Introduction

Glaciers and ice caps, exclusive of the Greenland and Antarctic ice sheets, contribute substantially to sea-level rise (Reference MeierMeier and others, 2007; Reference Bahr, Dyurgerov and MeierBahr and others, 2009). In recent decades, all around the world, these glaciers and ice caps (referred to as ‘glaciers’ in this paper) have rapidly disintegrated (Reference Oerlemans, Dyurgerov and van de WalOerlemans and others, 2007). These changes have not been homogeneous in time and space; growth and shrinkage have been observed simultaneously in different regions and at different time periods, making regional and global estimates of glacier change complicated. Furthermore, the large number of glaciers and their generally inaccessible locations require satellite or airborne remote sensing to monitor changes on a regional or global basis (Reference Raup, Racoviteanu, Khalsa, Helm, Armstrong and ArnaudRaup and others, 2007). This paper focuses on developing a simple and widely applicable measure of glacier changes that will be especially useful for observations over large areas using remote sensing.

Previous observations show that the accumulation–area ratio (AAR = S _c/S, where S _c is accumulation area and S is the total glacier area) changes in concert with annual (or net) glacier mass balance B (Reference Dyurgerov and MeierDyurgerov and Meier, 2005). This relationship suggests that the change in glacier accumulation areas can be quantified in relation to mass-balance changes and, to a certain extent, climate change, due to a statistically significant inverse relationship between B and annual global mean air temperature (Reference OerlemansOerlemans, 2001). This relationship has a physical background, as mass balance is generally a function of altitude, and the equilibrium-line altitude (ELA) separates the accumulation area (above) from the rest of the glacier (below). Air temperature and thus ice- and snowmelt rates generally decrease, and snow accumulation rates generally increase, with altitude. Thus ELA, AAR and B are related (Reference Shumsky and KotlyakovShumsky, 1947; Reference Meier and PostMeier and Post, 1962). We use this relationship to develop a single index for monitoring glacier changes.

Concept

The index is defined as the difference between two states of AAR for a glacier. One state is determined by averaging the observed time-varying AAR _i (where i is the year). We write this time average as 〈AAR〉. The second is a glacier’s hypothetical equilibrium AAR₀ which is the accumulation–area ratio of the glacier if and when it is in mass-balance equilibrium. In particular, the equilibrium state, AAR₀, is the value of AAR at B = 0 in a regression of AAR _i against B_i . Note that when making these averages and regressions for different glaciers, the periods of observations (ranges of i) will vary; this is a serious deficiency in mass-balance observations. It is impossible to compile a large sample of glaciers with both AAR data and simultaneous B data that all fall within the same range of years. Even so, the relationship between all mass-balance and AAR glacier time series is significant, with R ² = 0.55 (Fig. 1; data presented in Table 1). The α _d = 〈AAR〉 – AAR₀)/AAR₀ measures a glacier’s displacement from equilibrium and is a relatively simple index for monitoring glacier change that directly relates changes in mass balance B, ELA and accumulation area size S _c. Note that a_d represents an undelayed response to change in water and energy-balance components, whereas the change of the entire glacier area S, which occurs mainly below the ELA, is a delayed adjustment to mass-balance and climate change. Positive values of α _d suggest that a glacier’s accumulation area, and probably also its total area, have been expanding over the period of observations. Negative values of α _d suggest that a glacier has been or will be in retreat, thus probably decreasing S _c and S. Note that in general, S _c has been decreasing in recent years (e.g. Reference Haeberli, Hoelzle, Paul and ZempHaeberli and others, 2007). Thus, glaciers have an increasing deficit of accumulation area, closely related to climate warming (Reference Dyurgerov, Oerlemans and Tijm-ReijmerDyurgerov, 2007).

Fig. 1. The relationship between average mass balance 〈B〉 and average accumulation–area ratio 〈AAR〉 (R² = 0.55). Each point on the regression is the time average for one of the ninety-nine glaciers listed in Table 1, with time series from 5 to 45 years covering the period 1961–2004. Bars indicate the standard errors. The largest outlier is Ivory Glacier, New Zealand.

Table 1. Glacier data used in this analysis. 〈B〉 indicates an average over time and is measured in mm w.e.; 〈AAR〉 indicates an average over time and is measured in %; sterr is standard error = std dev. × N ^−1/2; years is the number of years of record; AAR₀ and α _d (%) are defined in the text

Table 1. Continued

Equivalently, the index α _d could be written as a ratio α _r = 〈AAR〉/AAR₀ (Reference Bahr, Dyurgerov and MeierBahr and others, 2009). These two formulations are linearly related and can therefore be used interchangeably. In some applications, the ratio α _r gives a more convenient measure of the percentage by which a glacier is out of equilibrium, and Reference Bahr, Dyurgerov and MeierBahr and others (2009) use this metric in a scaling analysis to estimate possible future changes in glacier volume and consequent sea-level rise. In other situations, the difference represented by α _d gives a more convenient measure of the actual change in AAR necessary for a glacier to reach equilibrium. The choice of α _d or α _r will depend on the application, and, because the formulations are otherwise equivalent, we reference only α _d in the remainder of this paper.

In order to use α _d (or α _r) as a quantitative measure of the observed state of glaciers relative to their equilibrium state, each glacier’s AAR and AAR₀ must be accurately observed or otherwise precisely determined. The major difficulty with existing AAR data is that the time series are incomplete, and no information on data accuracy has been reported in many publications. In the next section, we compile and analyze published AAR data with these constraints in mind.

Data Sources

The results of this study are based on available time series of B_i , S_i , S _ci and ELA _i for 99 glaciers from around the world during the time period 1961–2004 (Table 1). These data are mostly derived from Fluctuations of glaciers (e.g. Reference Haeberli, Zemp, Frauenfelder, Hoelzle and KääbWGMS, 2005) and the Glacier Mass Balance Bulletin (e.g. Reference Haeberli, Hoelzle and ZempWGMS, 2007), available at the website http://www.wgms.ch. The continuous time series of the above variables were also compiled in two reports (Reference DyurgerovDyurgerov, 2002; Reference Dyurgerov and MeierDyurgerov and Meier, 2005). AAR values are usually derived from glacier hypsometry and the mass-balance vs altitude function using ELA data (e.g. Reference Kjøllmoen, Andreassen, Engeset, Elvehøy, Jackson and GiesenKjøllmoen and others, 2006). In all, more than 200 annual time series were examined, and 99 time series with records of 5 years or longer were chosen for analysis.

These data are taken from observations spanning almost 50 years. The glacier surface area S_i changes with time and represents the ‘conventional’ area as defined by Reference Elsberg, Harrison, Echelmeyer and KrimmelElsberg and others (2001). The direct mass-balance method (Reference Mayo, Meier and TangbornMayo and others, 1972), with many modifications (Reference Østrem and BrugmanØstrem and Brugman, 1991), has been used for most observations. Some changes in methods and observing teams have occurred since observations began, so the time series may not be homogeneous (e.g. Storglaciären in Scandinavia (Reference Holmlund, Jansson and PetterssonHolmlund and others, 2005; R. Hock and others, unpublished information)). Non-homogeneity is a common source of uncertainty, which can only be estimated for a few glaciers.

Outliers were subjected to credibility assessment. The AAR data were sometimes found to be different in various publications (e.g. institutional reports, such as from the Norwegian Water Resources and Energy Directorate (NVE) or United States Geological Survey (USGS), compared to World Glacier Monitoring Service (WGMS) reports). Such data were excluded from the series or corrected by comparisons with the ELA results, where available. For all cases where the ELA is above the glacier’s highest elevation or below its lowest elevation, the AAR results are excluded from further analysis. This is a limitation of the method, especially when applied to a very small glacier where the ELA occasionally varies above and below the glacier (e.g. Glaciar Zongo in the Andes).

Relationship between AAR and Mass Balance

All 99 available time series of AAR _i were regressed against their respective B_i . The relationship between B and AAR is generally strong and linear (Fig. 2a). However, in some cases a linear regression is inappropriate.

Fig. 2. Examples of the relationship between AAR _i and B_i for different glaciers, showing (a) a linear regression (most common)(Storglaciären); (b) a higher-order polynomial regression with a concave-down relationship (Nigardsbreen); (c) a higher-order polynomialregression with a concave-up relationship (Wurtenkees); and (d) no significant relationship (Kara-Batkak). Each point is for a different year i.

Linear regression of AAR vs B. The majority of all regressions show a linear relation (69 of 99 glaciers) (e.g. Fig. 2a).

Non-linear regression of AAR vs B. A minority of the glaciers have non-linear regressions (e.g. Fig. 2b and c). We suggest that some of these non-linear regressions are caused by anomalous increases or decreases in snow accumulation in the upper or lower parts of the glaciers. Non-linearity has also been observed for some glaciers in maritime climates where glacier mass balance and volume change vs elevation are more sensitive to changes in the amount of snow accumulation (e.g. coastal Norway; Iceland). In general, these non-linear regressions can be best summarized as concave-down (e.g. Nigardsbreen in Scandinavia (Fig. 2b)) and concave-up (e.g. Wurtenkees in the European Alps (Fig. 2c)).

Large outliers with no regression. There are several cases with no statistically significant relationships between AAR and B. These may be related to the complicated basin topography and the resulting complex glacier shapes that affect snow-cover distribution. As a result, the spatial distribution of snow accumulation patterns does not show clear dependencies on elevation. Time series may also be short or AAR(B) may cover a limited range of AAR, so the relationship between AAR and B may not be reliable (e.g. Dunagiri and Changme-Khangpu glaciers in the Himalaya, Gråsubreen in Scandinavia, Kara-Batkak glacier in the Tien Shan, and Ivory Glacier in New Zealand (Fig. 2d; data in Table 1)). In these cases, the equilibrium state cannot be defined accurately, and 13 glaciers were removed from further analysis (these are indicated in Table 1 as ‘n.incl.’ or not included). A few glaciers with tongues calving in water (e.g. northwestern part of Devon Ice Cap in the Canadian Arctic, and Austdalsbreen in Scandinavia) show AAR₀ of about 70% or more because part of the ablation is by iceberg calving. AAR and B observations are very sparse for tidewater glaciers and outlets from ice caps, but the application of α _d to these glaciers deserves further investigation. Time series of mass balances and corresponding AAR for the 86 remaining glaciers were subjected to further analysis

AAR₀ has been computed for each glacier by using a linear or higher-order polynomial regression with a least-squares fit. To choose which regression should be used to calculate AAR₀ from the empirical data, the test 3σ ≥ [AAR₀ (linear) – AAR₀ (polynomial)] has been applied, where σ is the standard deviation of observed AAR. Based on this test, a higher-order polynomial regression of AAR₀ was adopted for 11 glaciers. The standard errors of AAR and mass balance are given in Table 1 , where n is the number of time series).

Results

Based on the data analysis and calculations of α _d, several results stand out.

1. 1. The widespread decreases in AAR appear to be due to rapid decreases in S _c relative to slower decreases of S (Fig. 3). Large changes in S _c may have resulted from quite small changes in the ELA because the area distribution (versus elevation) is generally largest above the average ELA. Changes in S occur more slowly as the glacier adjusts dynamically to climate and mass-balance changes.

2. 2. Faster decreases in S _c compared to S suggest that glaciers are committed to further retreat even if the climate were to stop changing. While changes to S _c could cease immediately, the ongoing dynamic response will cause continuing changes in glacier area S. If this committed retreat (α _d = −15.1 ± 12.2%) applies to all glaciers and ice caps, with total area about 763 × 10³ km² (Reference MeierMeier and others, 2007), the total glacier area has to decrease by about 115 × 10³ km². This decrease is apparently under way.

3. 3. α _d is decreasing with time (Fig. 4) due to decreasing globally averaged glacier mass balance (e.g. Reference Kaser, Cogley, Dyurgerov, Meier and OhmuraKaser and others, 2006) and increasing ELA (Reference Dyurgerov and MeierDyurgerov and Meier, 2005).

4. 4. The sample of glaciers available for analysis shows that negative α _d are dominant (Table 1). A majority of glaciers in the sample are far from a steady state and need to shrink to reach equilibrium with the recent climate. The largest negative α _d values are about −66% and −65% for the tropical glaciers Chacaltaya in South America and Lewis in East Africa respectively. These deficits suggest the forthcoming disappearance of many glaciers in the tropics (Reference Hastenrath and GreischarHastenrath and Greischar, 1997; Reference Kaser and OsmastonKaser and Osmaston, 2002). Mid-latitude and polar glaciers show less negative α _d values (Table 1).

5. 5. Some glaciers have a positive α _d, indicating that they need to expand their area by advancing. The maximum value is 49% for Storsteinsfjellbreen in Scandinavia. of 86 glaciers, 13 show positive mass balances (Fig. 5). These include 10 in Scandinavia, Sentinel Glacier in the Canadian Coast Ranges, Filleckkees in the European Alps, and Dyngjujökull, the periodically surging outlet of Vatnajökull in Iceland. Averaged over 86 time series for the entire 1961–2004 period, α _d is −15.1 ± 12.2%, and averaged for 99 glaciers B is −360 ± 42 mm a⁻¹ w.e. (Table 1). These averages, however, are over a long period, when most glacier balances were changing from near-equilibrium toward markedly negative values.

6. 6. The calculated AAR₀ show differences ranging from about 40% up to 80% for individual glaciers, as noted before (Reference Meier and PostMeier and Post, 1962). The most frequent values of AAR₀ in the sample used here are 50–60%, with an average value of 57.9 ± 0.9% (Table 1), the same as determined theoretically for mountain glaciers by Reference BahrBahr (1997).

7. 7. The equilibrium AAR₀ has not changed with time. The results shown in Figure 6 might suggest a slight decrease of AAR₀ with time, but applying the t-criteria between averaged AAR₀ for the periods 1961–75, 1976–90 and 1991–2004 shows that the differences are not significant at the 0.95 significance level. The Hydrologic Encyclopedia volume 1 (Reference Anderson and McDonnellAnderson and others, 2005, p. 387) gives the number for ‘Glacier equilibrium line area’ 0.65, which means AAR balanced. Our calculation gives 58%, which we recommend for use in paleoclimatic studies (e.g. in conjunction with reconstructions from trimlines and moraines).

8. 8. We also calculated AAR_0i using the average of all available AAR _i and B_i for all glaciers for each year i. These annual estimates of AAR_0i (averaged over all glaciers) ranged between 50.3% (in 1992) and 60.6% in 1986).

Fig. 3. The change in total area S and accumulation area S _c summed annually for 22 Northern Hemisphere glaciers during 1966–2001. These glaciers are designated by asterisks in Table 1. The trend lines show that S was decreasing at a rate of −0.21 km² a⁻¹ and S _c was decreasing at a rate of −0.87 km² a⁻¹.

Fig. 4. The change with time of α _d. For each glacier, a value for a_d was calculated for each year (AAR _i were calculated from annual observational data and AAR₀ was estimated from a linear or higher-order polynomial fit to the observed relationship between AAR _i and B_i ). Each point on the plot is the average a_d for all of the glaciers in the given year. The trend is −0.39% a⁻¹.

Fig. 5. The relationship between α _d and average mass balance (B) (R ² = 0.6). Each point represents an average for each of the 86 glaciers that can be assigned a value for α _d (Table 1). Averages are over the period of observation, 1961–2004, though most glaciers do not have data available for each year. Standard error bars are shown.

Fig. 6. The relationship between AAR and B for several different time intervals. Each point is the average AAR and average B for all glaciers in the dataset (Table 1) for a particular year. For 1961–75, the average (over all open circles) is 〈AAR〉 = 54% and 〈B〉 = −113 mm a⁻¹ w.e. (R ² = 0.6). For 1976–90, the average (over all crosses) is 〈AAR〉 = 49% and 〈B〉 = −238 mm a⁻¹ w.e. (R ² = 0.89). For 1991–2004, the average (over all filled circles) is 〈AAR〉 = 40% and 〈B〉 = −491 mm a⁻¹ w.e. (R ² = 0.83). Corresponding to these same time intervals, the average AAR₀ changed from 57.2% to 56.6% to 54.9%. For the entire 1961–2004 period (regression not shown), the average AAR₀ = 57.9%, 〈AAR〉 = 47% and 〈B〉 = −360 mm a⁻¹ w.e.

Conclusions

Changes in glaciers can be monitored with the index α _d. The index may be particularly useful with remotely sensed ELA and/or AAR data (S _c and Scan be determined remotely, with some known limitations) along with standard mass-balance observations. α _d quantifies the extent to which a glacier is out of equilibrium. The index α _d (and equivalently α _r) can be used in applications that need to know the extent to which glaciers are out of equilibrium, as has been done, for example, when calculating glacier contributions to sea level (Reference Bahr, Dyurgerov and MeierBahr and others, 2009). The index also suggests that AAR₀ = 57.9 ± 0.9% is an appropriate AAR for steady-state glaciers and this value should be used in paleo-glacier and paleoclimate reconstructions (instead of the more commonly used 66%).

Using α _d, our data show that the majority of glaciers are far from steady state and need to shrink to reach equilibrium with the recent climate (which can be monitored by the size of S _c). Faster decreases in S _c compared to S reflect the direct and undelayed response of glacier mass balance to climate warming. The delayed dynamic response of S suggests that the glaciers are committed to further retreat, even if the climate stops changing. With a worldwide total of 763 × 10³ km² of glacier area, the committed retreat will cause that total to decrease by 115 × 10³ km² before equilibrium can be re-established. This process is underway. Thus, even if the climate stops changing and does not become warmer, glaciers will continue to retreat.