Introduction

Field measurements of glacier mass change (e.g. Reference KasserKasser, 1967,Reference Kasser1973; Reference MüllerMüller,1977; Reference HaeberliHaeberli,1985; Reference Haeberli and MüllerHaeberli and Müller, 1988; Reference Østrem, Haakensen, Kjöllmoen, Laumann and WoldØstrem and others, 1991; Reference Haeberli and HoelzleHaeberli and Hoelzle, 1993; Reference Haeberli, Hoelzle, Suter and FrauenfelderHaeberli and others, 1998; Reference KrimmelKrimmel, 1999) are used to understand the response of glaciers to climatic variations (e.g. Reference HolmlundHolmlund,1996; Reference Dyurgerov and MeierDyurgerov and Meier, 2000; Reference McCabe, Fountain and DyurgerovMcCabe and others, 2000). It is also important to understand the contribution of glacier melt to regional stream-flow variations (e.g. Reference MeierMeier, 1969; Reference Braithwaite and OlesenBraithwaite and Olesen, 1988) and to global sea-level change (Reference MeierMeier, 1984; Reference Oerlemans and FortuinOerlemans and Fortuin, 1992; Reference Dyurgerov and MeierDyurgerov and Meier, 1997). However, mass-balance studies are time-intensive and can be applied to only a small fraction of the glaciers in any given region. Unfortunately, methods to rapidly assess regional glacier change have been elusive. Monitoring changes in terminus position is an attractive method because glacier termini are clearly visible and easily monitored by aerial and ground-based photographic surveys (Reference GilbertGilbert, 1904; Reference LaChapelleLaChapelle, 1962; Reference VeatchVeatch, 1969; Reference AniyaAniya, 1988) and by satellite imagery (Reference Williams and FerrignoWilliams and Ferrigno, 1993; Reference Duncan, Klein, Masek and IsacksDuncan and others, 1998; Reference PaulPaul, 2002). Inferring climate change or glacier mass change can be difficult because one must separate the dynamic response of the glacier from the changes in glacier mass (Reference NyeNye, 1962). Recent advances in understanding the time-scale of glacier responses to climate change (Reference Jóhannesson, Raymond and WaddingtonJóhannesson and others, 1989) have increased our capability to interpret the lag time between climate variation and terminus response. However, to track climate change through changes in terminus position is still problematic because of kinematic waves (Reference NyeNye, 1965; Reference Van de Wal and OerlemansVan de Wal and Oerlemans, 1995). To apply time-scale response methods also requires estimates of glacier thickness and some knowledge of mass change at the terminus, which are unknown for most glaciers.

One appealing alternative to measuring terminus position is to find the equilibrium-line altitude (ELA). The position of the ELA is controlled entirely by climatic processes (Reference KuhnKuhn, 1981; Reference Ohmura, Kasser and FunkOhmura and others, 1992; Reference SeltzerSeltzer, 1994; Reference Fountain, Lewis and DoranFountain and others, 1999). Up-glacier from this “line”, the accumulation zone gains more mass annually than it loses. Down-glacier, in the ablation zone, the glacier loses more mass than it gains. At the ELA, the annual mass change is zero (Reference PatersonPaterson,1994). The most accurate method to determine the ELA is by field measurements (e.g. Reference Østrem and BrugmanØstrem and Brugman, 1991). One approach is to contour the point measurements of mass balance on a map to determine the glacier’s ELA for that year. Such field methods are slow and expensive and therefore are only applied to a few glaciers. These constraints have led to a variety of remote monitoring techniques to find the position of a glacier’s ELA. Assuming that the snowline at the end of the summer is a good approximation of the ELA, then aerial and ground-based photography are attractive methods to monitor inferred changes in ELA (Reference LaChapelleLaChapelle, 1962). This method is difficult in practice because the timing of the photography is critical. Observations have to be collected just prior to the start of the accumulation season, and early snowfall or late warming events may require repeated imaging. Also, poor weather, common to mountainous regions, can delay image acquisition for weeks.This method cannot easily be applied to environments where accumulation and ablation seasons do not neatly fit into the climatic seasons of winter and summer, respectively. In the dry valleys of Antarctica, for example, snowfall occurs year-round on all surfaces and distinct accumulation/ablation seasons do not exist (Reference Fountain, Dana, Lewis, Vaughn, McKnight and PriscuFountain and others,1998). In the Himalaya, the accumulation and ablation seasons occur simultaneously during the monsoonal season (Reference Ageta and HiguchiAgeta and Higuchi, 1984), and in the equatorial regions multiple seasons of accumulation and ablation exist (Reference KaserKaser,1999).

An alternative method for determining ELA is based on the inflection of elevation contour lines on a topographic map of a glacier (Reference ØstremØstrem, 1966; Reference PorterPorter, 1975). This method was first introduced by Reference HessHess (1904) to determine firn-line locations, and in the European scientific literature it is sometimes referred to as the “Hess method”. The accumulation zone of the glacier is bowl-shaped, because snow accumulating on the margins of the glacier advects toward the center. Below the ELA, in the ablation zone, the contour lines are convex, because mass is lost from all sides and ice is advected away from the glacier center towards the margins. The transition or inflection from a concave to convex contour is a relatively flat surface that should be close to the location of the long-term averageELA (Fig.1). If this is true, then the topography of the glacier surface can be used to infer the average location of the ELA. Because the ELA position determined by the contour method results in part from glacier motion, we term this ELA the “kinematic ELA” to distinguish it from the year-to-year ELA determined by observations of glacier surface mass changes, or “observed ELA”.

Fig. 1. Contour map of South Cascade Glacier, Washington, U.S.A., from the United States Geologic Survey Dome Peak 7.5′ quadrangle. According to the map, the glacier was field-checked in1965.

This paper examines the correspondence between the kinematic and observed ELA. To our knowledge, the “kinematic” (“Hess”) method has not been empirically tested in this way. If these two values correspond, historical maps of glaciers can be used to estimate ELA change since the turn of the 20th century, when reliable topographic maps of alpine regions became available, and modern maps can be used to estimate ELAs for glaciers on which mass-balance studies are not performed.We also evaluate the correlation between ELA and mean glacier elevation. The normalized form of mean glacier elevation, which we use, is similar to the toe–headwall altitude ratio (THAR), a procedure for estimating ELA in the glacial geologic literature (Reference MeierdingMeierding, 1982; Reference HawkinsHawkins, 1985; Reference Torsnes, Rye and NesjeTorsnes and others,1993).

Table 1. The glaciers used in this study

Methods

We collected topographic maps of glaciers for which observed ELAs have been recorded by the World Glacier Monitoring Service (WGMS). The 40 glaciers we examined are listed in Table 1. Our dataset is biased towards glaciers in the Northern Hemisphere, but that should not change our results. We selected the contour that best represented the inflectionbetween the regions of surface concavity (accumulation zone) and convexity (ablation zone). The altitude represented by this contour is defined as the kinematic ELA. In some cases it is difficult to determine the specific contour defining the inflection of the glacier surface because the contour line itself may contain concave and convex segments. In these situations the line that shared the most symmetrical distribution of concavity and convexity was selected. The error of the selection of the contour line is, at minimum, the error of the contour map, typically half of the contour interval. In those cases where several adjacent contour lines seemed suitable, we chose the average. The date of the glacier surface depicted was determined from either the date of photography used to make the map or the date the surface was surveyed. If a “field check” was performed on the glacier, the map’s date was set to that of the fieldwork. Information on the date of aerial photography or field checking is typically included in the map legend.

The highest and lowest elevations of each glacier were recorded from the maps and checked against values reported to theWGMS. We used them to calculate the mean elevation of each glacier (maximum elevation plus minimum elevation divided by two). This is not the area-weighted mean discussed by Reference Braithwaite and MüllerBraithwaite and Müller (1980), rather it corresponds to their “E _CR”. Reference KurowskiKurowski (1891) introduced this method of estimating a glacier’s firn line, and Reference PorterPorter (1975) suggested the mean elevation of a glacier is a good estimate of the ELA, but neither author presents data to support this assertion.

ELA data were obtained from the WGMS reports (Reference KasserKässer, 1967, Reference Kasser1973; Reference MüllerMüller, 1977; Reference HaeberliHaeberli, 1985; Reference Haeberli and MüllerHaeberli and Müller, 1988; Reference Haeberli and HoelzleHaeberli and Hoelzle, 1993; Reference Haeberli, Hoelzle, Suter and FrauenfelderHaeberli and others, 1998). We averaged the observed ELAs reported for the years 1965–95, and compared this average with the kinematic ELA for a date within this time period. We did not calculate steady-state ELAs because most glaciers are not in a steady state.We chose to average the whole record, rather than select a smaller time interval, both because the kinematic ELA results from long-term climate trends and because we did not want abnormally high or low ELAs to bias the average.Typically, the maps were prepared at the beginning of the mass-balance program and precluded averaging the observed ELA record for the years preceding the map date, as might be suggested by response-time theories (Reference Jóhannesson, Raymond and WaddingtonJóhannesson and others,1989).

Results

Table 2 summarizes the results of our kinematic ELA estimates and the observed ELAs and mean elevation.The correlation between the kinematic and observed ELAs is very good (Fig. 2a), with a correlation coefficient, r ², of 0.99 (Table 3). The slope of the least-squares fit is 1.05, only slightly steeper than a one-to-one ratio. The intercept of the equation is +10.8 m, indicating the observed ELA is a bit higher than the kinematic ELA. The relation between mean glacier altitude and observed ELA (Fig. 2b) is similar to Figure 2a, with a slope slightly smaller (0.99) and a larger intercept, 46 m, with the observed ELA higher than the mean altitude. Correlations were also calculated between the observed ELA for only that year in which the map was made, and the kinematic ELA and mean altitude (Fig. 2c and d). Results were similar to those of the previous findings, but with lower correlation coefficients and larger intercepts, as one might expect from the year-to-year variability of the observed ELA relative to the mean ELA.

Fig. 2. Relations between the observed ELA and the kinematic ELA and mean elevation: (a) kinematic ELA vs mean observed ELA; (b) mean glacier elevation vs mean observed ELA; (c) kinematic ELA vs same-year observed ELA; (d) mean glacier elevation vs same-year observed ELA.

Table 2. The data used in this study

Table 3. Regression equations for the studied ELA relationships corresponding to Figures 2 and 3

The correlation between kinematic and observed ELAs should be good, since the two values are constrained to the altitude range of each glacier, while the range itself varies greatly among the glaciers. Our sample is also dominated by relatively small alpine glaciers, on which many mass-balance studies are based. To remove the spurious correlation induced by the wide range of altitudes, we normalized the data by the altitude range of the glacier,

(1)

where E is the normalized value, subscript i indicates either the kinematic (k) or observed values (o), ELA is observed or kinematic ELA, z ₀ is the minimum glacier altitude, and, z _m is the maximum glacier altitude. The normalization limits the altitude of each ELA to a range of 0–1. The mean elevation for all glaciers becomes 0.5. Averages of the normalized data showed that the kinematic ELA was about 20% lower than the observed ELA. An F test on the normalized data showed that the kinematic and observed ELAs were not statistically different. The mean elevation data cannot be analyzed in the same way against the observed ELAs, as this would compare a single point (0.5) with a variable dataset. As can be seen in Figure 3c and d, there is no correlation between the normalized mean elevation and normalized observed ELA.

Fig. 3. The relation between the normalized values of the kinematic and mean observed ELAs and the mean elevation and mean observed ELAs: (a) normalized kinematic ELA vs mean observed ELA with all glaciers included; (b) normalized kinematic vs mean observed ELA for glaciers with >15 years of data; (c) normalized mean elevation vs mean observed ELA for all glaciers; (d) normalized mean elevation vs mean observed ELA for glaciers with >15 years of observed ELA data.

The relation between the normalized observed and kinematic ELA was poor (Fig. 3a; r ² = 0.1). Concerned that short time series of ELAs could bias the relation, we replotted our data after removing all glaciers with observation records of <15 years. The results for these long-record glaciers are much better (Fig. 3b; Table 3: r ² = 0.59).

Glaciers with kinematic waves produce large variations in terminus position that might strongly affect the application of the proposed method. We compared the kinematic ELA with the mean glacier elevation for a series of 12 maps of Nisqually Glacier, Mount Rainier, Washington, dating from 1913 to 1994. Nisqually Glacier is well known for its kinematic waves (Reference MeierMeier, 1962), but its mass balance has never been measured due to the difficulty of the glacier terrain. Our results show that the mean elevation is far more variable than the kinematic ELA (Fig. 4). The change in kinematic ELA is coincident with extensive glacier retreat from 1913 to 1940 (T. Reference NylenNylen, 2003). Since 1940 the ELA has been relatively stable and, relative to the errors in the method, the ELA can be considered constant. We believe the pre-1940 rise in ELA is real and is perhaps an artifact of the Little Ice Age in this region. After 1940, the variation in the mean elevation is large compared to the kinematic ELA and is a result of kinematic waves which temporarily lengthen the glacier. In addition, defining the terminus of such glaciers is problematic because of the large portions of “dead” ice that accompany large advances from the kinematic waves (T. Nylen, unpublished information).

Fig. 4. Estimates of the ELA with time using the mean elevation and topographic methods for Nisqually Glacier which regularly experiences kinematic waves. No kinematic ELA was recorded for the years 1925–45 because the maps from those years only included the ablation area.

Discussion

The correlation between the kinematic and observed ELA is good, but we were initially puzzled by the offset (20% of the total glacier elevation in Fig. 3b) between the normalized kinematic and observed ELAs. Reference HookeHooke (1998) points out that an offset should be expected because glaciers move down slopes. For a steady-state glacier, the net balance (b _n) is directly related to the emergence (vertical) velocity (w _s) at the surface. For a point on a surface with slope α and horizontal velocity u _s,

(2)

At the observed ELA, b _n = 0, and since u _s is positive and the slope is negative, the emergence velocity w _s is negative, or into the glacier. Thus, at the observed ELA, mass is being advected into the glacier and the kinematic ELA of zero emergence velocity is located down-glacier. Alternatively, where w _s is equal to zero (kinematic ELA), the balance is negative, placing the kinematic ELA lower on the glacier than the observed ELA, in the ablation zone, as shown by our results. Another possible cause of this offset is that the map was made before field observations were collected and the ELA has risen in elevation since that time.

The mean elevation of a glacier correlates well with the observed ELA (Fig. 2b and d) when elevations above sea level are used. This is spurious because the relation is biased by the large range in glacier elevations. Moreover, as previously described, the normalized ELA shows great scatter about the normalized mean elevation, fixed at 0.5 (Fig. 3c and d). Normalized mean elevation is similar to the THAR method (Reference MeierdingMeierding, 1982; Reference HawkinsHawkins, 1985; Reference Torsnes, Rye and NesjeTorsnes and others, 1993). The THAR method is used principally in glacial geology to determine a former glacier’s ELA based on valley morphology. The difference in elevation between the former glacier’s terminus, or toe, (usually determined from moraine location) and the elevation of the former glacier’s headwall is multiplied by the THAR to estimate the elevation of the former equilibrium line relative to the terminus. ATHAR of 0.4 means that the ELA is 40% of the glacier’s elevation range higher than the terminus. Both Reference MeierdingMeierding (1982) and Reference Torsnes, Rye and NesjeTorsnes and others (1993) compared the THAR of 0.4 with the accumulation–area ratio (AAR) of 0.6, an equilibrium value for “normal” alpine glaciers (Reference Meier and PostMeier and Post, 1962), and found a good correlation.

For the small mountain glaciers used in this study we do not observe a constant THAR, as the normalized ELA is quite variable from glacier to glacier. This brings into question the validity of the THAR method for small alpine glaciers. However, it may be useful in the broadest of contexts because the average of the normalized observed ELAs is 0.55, close to the normalized value of glacier elevation (THAR) of 0.5.

A strong word of caution: our methods to estimate the ELA from maps or average glacier altitude and THAR apply only to alpine glaciers that terminate “normally” on dry land. Like the application of the AAR, these two methods do not apply to glaciers that terminate in marine or lacustrine waters or to glaciers that terminate on cliffs. In these situations, the relation between mass balance and altitude is strongly biased by calving at the glacier terminus. The calving flux is largely controlled by processes at or near the terminus and, for marine and lacustrine systems, is largely independent of annual climate (Reference Brown, Meier and PostBrown and others, 1982; see papers inReference Van der VeenVan der Veen, 1997).

For “normal” glaciers the kinematic ELA will provide a reasonable approximation of a glacier’s average ELA at far less expense than a traditional mass-balance study. Recent advances in remote-sensing techniques for mapping glacier surfaces (Reference EchelmeyerEchelmeyer and others, 1996; Reference AllenAllen, 1998; Reference Favey, Geiger, Gudmundsson and WehrFavey and others,1999; Reference Kääb and FunkKääb and Funk,1999; Reference HubbardHubbard and others, 2000) will make the kinematic ELA method even more practical.

Conclusions

The “kinematic” method of estimating the ELA, by selecting the topographic contour representing the inflection between the concave and convex surfaces on a glacier, is a reasonable approximation for the actual, long-term, ELA acquired through traditional mass-balance studies. The kinematic ELA occurs lower on the glacier than the observed ELA, consistent with theoretical models of glacier flow and with a climate-driven rise in the position of the ELA during the period of observed ELA data collection. Distortions to the glacier surface by kinematic waves do not seem to affect the results of this method. Application of this method is limited to those “normal” alpine glaciers that do not terminate in cliffs or in water.

The mean elevation of a glacier does not provide a reasonable approximation for the observed ELA but may be useful as the most general of indices. The THAR method, an approach common to glacial geological studies, may be applicable in a broad sense but should not be used with individual glaciers. Our work suggests an appropriateTHAR for modern alpine glaciers (using the highest elevation on each glacier as the headwall altitude) is 0.55, much higher than the value of 0.4 commonly used.

These methods allow the examination of the spatial and temporal variation in ELA starting in the early 20th century when reliable topographic maps of alpine regions became available. With increasing access to airborne and satellite imagery, practical application for constructing and revising topographic maps will increase in time. By applying the kinematic method to these topographic maps, we can remotely infer ELAs for many glaciers over large regions and track changes in ELA through time. This will increase our understanding of variations in glacier activity in relation to climatic variations and local topographic settings.