1980; Young, 1981; Schytt, 1981) established
empirical equations linking the ELAt, in the year t, to the mean
specific balance b¯t in the same year:
ELAt=α+β×b¯t,
where α is the intercept and ß is the slope of the equation. By
definition, the balanced-budget ELA0=α. We can therefore
calculate balanced-budget ELA0 from mass-balance data, using Eq. (7) as a regression equation as long as we have a few years of parallel data
for surface mass balance and ELA to calibrate αandβ. In the absence of long data series, Østrem and Liestøl (1961)
calculated balanced-budget ELA for a number of glaciers using a
balance–altitude curve from a single year of mass-balance observations.
The two-yearly Glacier Mass Balance Bulletin published by WGMS
(http://wgms.ch/products_gmbb) since 1988 lists balanced-budget ELA and AAR
statistics for a steadily increasing number of glaciers, i.e. 29 glaciers in
the 1988–1989 bulletin to 77 glaciers in the 2008–2009 bulletin. The
selection criterion in the WGMS reports (http://wgms.ch/products_gmbb) seems to be N≥ 6 of record.
ELA varies greatly from year to year on any glacier. Figure 2 illustrates ELA
variations on Hintereisferner as an example. This large year-to-year
variation, with a standard deviation of ±129 m for Hintereisferner,
means that at least a few years of ELA measurement are needed to calculate a
reliable mean ELA. The mean ELA for the 55 years of record in Fig. 2 is 3037 m a.s.l. This mean ELA is slightly biased as a climatological index because
it excludes the 3 years (warmest years?) when the ELA was above the
maximum altitude of the glacier.
There is an obvious multi-decadal variation in ELA for Hintereisferner with
a slight downward trend until the late 1970s followed by a rising trend up
to the year 2010, with an increasing number of single years with ELA above
the maximum altitude of the glacier. The mean ELA for the whole record (3037 m a.s.l.) is therefore too high to represent the first 3 decades and too
low to represent the last 3 decades. The mean ELA does not itself say
much about the overall “health” of the glacier over the nearly 6 decades
of record. A more meaningful index is the deviation of ELA from the
balanced-budget ELA, i.e. the ELA needed to keep the glacier (with its
current area distribution) in an overall condition of zero mass balance. The
latter concept is illustrated in Fig. 3 for Hintereisferner where yearly
values of ELA are plotted against mean specific balance.
Year-to-year variations in equilibrium line altitude (ELA) at
Hintereisferner, Austrian Alps, as measured in a surface mass-balance
programme for 1953–2010. Gaps in the record after 2000 refer to years where
ELA was above the maximum altitude of the glacier.
Equilibrium line altitude (ELA) plotted against mean specific mass
balance of Hintereisferner, Austrian Alps. Dashed lines denote 95 %
confidence interval around the regression line according to Student's
t-statistic.
Figure 3 shows a strong negative correlation between ELA and mass balance for
Hintereisferner (correlation coefficient r=-0.93 with sample size 55).
The ELA–balance relation in this case is represented by the regression line,
whose reliability is expressed by the 95 % confidence interval. The
balanced-budget ELA is 2923 m a.s.l. where the regression line coincides
with zero mass balance, and the associated 95 % confidence interval has a
width of ± 17 m for zero mass balance. From Figs. 2 and 3, the observed
ELA for 1953–1980 is often lower than the balanced-budget ELA while it is
never lower after 1980, suggesting that the present altitude–area
distribution of Hintereisferner is increasingly out of equilibrium with
climate.
Values of the various altitude concepts for Hintereisferner, discussed in
Sect. 2 or above, are summarized in Table 1, clearly showing that they are
clustered near the middle reaches of the glacier, i.e. around 3050 m a.s.l.,
while balanced-budget ELA is somewhat lower. The clustering of the
topographic parameters will occur for any other glacier that is somehow “fat
in the middle”, although topographic “anomalies” can occur for other
glaciers.
Summary of glacier altitudes for Hintereisferner, Austrian Alps,
based on area–altitude data for the year 2001.
ConceptSymbolAltitude(m a.s.l.)Mid-range altitudeHmid3064Median (50 %) altitudeH503056Kurowski mean altitude (area-weighted mean)Hmean3038Mean ELA for 1953–2010ELA‾3037Balance-budget ELA (intercept in ELA–balance regression equation)ELA02923
Of the total of 371 glaciers in my database with ≥ 1 year of mass-balance data, there are 137 glaciers (37 % of total) with no ELA data (N= 0), either because ELA measurements are not part of the observation
programme or because ELA was above the glacier (ELA ≥ hmax) for
the whole period of record. There are a further 84 glaciers (23 % of
total) with less than 5 years of record for both ELA and balance (5 > N≥ 1). This means that data from only 150 glaciers
(40 % of total) are potentially available to calculate balanced-budget ELA
if we regard N≥ 5 as sufficient for calculating reliable statistics
(reduced to 85 glaciers if we use the stricter criterion N≥ 10 years).
For these 150 glaciers with the necessary data, there are generally high
correlations between ELA and mass balance (Fig. 4). For example, there are
143 glaciers with “good correlations” (correlation coefficient r≤-0.71), i.e. where the dependent variable “explains” at least half the
variance of the independent variable. There are, however, seven glaciers
with “poor correlations” (r>-0.7). For a correlation
coefficient approaching zero, the slope of the regression equation will also
approach zero as both slope and correlation coefficient depend on the
covariance of mass balance and ELA. As the slope of the regression equation
approaches zero, the intercept approaches the mean of the ELA. Although low
correlations between mass balance and ELA should cause errors, I did not
exclude results for these seven glaciers from further analysis because I wanted
to see their possible effects on final results (discussed in Sect. 6).
Rea (2009) calculated balanced-budget ELA for 66 glaciers but only includes
glaciers with at least 7 years of record (N≥ 7) up to 2003, and
excludes very small glaciers (< 1 km2). The agreements between
my estimates of balanced-budget ELA and his are very close for the 66 glaciers common to both studies, i.e. with mean and standard deviation of
+3 m and ±25 m for the differences between the two studies.
Glacial geomorphologists like to claim that their single-glacier results
“represent” conditions in a wide region around the measured glacier, i.e.
the result is similar to what the results would be from other glaciers if
they were measured. The question of spatial representativeness of the
surface mass-balance data considered here is beyond the scope of the present
paper, e.g. see Gardner et al. (2013), but it is also important to note that
the available mass-balance data include relatively few glaciers with heavy
debris cover or with tongues calving into lakes or oceans. We may also
doubt whether anybody chooses to measure the surface mass balance of a
glacier fed by frequent avalanches onto the accumulation area, so the
available data are biased against this type of glacier. The available data
cannot therefore be completely representative of conditions in the real
world where debris cover, calving, and substantial accumulation by snow
avalanching are common, especially in the high mountain environments of Benn
and Lehmkuhl (2000).
Balanced-budget ELA and Kurowski mean altitude
Liestøl (1967) calculated balanced-budget ELA by regression of ELA on
measured mass balance and compared it with mean altitude for one glacier
(Storbreen, Norway), and Braithwaite and Müller (1980) did the same for
32 glaciers in different parts of the world.
Histogram showing number of glaciers vs. correlation coefficients
between equilibrium line altitude (ELA) and mean specific balance. The bold line
denotes a Gaussian curve with the same mean and standard deviation as the plotted
data.
According to Sect. 5, balanced-budget ELAs are available for 150 glaciers in the 371-glacier data set and the Kurowski mean altitude should
be estimated for as many of these glaciers as possible. Detailed
area–altitude data were identified in the published metadata for 148 out
of the 371 glaciers. For most of these glaciers, area–altitude tables are
given for every year of record (together with mass balance as a function of
altitude) and area–altitude data for 2001 were selected, if available,
for the calculation of Kurowski mean altitude. Otherwise, data for the year
closest to 2001 were selected. For a few glaciers, the area–altitude
distribution is very out of date but nothing better is available. For
Hintereisferner, the Kurowski mean altitude varies from 3010 m a.s.l. in
1965 to 3038 m a.s.l. in 2001 so errors of several decametres can occur if
there is a large time difference between area–altitude and mass-balance
data.
When combining the data sets for ELA0 (150 out of 371 glaciers) and
Kurowski mean altitude Hmean (148 out of 371 glaciers), it was found
that many glaciers had one kind of data and not the other kind, so there are
in total only 103 glaciers with data for both ELA0 and Hmean, Of
these 103 glaciers there are now only three with poor correlations (r>-0.71).
Available surface mass-balance data for the present analysis from
WGMS (www.wgms.ch/products_fog.html) and some other sources.
No.Name of variableGlaciers% of totalwith data1≥1 year of mass-balance measurements up to year 2010371100 %2≥1 year of ELA measurements up to year 201023463 %3≥5 years of mass balance and ELA measurements up to year 201015040 %4Hypsographic (area–altitude) data for ≥ 1 year allowing calculation of Kurowski mean altitude14840 %5Combining cases (3) and (4)10328 %
The data availability is summarized in Table 2. It is sad to see how easily
371 glaciers with some mass-balance data have been reduced to only 103 glaciers (28 % of total) with all the information that we need for the
present study. There is little that can be done about the shortness of most
surface mass-balance series as such work is generally not well funded or
resourced with the honourable exceptions of some studies in the Alps and in
Scandinavia. However, the lack of published area–altitude data for some
glaciers is less excusable as such data are almost certainly available to
the data collectors. I hope that my paper will encourage workers to publish
their missing area–altitude data although third parties could probably
obtain this data using available glacier outlines from satellite images and
digital elevation models. With more area–altitude data, the number of
glaciers in the study could be increased to 40 % of the total. Even the
single digits for “primary classification” and “frontal characteristics” are
not available for all observed glaciers (http://wgms.ch/products_fog).
The most obvious way of comparing balanced-budget ELA and Kurowski mean
altitude is to plot an X–Y scatter graph, and Fig. 5 shows the extremely
high correlation between the two variables. The 95 % confidence interval
is not plotted here because it is too close to the regression line to make a
neat figure. This high correlation is by no means “spurious” (Leonard and
Fountain, 2003) but it is not very useful because the scale of variations of
the dependent and independent variables is so large compared with
differences between the variables. Plotting balanced-budget ELA against
other topographic variables also shows extremely high correlations. In an
attempt to find a more meaningful correlation, I follow Leonard and Fountain (2003) and Braithwaite and Raper (2009) and “normalize” both variables with
respect to the altitude range of the glaciers before re-plotting (Fig. 6).
The normalization involves subtraction of Hmin from each variable and
then division by (Hmax-Hmin). The correlations between balanced-budget
ELA0 and Kurowski mean altitude in normalized form (Fig. 6) is lower than
in Fig. 5 but is still high enough to show a satisfactory agreement (r=+0.83 for 103 glaciers) between the two variables. The regression line in
Fig. 6, with its 95 % confidence interval, is slightly lower than the
1:1 line expected for ELA0=Hmean. However, Cox (2004) points out that
plots like Fig. 6 may also be misleading because it is the absolute
difference (in metres) between the balanced-budget ELA and Kurowski mean
altitude that we wish to see. It is convenient to define a new variable:
Emean=ELA0-Hmean,
where Emean is the error in estimating ELA0 from the Kurowski
mean altitude Hmean. Similar errors could be defined for other ways of
estimating ELA0, e.g. the error E50 using H50 or Emid
using Hmid but this is beyond the scope of the present paper.
The error ELAmean for each glacier has little relation to the
correlation between ELA and balance referred to in Sect. 5, thus
justifying the inclusion of several glaciers with poor ELA–balance
correlations. The three glaciers with poor ELA–balance correlations (r>-0.71) have small values for error Hmean, and the glaciers
with largest (+ive) difference (Goldbegkees) and smallest (-ive)
difference (Bench) both have good ELA–balance correlations.
Balanced-budget ELA vs. Kurowski's mean altitude for 103 glaciers.
Balanced-budget ELA vs. Kurowski's mean altitude for 103 glaciers
with both variables normalized to the altitude range of the glaciers. The
normalization of each variable involves subtraction of Hmin and division
by (Hmax–Hmin). Dashed lines denote 95 % confidence interval around
the regression line according to Student's t-statistic.
Histogram of errors between balanced-budget ELA and Kurowski mean
altitude for 103 glaciers. The bold line denotes a Gaussian curve with the same mean
and standard deviation as the plotted data.
The error Emean is plotted in the histogram in Fig. 7. Differences of
between +212 and -195 m occur but overall the differences have mean - 36 m
and standard deviation ±56 m, indicating general agreement within a
few decametres. The distribution is somewhat skewed with more negative
values than positive values so that extremely negative values in Fig. 7 are
perhaps not so noteworthy. The very high positive value in Fig. 7 (for
Goldbergkees in the Austrian Alps) is isolated and can therefore be regarded
as an “anomaly”. Braithwaite and Müller (1980) found a mean and standard
deviation of -40 ± 40 m for the differences for 32 glaciers, not
including Goldbergkees as there were then no data from that glacier, which
is not very different from present results.
In his review of my discussion paper Rabatel (2015) raises the question of
the performance of the Kurowski model in different parts of the world, e.g.
one might expect higher errors on glaciers at high latitudes where
superimposed ice may be expected, on Himalayan glaciers, and on glaciers in
tropical South America. The data for the 103 glaciers were resampled into
seven subsets roughly representing different regions (Table 3). The high latitudes
data set (eight glaciers) are from islands in the North American and Eurasian
Arctic plus McCall Glacier in the Brooks Range, where one would expect
significant superimposed ice. The Asia data set (18 glaciers) includes glaciers
from various ranges, including the Caucasus, but with no data from the
Himalaya because there is no glacier with the necessary 5 years of
ELA–balance to be included in this study. The tropics data set (5) includes four
glaciers from tropical South America and one from east Africa. The overall
pattern in Table 3 is for a low range of variations between groups and
within groups, indicating similar performance of the Kurowski mean altitude
for the different region. The Scandinavia group has smallest errors for both
mean and standard deviation, indicating the region with best performance of
the Kurowski model.
One might expect the Kurowski mean altitude Hmean to perform
differently for glaciers of differing morphology. This is tested with the
box plot in Fig. 8 where mean and 95 % confidence intervals for the error
Emean are plotted against primary classification of the glaciers using
metadata from the World Glacier Monitoring website (http://wgms.ch/products_fog/). According to the definitions in TTS (1977) the
digits and their definitions are as follows: 3 Ice cap represents dome-shaped ice mass with radial flow; 4 Outlet glacier drains an
ice-field or ice cap, usually of valley glacier form – the catchment area may not be clearly delineated;
5 Valley glacier flows down a valley – the catchment area is in most cases well
defined; and 6 Mountain glacier represents any shape, sometimes similar to a valley glacier but much smaller, frequently located in a cirque or niche.
Mean and standard deviations of Kurowski error Emean for
glaciers in various regions.
GlaciersMeanSD(m a.s.l.)(m a.s.l.)High latitudes8-52±41Mainland N. America19-63±62Scandinavia29-16±29Alps22-16±71Asia18-54±45Tropics5-54±86Other2––Full data set103-36±56
Box plot of mean balanced-budget ELA minus Kurowski mean altitude
(ELA0–Hmean) vs. primary classification of glaciers. Error
bars represent 95 % confidence intervals of the means according to
Student's t-statistic. Number of glaciers in each group are given in the
lower part of the diagram.
Primary classification is missing for four out of the 103 glaciers. It is
difficult to draw any conclusions for ice caps as there are only four cases
and the confidence interval is very large (and unreliable). For the other
morphologies, it is clear that the Kurowski mean altitude significantly
overestimates (at 95 % level) the balanced-budget ELA0 for outlet
glaciers (mean and standard deviation of -40 and ±42 m for 19 glaciers) and for valley glaciers (-50 ± 52 m for 42 glaciers).
However, for mountain glaciers the overestimation is insignificant with a
mean and standard deviation of -8 ± 59 m for 34 glaciers.
Errors found here for Kurowski's mean altitude may be tolerable for some
applications, e.g. reconstructions of temperature and precipitation from
traces of former glaciers (Hughes and Braithwaite, 2008). In this case, one
could simply calculate the Kurowski mean altitude for a reconstruction of
the former glacier's topography and then apply the appropriate “correction”
according to the primary classification of the glacier. For a “standard”
vertical lapse rate of temperature (-0.006 K m-1) and an error of
±50 m a.s.l. in estimated ELA, the resulting error in estimating
summer mean temperature would only be of the order ±0.3 K. This is
fairly small compared with the uncertainties in the relation between
accumulation and summer mean temperature at the ELA reported by several
workers: see Braithwaite (2008) for references to such studies going back to
1924.
Discussion
If we return to Kurowski's theoretical treatment, his only real assumption
is that balance gradient is constant over the whole glacier. It has long
been supposed that this is not exactly correct (Hess, 1904; Reid, 1908;
Lliboutry, 1974; Braithwaite and Müller, 1980; Kuhn, 1984; Furbish and
Andrews, 1984; Kaser, 2001) although Kurowski (1891) assessed the possible
error as small. Osmaston (2005) and Rea (2009) extend the Kurowski method to
account for different balance gradients but do not assess the error in the
original Kurowski mean altitude.
In the original theory, the vertical gradient of mass balance is constant
over the whole glacier:
dbdhglacier=Constant.
In a recent modification of the theory (Osmaston, 2005; Rea, 2009) balance
gradients are different for ablation and accumulation areas, as expressed by
the balance ratio (BR), where
BR=db/dhabldb/dhacc.
According to Rea (2009), balance ratio greater than unity would lower the
theoretical ELA, i.e. make the error Emean negative, and balance ratio
less than unity would make the error positive. For the present data set, the
error is negative for 84 glaciers (82 % of 103 glaciers) and positive for
19 glaciers (18 %), suggesting that balance ratio BR is commonly greater
than unity but not always. Rea (2009) calculates balance ratio for 66 glaciers using published data for observed surface mass balance vs.
altitude, and I can compare his balance ratios with the Kurowski error
Emean. There is a strong correlation (Fig. 9) between Rea's balance
ratio and the Kurowski error, i.e. r=-0.83. The very high balance ratio
(BR > 5) in Fig. 9 for Zongo Glacier is an obvious anomaly
although there may be good grounds to expect a reasonably large BR for
tropical glaciers like this one (Kaser, 2001; Sicart et al., 2011; Rabatel et al., 2012), and the regression line in Fig. 9 does suggest a BR value a little
greater than 3 for Zongo. I took the BR value for Zongo from Table 3 in Rea's
paper but in a footnote to the table referring to this point and others he
notes that it “indicates a glacier where either, or both, the net balance accumulation or ablation gradient
is not approximated by a linear relationship. AABRs for these glaciers should be treated with caution.
These glaciers were not used to calculate the global AABR”. Soruco et al. (2009) report
a significant revision of data from
Zongo Glacier but this would have been too late for the Rea (2009) analysis.
Figure 9 validates the reluctance of Rea (2009) to use Zongo data in his
global AABR.
This strong correlation in Fig. 9 supports the validity of the balance ratio
approach. However, it is clear that the 66 glaciers in Fig. 9 show a lower
proportion of glaciers with positive Kurowski error than the full data set of
103 glaciers. The box plot in Fig. 10 shows means and 95 % confidence
intervals of Rea's balance ratio for different types of glaciers. The solid
dots refer to results from the original data (66 glaciers) of Rea (2009),
while the open circles refer to an “augmented” data set (103 glaciers) where
balance ratios for the 37 excluded glaciers are estimated from the
regression equation in Fig. 9. Leaving aside the unspecified and ice cap
classes for which there are too few data, the plots show higher balance
ratios for outlet glaciers and valley glaciers (not significantly different
from BR = 2 with 95 % confidence), and lower balance ratios (not
significantly different from BR = 1) for mountain glaciers. The increased
sample size using the regression equation has doubled the number of mountain
glaciers from 17 in the original data to 34 and this has reduced the width
of the 95 % confidence interval for mean balance ratio for mountain
glaciers but still does not exclude BR = 1.
Balance ratio (Rea, 2009) plotted against Kurowski error (ELA0–Hmean) for 66 glaciers. Dashed lines denote 95 % confidence
interval around the regression line according to Student's t-statistic.
The pattern in Fig. 10 does not support the global validity of a balance
ratio of much greater than unity, i.e. 1.75 ± 0.71 according to Rea (2009). Rather, balance ratios are generally greater than unity for outlet
glaciers and valley glaciers, consistent with the negative error in equating
balanced-budget ELA0 to Kurowski mean altitude for these glacier
types. For mountain glaciers, balance ratios are closer to unity and the
average error in the Kurowski altitude is correspondingly less.
Balance ratio (Rea, 2009) vs. primary classification of glacier.
Version 1 is for the original data (66 glaciers) and version 2 is for an
augmented data set (103 glaciers) using a regression line in Fig. 9. Error
bars represent 95 % confidence intervals of the means according to
Student's t-statistic.
Outlet and valley glaciers in the present data set have larger altitude
ranges between highest and lowest points, with mean altitude ranges of 960 m
(with standard deviation ±405 m) and 978 m (with standard deviation
±499 m) respectively, compared with mountain glaciers with a mean
altitude range of 570 m (with standard deviation ±249 m). A larger
altitude range might allow enough contrast in balance gradients between
accumulation and ablation zones to significantly lower the
balanced-budget ELA0 while a more restricted altitude range might
not allow such a large contrast in balance gradients, and ELA0 will
therefore be in better agreement with Hmean for mountain glaciers.
Kern and László (2010) relate their “steady-state accumulation–area
ratio” to glacier size but, from present results, I suggest their relation
between AAR0 and glacier size reflects the dependence on primary
classification of the glacier, as I show here for (ELA0–Hmean).
The most likely physical explanation for different balance gradients in
ablation and accumulation areas is the vertical variation in precipitation
and/or accumulation across glaciers (Jarosch et al., 2012). Aside from the
possible expansion of the balance ratio data set (Rea, 2009) to include small
glaciers, some further insights into balance ratios could be gained from
glacier-climate modelling. For example, my group have in the past tuned
mass-balance models in two different ways. Method one (Braithwaite and
Zhang, 1999; Braithwaite et al., 2002) involved varying precipitation to fit
the modelled mass balance to observed mass balance over the whole altitude
range of the glacier. Method two (Raper and Braithwaite, 2006; Braithwaite
and Raper, 2007), involved varying precipitation at the assumed ELA to make
the model mass balance at the ELA equal to zero. In method one the model
gives precipitation across the whole altitude range of the glacier, while
method two only gives model precipitation at the ELA.
For method one, model precipitation increases with elevation for some
glaciers, e.g. see Fig. 2 in Braithwaite et al. (2002), but not for others.
For method two, modelled balance gradients are consistently lower in the
accumulation zone compared with the ablation zone (Raper and Braithwaite,
2006, Fig. 2; Braithwaite and Raper, 2007, Fig. 5). Results from method one
are consistent with a range of values for balance ratio, while method two
indicates higher values of balance ratio, presumably reflecting the fact
that our mass-balance model uses a higher degree-day factor for melting ice
than for melting snow.
On real-world glaciers, precipitation may increase due to orographic or
topographic channelling effects, or the “effective” precipitation at the
glacier surface may be augmented by snow drifting or avalanching from
surrounding topography. These effects are probably more likely to be
important for mountain glaciers that are more constrained by topography than
for outlet and valley glaciers. For example, two mountain glaciers in the
Polar Ural (IGAN and Obrucheva) have excellent agreement between
balanced-budget ELA0 and Kurowski mean altitude and are known to depend
upon topographic augmentation of precipitation (Voloshina, 1988).
The above discussion of modelling results cannot be definitive but it
suggests that earlier degree-day modelling work with method one
(Braithwaite et al., 2002) ought be repeated and expanded with more explicit
emphasis on precipitation variations and balance ratios. Without further
progress and insights, we must be satisfied with present results that
balanced-budget ELA can be approximated by Kurowski mean altitude with a
mean error of only a few decametres.
Kurowski (1891) is a good example of a glacier-centred approach to snow line,
avoiding problematic discussions of climatic and orographic snow lines as
proposed by Ratzel (1886). Hess (1904, p. 68) suggests that glacier-based
snow line refers to climatic snow line but most glaciers are influenced to
some degree by local topography so balanced-budget ELAs generally have the
nature of orographic rather than climatic snow line. Some glaciers, e.g.
many of the mountain glaciers in the present study, may be more affected by
local precipitation variations than most of the outlet and valley glaciers
in the present study. The distinction between two types of ELA, i.e. TP-ELA
where ELA depends on temperature and precipitation conditions, and TPW-ELA
where ELA depends on additional effects of wind-transported precipitation,
proposed by Bakke and Nesje (2011), might be relevant here. I have no space
to discuss their arguments in detail but Bakke and Nesje (2011) believe that
wind-transported snow lowers ELA on “cirque glaciers” compared with
“plateau glaciers” under otherwise similar conditions.
Conclusions
The estimation of balanced-budget ELA by the mean altitude of a glacier,
suggested by Kurowski (1891), has been widely misquoted in the literature
but not properly tested. There is a high correlation between balanced-budget
ELA and Kurowski mean altitude for the 103 glaciers for which the necessary
data are available, with a small mean difference of -36 m between the two
altitudes with standard deviation ±56 m. Balanced-budget ELA is
significantly lower (at 95 confidence level) than Kurowski mean altitude for
outlet and valley glaciers, and not significantly lower for mountain
glaciers. The agreement between balanced-budget ELA and Kurowski mean
altitude is very impressive for a method proposed more than 120 years ago
and now tested against modern mass-balance data.
Acknowledgements
This research benefits from the thousands of persons who have measured
glacier surface mass balance under arduous field conditions and then made
their results available for further research via the World Glacier
Monitoring Service (WGMS). The School of Environment, Education and
Development (SEED) at the University of Manchester, provided me with an office,
IT and the UK's best library facilities via an honorary senior research
fellowship. Hans-Dieter Schwartz, honorary research associate in
glaciology at The Bavarian Academy of Sciences and Humanities, Munich,
tracked down digital copies of some nearly forgotten articles from the
nineteenth century when German was probably the main language of advanced
glacier research. Dr Johannes Seidl, Head of Archives at the University of
Vienna, provided biographical data on Ludwig Kurowski (1866–1912). Etienne
Berthier (the editor) and two referees (Graham Cogley and Antoine Rabatel)
made many helpful comments.
Edited by: E. Berthier
ReferencesAbermann, J., Lambrecht, A., Fischer, A., and Kuhn, M.: Quantifying changes
and trends in glacier area and volume in the Austrian Ötztal Alps (1969–1997–2006),
The Cryosphere, 3, 205–215, 10.5194/tc-3-205-2009, 2009.
Anonymous: Mass-balance terms, J. Glaciol., 52, 3–7, 1969.
Armstrong, T., Robert, B., and Swithinbank, C.: Illustrated glossary of snow
and ice (2nd ed.), Scott Polar Research Institute, Cambridge, 60 pp.,
1973.
Baird, P. D.: The glaciological studies of the Baffin Island Expedition,
1950. Part 1, Methods of nourishment of the Barnes Ice Cap, J. Glaciol., 2, 17–19, 1952.
Bakke, J. and Nesje, A.: Equilibrium-Line Altitude (ELA), in: Encyclopedia of
Snow, Ice and Glaciers, edited by: Singh, V., Singh, P., and Haritashya, U.,
Springer, the Netherlands, 268–277, 2011.
Benn, D. I. and Evans, D. J. A.: Glaciers and glaciation, Abingdon, Hodder
Education, 802 pp., 2010.
Benn, D. I. and Lehmkuhl, F.: Mass balance and equilibrium line altitudes of
glaciers in high-mountain environments, Quatern. Int., 65/66,
15–29, 2000.
Benn, D. I., Owen, L. A., Osmaston, H. A., Seltzer, G. O., Porter, S. C., and
Mark, B.: Reconstruction of equilibrium-line altitudes for tropical and
sub-tropical glaciers, Quatern. Int., 138–139, 8–21, 2005.
Braithwaite, R. J.: Glacier mass balance: the first 50 years of
international monitoring, Prog. Phys. Geogr., 26, 76–95, 2002.
Braithwaite, R. J.: Temperature and precipitation climate at the
equilibrium-line altitude of glaciers expressed by the degree-day factor for
melting snow, J. Glaciol., 54, 437–444, 2008.
Braithwaite, R. J.: After six decades of monitoring glacier mass balance we
still need data but it should be richer data, Ann. Glaciol., 50,
191–197, 2009.Braithwaite, R. J.: From Doktor Kurowski's Schneegrenze to our modern glacier
equilibrium line altitude (ELA), The Cryosphere Discuss., 9, 3165–3204,
10.5194/tcd-9-3165-2015, 2015.
Braithwaite, R. J. and Müller, F.: On the parameterization of glacier
equilibrium line, IAHS Publication 126, Riederalp Workshop 1978 – World
Glacier Inventory, 263–271, 1980.
Braithwaite, R. J. and Raper, S. C. B: Glaciological conditions in seven
contrasting regions estimated with the degree-day model, Ann. Glaciol., 46, 297-302, 2007.Braithwaite, R. J. and Raper, S. C. B:
Estimating equilibrium-line altitude (ELA) from glacier inventory data, Ann. Glaciol., 50, 127–132, 2009.
Braithwaite, R. J. and Zhang, Y.: Modelling changes in glacier mass balance
that may occur as a result of climate changes, Geogr. Ann., 81A, 489–496, 1999.
Braithwaite, R. J., Zhang, Y., and Raper, S. C. B: Temperature sensitivity of
the mass balance of mountain glaciers and ice caps as a climatological
characteristic, Zeitschrift für Gletscherkunde und Glazialgeologie, 38,
35–36, 2002.
Brückner, E.: Die hohen Tauern und Ihre Eisbedeckung, eine orometrische
Studie, Z. Deut. Österreich. Alpenver., 17, 163–187, 1886.
Carrivick, J. L. and Brewer, T. R.: Improving local estimations and regional
trends of glacier equilibrium line altitudes, Geogr. Ann.,
86A,
67–79, 2004.
Chinn, T. J. H.: Glacier fluctuations in the Southern Alps of New
Zealand determined from snowline elevations, Arctic Alpine Res., 27,
187–198, 1995.
Cogley, J. G. and McIntyre, M. S.: Hess altitudes and other morphological
estimators of glacier equilibrium lines, Arct. Antarct. Alp. Res., 35, 482–488,
2003.
Cogley, J. G., Hock, R., Resamussen, L. A., Arendt, A. A.,
Braithwaite, R. J., Jansson, P., Kaser, G., Möller, M., Nicholson, L., and
Zemp, M.: Glossary of Glacier Mass Balance and Related Terms, IHPO-VII
Technical Documents in Hydrology No. 86, IACS Contribution No. 2,
UNESCO-IHP, Paris, 114 pp., 2011.
Cox, N.: Speaking Stata: graphing agreement and disagreement, Stata
J., 4, 329–349, 2004.
Dobhal, D. P.: Median elevation of glaciers. Encyclopedia of Snow, Ice and
Glaciers, edited by: Singh, V., Singh, P., and Haritashya, U., Springer,
the Netherlands, p. 726, 2011.
Dyurgerov, M.: Glacier mass balance and regime: data of measurements and
analysis, Boulder, CO, University of Colorado, Institute of Arctic and Alpine Research, INSTAAR Occasional Paper 55, 2002.
Dyurgerov, M. and Meier, M. F.: Glaciers and the changing Earth system: a
2004 snapshot. Boulder, CO, University of Colorado. Institute of Arctic and Alpine Research, INSTAAR Occasional Paper 58, 2005.
Dyurgerov, M., Meier, M. F., and Bahr, D. B.: A new index of glacier area
change: a tool for glacier monitoring, J. Glaciol., 55,
710–716, 2009.Escher, H.: Die Bestimmung der klimatischen Schneegrenze in den Schweizer Alpen, Geogr. Helv., 25, 35–43, 10.5194/gh-25-35-1970, 1970.
Evans, I. S.: World-wide variations in the direction and concentration of
cirque and glacier aspects. Geogr. Ann., 59A, 151–175, 1977.
Evans, I. S.: Local aspect asymmetry of mountain glaciation: a global survey
of consistency of favoured directions for glacier numbers and altitudes,
Geomorphology, 73, 166–184, 2006.
Fischer, M., Huss, M., Barboux, C., and Hoelzle, M.: The new Swiss Glacier
Inventory SG12010: relevance of using high-resolution source data in areas
dominated by very small glaciers, Arct. Antarct. Alp. Res., 46,
933–945, 2014.
Furbish, D. J. and Andrews, J. T.: The use of hypsometry to indicate
long-term stability and response of valley glaciers to changes in mass
transfer, J. Glaciol., 30, 199–211, 1984.Gafurov, A., Vorogushyn, S., Farinotti, D., Duethmann, D., Merkushkin, A.,
and Merz, B.: Snow-cover reconstruction methodology for mountainous regions
based on historic in situ observations and recent remote sensing data, The
Cryosphere, 9, 451–463, 10.5194/tc-9-451-2015, 2015.
Gardner, A. S., Moholdt, G., Cogley, J. G., Wouters, B., Arendt, A. A., Wahr,
J., Berthier, E., Hock, R., Pfeffer, W. T., Kaser, G., Ligtenberg, S. M., Bolch, T.,
Sharp, M. J., Hagen, J. O., van den Broeke, M. R., and Paul, F.: A reconclided
estimate of glacier contributions to sea level rise: 2003 to 2009, Science
340, 852–857, 2013.
Haeberli, W., Hoelzle, M., Paul, F., and Zemp, M.: Integrated monitoring of
mountain glaciers as key indicators of global climate change: the European
Alps, Ann. Glaciol., 45, 150–160, 2007.
Hawkins, F. F.: Equilibrium-line altitudes and paleoenvironments in the
Merchants Bay area, Baffin Island, N.W.T., Canada, J. Glaciol., 31,
205–213, 1985.
Heim, A.: Handbuch der Gletscherkunde, Stuttgart, Verlag von J. Engelhorn,
560 pp.,
1885.
Hess, H. H.: Die Gletscher, Braunschweig, Friedrich Vieweg, 426 pp., 1904.
Heyman, J.: Paleoglaciation of the Tibetan Plateau and surrounding mountains
based on exposure ages and ELA depression estimates, Quaternary Sci. Rev., 91, 30–41, 2014.
Hoinkes, H.: Methoden und Möglichkeiten von Massen-Haushaltsstudien auf
Gletschern: Ergebnisse der Messreihe Hintereisferner (Ötztaler Alpen)
1953–1968, Zeitschrift für Gletscherkunde und Glazialgeologie VI, 1–2,
37–89, 1970.
Hughes, P. D. and Braithwaite, R. J.: Application of a degree-day model to
reconstruct Pleistocene glacial climates, Quaternary Res., 69,
110–116, 2008.
Ignéczi, A. and Nagy, B.: Determining steady accumulation-area
ratios of outlet glaciers for application of outlets in climate
reconstructions, Quatern. Int., 293, 268–274, 2013.
Jania, J. and Hagen, J. O.: Mass balance of Arctic glaciers. Sosnowiec/Oslo,
International Arctic Science Committee, Working Group on Arctic Glaciology,
IASC Report 5, 1996.
Jarosch, A. H., Anslow, F. S., and Clarke, G. K. C.: High-resolution
precipitation and temperature downscaling for glacier models, Clim. Dynam., 38, 391–409, 2012.
Kaemtz, L. F.: A complete course of meteorology, translated from German by
Walker, C. V., London, Hippolyte Bailliére, 598 pp., 1845.Kaser, G.: Glacier–climate interaction at low latitudes, J. Glaciol., 47, 195–204. 10.3189/172756501781832296, 2001.
Kaser, G. and Osmaston, H.: Tropical glaciers, Cambridge University Press,
Cambridge, 207 pp., 2002.
Kern, Z. and László, P.: Size specific steady-state
accumulation-area ratio: an improvement for equilibrium-line estimation of
small palaeoglaciers, Quaternary Sci. Rev., 29, 2781–2787, 2010.
Klengel, F.: Die Historische Entwickelung des Begriffs der Schneegrenze von
Bouguer bis auf A. v. Humboldt, 1736–1820, Leipzig, Verein der Erdkunde,
87 pp.,
1889.
Kotlyakov, V. M. and Krenke, A. N.: Investigations of the hydrological
conditions of alpine regions by glaciological methods, IAHS Publication No.
138, Symposium of Exeter 1982 – Hydrological Aspects of Alpine and High
Mountain Areas, 31–42, 1982.
Kuhn, M.: Mass budget imbalances as criterion for a climatic classification
of glaciers, Geogr. Ann., 66A, 229–238, 1984.
Kurowski, L.: Die Höhe der Schneegrenze mit besonderer
Berücksichtigung der Finsteraarhorn-Gruppe, Pencks Geographische
Abhandlungen 5, 119–160, 1891.
Leonard, K. C. and Fountain, A.: Map-based methods for estimating glacier
equilibrium-line altitudes, J. Glaciol., 49, 329–336, 2003.
Liestøl, O.: Storbreen glacier in Jotunheimen, Norway, Nor. Polarinst.
Skr., 141, 63 pp., 1967.
Lliboutry, L.: Multivarioate statistical analysis of glacier annual
balances, J. Glaciol., 13, 371–392, 1974.
Manley, G.: The late-glacial climate of north-west England, Geol. J., 2, 188–215, 1959.
Mathieu, R., Chinn, T., and Fitzharris, B.: Detecting the equilibrium-line
altitudes of New Zealand glaciers using ASTER satellite images, New Zealand,
Journal of Geology and Geophysics, 52, 209–222, 2014.
Meierding, T. C.: Late Pleistocene glacial equilibrium-line altitudes in the
Colorado Front Range: a comparison of methods, Quaternary Res., 18,
289–310, 1982.
Mousson, A.: Die Gletscher der Jetztzeit, Zurich, Druck und Verlag Fr.
Schulhess, 216 pp., 1854.
Müller, F.: Present and late Pleistocene equilibrium line altitudes in
the Mt Everest region – an application of the glacier inventory, IAHS
Publication 126, Riederalp Workshop 1978 – World Glacier Inventory, 75–94,
1980.
Nesje, A. and Dahl, O.: Glaciers and environmental change, London, Arnold,
203 pp.,
2000.
Osmaston, H.: Models for the estimation of firnlines of present and
Pleistocene glaciers, in: Peel, R., Chisholm, M., and Haggett, P.,
Processes in Physical and Human Geography, Bristol Essays, London, Heinemann
Education Books, 218–245, 1975.Osmaston, H.: Estimates of glacier equilibrium line altitudes by the Area × Altitude,
the Area × Altitude Balance Ratio and the Area × Altitude Balance Index methods
and their validation, Quatern. Int., 138–139, 22–31, 2005.
Østrem, G.: ERTS data in glaciology – an effort to monitor glacier mass
balance from satellite imagery, J. Glaciol., 15, 403–415, 1975.
Østrem, G. and Liestøl, O.: Glasiologiske undersøkelser i Norge
1963, Norsk Geografiske Tidsskrift, Norwegian Journal of Geography, 18,
282–340, 1961.
Paschinger, V.: Die Schneegrenze in verschiedenen Klimaten, Petermanns
Mitteilungen, 173, 94 pp.,
1912.
Paterson, W. S. B.: The physics of glaciers, 3rd ed., Pergamon, Oxford,
480 pp.,
1994.
Porter, S. C.: Equilibrium-line altitudes of late Quaternary glaciers in the
Southern Alps, New Zealand, Quaternary Res., 5, 27–47, 1975.
Rabatel, A., Dedieu, J. P., and Vincent, C.: Using remote-sensing data to
determine equilibrium-line altitude and mass-balance time series: validation
on three French glaciers, 1994–2002, J. Glaciol., 51, 539–546,
2005.
Rabatel, A., Bermejo, A., Loarte, E., Soruco, A., Gomez, J., Leonardini, G.,
Viincent, C., and Sicart, J. E.: Can the snowline be used as an indicator of the
equilibrium line and mass balance for glaciers in the outer tropics? J. Glaciol., 58, 1027–1036, 2012.Rabatel, A., Letréguilly, A., Dedieu, J.-P., and Eckert, N.: Changes in
glacier equilibrium-line altitude in the western Alps from 1984 to 2010:
evaluation by remote sensing and modeling of the morpho-topographic and
climate controls, The Cryosphere, 7, 1455–1471, 10.5194/tc-7-1455-2013,
2013.
Radok, U.: Climatic background to some glacier fluctuations. IAHS
Publication 126, Riederalp Workshop 1978 – World Glacier Inventory,
295–304, 1980.
Raper, S. C. B. and Braithwaite, R. J.: Low sea level rise projections from
mountain glaciers and ice caps under global warming, Nature, 439,
311–313, 2006.
Ratzel, F.: Die Bestimmung der Schneegrenze, Der Naturforscher 1886, Verlag der H. Laupp'schen Buchhandlung in Tübingen, 12 June
1886.
Rea, B. R.: Defining modern day area-altitude balance ratios (AABRs) and
their use in glacier-climate reconstructions, Quaternary Sci. Rev.,
28, 237–248, 2009.
Reid, H. F.: A proof of Kurowski's rule for determining the height of the
neve-line on glaciers. Zeitschrift für Gletscherkunde, für
Eiszeitforschung und Geschichte des Klimas 3 (1908/1909), 2, 142–144, 1908.
Schytt, V.: The net mass balance of Storglaciäeren, Kebnekaise, Sweden,
related to the height of the equilibrium line and to the height of the 500
mb surface. Geogr. Ann., 63A, 219–223, 1981.Sicart, J. E., Hock, R., Ribstein, P., Litt, M., and Ramirez, E.:
Analysis of seasonal variations in mass balance and meltwater discharge of
the tropical Zongo Glacier by application of a distributed energy balance
model, J. Geophys. Res., 116, D13105. 10.1029/2010JD015105, 2011.
Sissons, J. B.: A late-glacial ice cap in the Central Grampians, Scotland,
T. I. Brit. Geogr., 62, 95–114, 1974.
Soruco, A., Vincent, C., Francou, B., Ribstein, P., Berger, T., Sicart, J. E.,
Wagnon, P., Arnaud, Y., Favier, V., and Lejeune, Y.: Mass balance of Glaciar Zongo,
Bolivia, between 1956 and 2006, using glaciological, hydrological and
geodetic methods, Ann. Glaciol., 50, 1–8, 2009.
Sutherland, D. G.: Modern glacier characteristics as a basis for inferring
former climates with particular reference to the Loch Lomond stadial,
Quaterary Sci. Rev., 3, 291–309, 1984.
Tang, Z., Wang, J., Li, H., Liang, J., Li, C., and Wang, X.: Extraction and
assessment of snow line altitude over the Tibetan plateau using MODIS
fractional snow cover data (2001 to 2013), J. Remote Sens., 8, 084689, 1–13,
2014.
TTS: Instructions for the compilation and assemblage of data for a world
glacier inventory, compiled by: Müller, F., Caflisch, T. A., and
Müller, G., Temporary Technical Secretariat (TTS) for the World Glacier
Inventory, Zürich, ETH Zürich, 29 pp., 1977.
Voloshina, A. P.: Some results of glacier mass research on the glaciers of
the Polar Urals, Polar Geography and Geology, 12, 200–211, 1988.
Young, G. J.: The mass balance of Peyto Glacier, Alberta, Canada, 1965 to
1978, Arctic Alpine Res., 13, 307–318, 1981.