Introduction
The mass-balance record of White Glacier. Axel Heiberg Island, Northwest Territories, Canada (Fig. 1), now extends over 32 years, with one 3 year gap. Here we document the record, assess its accuracy, consider alternatives to a stake-based measurement programme and assess the value of White Glacier as an index of climatic change. All of the raw data for this study, and further analytical material, will be found in Reference Cogley,, Adams,, Ecclestone,, Jung-Rothenhäusler and Ommanncey.Cogley and others (1995).
Fig. 1. White Glacier. Longitude, latitude marks at left are each 0.02° long; contour interval 100 m. Stakes shown are those of 1987. Inset: location of White Glacier and other places mentioned in test; StPB: St Patrick Bay ice caps.
Setting and History of Measurements
The mean annual precipitation at Eureka, 120 km northeast of White Glacier, is 58 mm. At Isachsen, 280 km to the southwest, it is 117 mm. Reference Koerner,Korner’s (1979) measurements on the eastern and northern sectors of the Müller Ice Cap indicate accumulation of about 160 mm a−1. Reference Ohmura, and Müller,Ohmura and Müller (1977) report a 3 year mean for summer precipitation of more than 60 mm, and an increase with elevation at a rate of 72 mm km−1, for the White Glacier area. Reference Müller, and Müller,F. Müller (1963d) reports a mean annual accumulation, probably not regionally representative, of about 370 mm for a 41 year record of annual layers studied in an ice shaft near the highest point on the Müller Ice Cap, 40 km north of White Glacier. In short, the neighbourhood of White Glacier appears to receive more precipitation than does the nearest weather station at Eureka. The mean annual temperature near sea level at Eureka is −19.7°C.
The ice cover of Axel Heiberg Island is described in detail in Reference Ommanney,Ommanney (1969). White Glacier is a compound valley glacier, about 15 km long and 38.7 km2 in area (Fig. 1). The surface elevation above sea level ranges from 1782 m down to 75 m, with moraine-covered ice to 53 m. The tongue is about 1 km wide. Thicknesses, typically 200 m and up to 400 m, are reported by Reference Becker,Becker (1963), Reference Redpath,Redpath (1965) and (Reference Blatter,Blatter 1985, Reference Blatter.1987). White Glacier is polythermal, with a cold upper shell and, beneath most of the tongue, a temperate sole (Reference Blatter,Blatter, 1985, Reference Blatter.1987). Surface velocities measured a few kilometres up-glacier from the snout (Reference Müller, and Müller,F. Müller, 1963e) are about 20–30 m a−1. The snout is retreating at from 3 ma−1 (1959–62; F. Müler and others, 1963) to 5 ma−1 (1960–70; Reference Arnold,Arnold, 1981). Information in Reference Moisan, and Pollard,Moisan and Pollard (1992) for 1960–88 is consistent with these estimates. Thompson Glacier (area ∼280 km2). contiguous with the snout of White Glacier, is advancing at 20 ma−1.
Mass-balance measurements have been made on While Glacier sinee 1359. Relevant processes have been studied by, among others, Reference Havens,, Müller and Wilmot.Havens and others (1965: meteorology, climatology), Reference Iken,Iken (1974: surface velocities), (Reference Blatter,Blatter 1985, Reference Blatter.1987: thermal regime) and Reference Adams,, Cogley, and Ecclestone,Adams and others (1995: equilibrium-zone processes). (Reference Ommanney, and Adams,Ommanney 1987a, Reference Ommanney, and Adams,b) gives a bibliography of glaciological research in the area.
In 1960, an extensive pattern of stakes was established. (Reference Müller,Müller 1962. Reference Müller, and Müller,1963c), Reference Adams,Adams (1966), Reference Müller, and Keeler,Müller and Keeler (1969). Reference Young. and Müller.Young (1972) and Reference Arnold,Arnold (1981) have exploited the detailed know ledge of the mass balance for the early years 1960–62.
During 1963–79, annual measurements were made on fewer stakes (varying from 23 to 80). These were documented, with annual commentaries, in Reference Weiss,Weiss (1984). Special projects produced additional measurements in some years (e.g. Reference Arnold,Arnold. 1981).
There is a gap in the record following the death of F. Müller, the principal investigator. The missing years, 1980, 1981 and 1982, are addressed below.
In 1983, measurements were resumed by groups from Treni University, Canada, and these have continued. The balance for the first year of resumed measurements is probably more uncertain than later estimates. The number of stakes (Fig. 2) has been about 30 per year or 0.6–0.9 per km2, to be compared with 2–3 per km2 in some earlier years.
Fig. 2. Density of slakes in each stratigraphic-balance year.
Methods Of Measurement
Definitions
We use b to denote the net mass balance at any one stake for any one year. The symbol b(h), on the other hand, denotes a 1 year mass-balance estimate made by averaging all the stake measurements in the small range of elevations between h − ½dh and h + ½dh. The annual mass balance for the whole glacier is denoted by B. The long-term average of Bis represented by 〈B〉 and is called the mass-balance normal. These quantities are quoted as specific balances, in units of mass per unit area per unit time (kg m−2a−1, or mm water a−1). We refer to Strati-graphic years by the calendar year in which they end. Thus, 1960 refers to the mass-balance year 1959–60.
Field methods
The standard approach to mass-balance work, adopted here, is described in Reference Paterson,Paterson (1981) Reference Østrem, and BrugmanØstrem and Brugman (1991). We have relied on the siratigraphic system, but with a modification. Resources allow only a single field visit each year, and it is not possible to estimate separate winter and summer balances. The annual visit is therefore made in spring (late May or early June). This allows us to determine with good accuracy the winter accumulation of the current (Septembcr-to-September) stratigraphic balance year, while in the ablation zone we measure the loss of ice during the previous balance year. The lowering of the ice surface is determined with reference to a mark on a stake (or a thermistor cable) inserted vertically into the glacier. To obtain the stake mass balance b we assume that the lost ice had a density of 900 kg m−3.
On polar glaciers, there may be a more-or-less extensive superimposed-ice zone above the equilibrium line (Reference Adams,, Cogley, and Ecclestone,Adams and others, 1995). Especially in hollows, the initial melt refreezes to form superimposed ice at the base of the winter snowpack. At lower elevations the superimposed ice eventually melts, but higher up some of it persists and contributes to net accumulation. Where there is superimposed ice, net accumulation is measured as the rise of the glacier-ice surface with reference to a mark on a stake (times 0.9 to obtain water-equivalent mass, which introduces an error since the density of the superimposed ice is usually less than 900 kg m−3). Just above the superimposed-ice zone, in the lowest part of the percolation zone, net accumulation is often a combination of snow (measured in pits) and, beneath the snow, “concealed superimposed ice” (measured as a rise of the ice surface against stakes located close to the pits). An added complication is that spatial variations in total accumulation, total melt and formation of superimposed ice produce outliers of accumulation and inliers of ablation.
Above the superimposed-ice zone, pits are dug to identify mass added between the end of one summer and the end of the next. The density of the accumulated volume is determined by identifying distinct layers in the walls of the pit and measuring the density of each layer. The average density is the layer-depth weighted sum of the measured densities. (Before 1983, a complete column of accumulated snow was weighed.)
Calculations
The net mass balance B for the whole glacier is, by definition,
where A is the area covered by the glacier and b ≡ b(x, y, Δt) represents the mass balance at each point (x, y) on the glacier over the lime-span Δt. The area is assumed not to change during Δt. We assume that elevation h is the dominant influence over b, and extrapolate from the stakes only in the vertical direction. Measurements of b are grouped into 100 m high elevation bands, and a function, β(h) say, is fitted to b(h), the set of elevation-band averages of the measured b. This reduces bias from the uneven distribution of stakes and is consistent with the weighting of elevation-band estimates by their areas a(h):
where the range of summation H is from the glacier’s minimum elevation to its maximum elevation.
As shown below (see also appendix B of Reference Cogley,, Adams,, Ecclestone,, Jung-Rothenhäusler and Ommanncey.Clogley and others, 1995), the structure of the relationship between b and h is relatively simple. β(h) is evaluated at the midpoint of every 25 m elevation band, a relatively fine resolution which takes advantage of hypsometric estimates of the area a(h) derived from a digital elevation model (DEM) by a simple binning routine. The DEM (Reference Cogley,Cogley, 1992a; National Research Council of Canada, 1962a, b, 1965) has a horizontal resolution of 50 m. Equation (2) is applied to obtain B.
The polynomial β(h) always fits the data very well (Reference Cogley,, Adams,, Ecclestone,, Jung-Rothenhäusler and Ommanncey.Clogley and others, 1995). Because polynomials diverge, it is risky to extrapolate them. Beyond the elevation range of the data, therefore, β(h) is set to its value at the highest or lowest h at which there is a measurement. The effect of this assumption can be seen in the vertical line segments with which graphs of β(h) (shown below) begin and end.
The ELA h0 and accumulation area ratio (AAR) are Computed by locating the first zero-crossing of the polynomial β(h), which is accepted as an estimate of h0 . AAR is simply the ratio of the area above the zero-crossing to total area. The risk of there being more than one zero-crossing, and more generally of the polynomial predicting negative mass balance at high elevations, has to be checked visually. None of the mass-balance years in the current record exhibits this quirk.
Reassessment
We have detailed stake diagrams and calculations for all years since 1983. All of these have been checked carefully three times, with special attention paid to outliers. Some but not all of the outliers were found to be in error and were corrected or rejected; not all of the detected errors were outliers. We also corrected a few errors in earlier tabulations and transcriptions, including a number of incompatible estimates of stake elevation, which we altered to make the vertical stake trajectories from year to year plausible.
All stake mass balances b are presented by Reference Cogley,, Adams,, Ecclestone,, Jung-Rothenhäusler and Ommanncey.Cogley and others (1995), as are elevation-band averages and standard deviations. Information before 1983 comes from Reference Weiss,Weiss (1984), for 1963–79; from diagrams and tables in (Reference Müller, and Müller,F. Müller 1963a, Reference Müller, and Müller,b) for 1962; and from diagrams in Reference Adams,Adams (1966) for 1960–61, including some averages for stake cross-profiles.
Errors
Measurement errors
Stake mass balances are built up from simple measurements of lengths and weights, which in themselves are rather accurate. Reference Adams,Adams (1966) shows that stake readings are accurate to ±5 mm. in the sense that they are reproducible within this range and are representative for an area of several square metres. Measurements of snow density are less accurate (see below), while some uncertainty must arise from the assumption that the density of glacier ice is 900 kg m−3. If this is an overestimate, as is possible in the superimposed-ice zone, the ensuing calculations will be biased. We cannot assess this possibility, and a bias in b of up to a few tens of millimetres is quite possible for some stakes.
Differences between stake readings are less accurate than single readings, especially if the stake tilts or settles between the two measurement epochs, or was not vertical to start with. Special problems arise in the superimposed-ice zone, where the glacier gains mass in the form of refrozen meltwater. Here differences between stake readings may be misinterpreted because the stake appears m have sunk into the glacier. In the lower percolation zone (Reference Adams,, Cogley, and Ecclestone,Adams and others, 1995), next above, an unknown fraction of the added mass is lost to internal accumulation when it melts and percolates, often via pipes of restricted extent, to depths below the current year’s snow. This effect is very difficult to quantify, but it must lead to an underestimate of accumulation. We have tried to estimate the losses by inserting pans into the snowpack, but abandoned the method because it was time-consuming and gave poor and doubtful results. We have also tried to measure year-over-year changes of density in the snow beneath that of the current year, but sampling errors tend to be very large. Reference Trabant, and Mayo,Trabant and Mayo (1985) deduced internal accumulation by measuring the anomalous heating at depth due to the refreezing meltwater, but appropriate measurements are not available for White Glacier.
In the accumulation zone we obtain two estimates of depth, one from the year-to-year difference in stake readings and the other from identification of the previous year’s surface in a nearby snow pit. The two depths may be assumed to sample randomly the local variability, which generates uncertainty. For ten such pairs in 1990 and 1991 the standard error was 68 mm of snow, or on average about 10% of the average of the two depths. We cannot repeat this exercise for the densities measured in the pit walls, but it seems reasonable to assume an error of the same order; the spring balances used in the field have been checked against a laboratory balance and found to be accurate to better than ±2%, but the error in the sampled volume remains undetermined. Similar errors probably affect the whole-column weights measured before 1983. A ± 10% error in positive values of b will typically be about ±(20–30) mm water equivalent.
Since 1987 the elevations of stakes have been measured by altimeter (a portable aneroid barometer, calibrated by measuring atmospheric pressure at the known elevation of the base camp). These elevations are considered accurate to within ±5 m. For 1960–62, elevations were estimated from recent maps, and similar estimates seem to have been made up to 1967, from which date the elevations were based on occasional accurate triangulation. Earlier elevations are regarded as accurate to no better than ±20 m. The impact of this uncertainty is probably negligible, since the stake readings are grouped into 100 m elevation-band averages before the balance calculations proceed.
Sampling errors
The greatest uncertainty in B is probably due to undersampling. i.e. to the unrepreseniativeness of measurements of b as a sample of the glacier surface. At the local scale (tens to hundreds of metres) glacier surfaces have noticeable microtopography. Prominences must have more negative mass balances than depressions, otherwise the irregular surface, which is steadily regenerated by the internal dynamics of the glacier, would become steadily more irregular. The microclimatic effects of local variations of slope and aspect also amplify the spatial variability of mass balance. On the glacier-wide scale, it is impossible to sample areas which are inaccessible or dangerous, and undersampling is likely as well, due for example to uneven coverage of areas exposed to and sheltered from wind or sun.
Reference Lliboutry,Lliboutry (1974) modelled ablation-zone stake time series as the sum of a constant stake-dependent term, a time-varying stake-independent term and a Gaussian error term. The error term (i.e., the standard error of any one b) was about ±200 mm. Reference Young,Young (1981) claims “conserv ative” Standard errors of 20 mm in the ablation zone and 50 mm in the accumulation zone of Pcyto Glacier. Alberta, Canada, although apparently on a subjective basis. Reference Rogerson,Rogerson (1986) estimates an error in B of ± 100 mm for glaciers in Labrador, again subjectively. Reference Pelto,Pelto (1988) estimated errors of ±(150–200) mm for densely sampled small glaciers in the northwestern United States. Reference Braithwaite, and Olesen,Braithwaite and Olesen (1989) give errors of ±230 mm, which is the standard deviation of three stakes a few metres apart, and ±280 mm from Reference Lliboutry,Lliboutry’s (1974) model, for an ablation zone in southwest Greenland.
Reference Adams,Adams (1966) describes White Glacier cross-profiles at various elevations, which sample the (roughly) horizontal variability at hectometre-to-kilometre scales which is not sampled by the stake networks of most later years (Table 1). Standard deviations in the table average 196 mm. Ridges Profile was deliberately established to probe variability in an area of rough microrelief and difficult access, and its standard deviation of 314 mm may be accepted as an extreme value. The large standard deviation at Moraine Profile in 1961 is due to slush avalanches which introduced superimposed ice into parts of the profile.
Table 1. Error measures from cross-profiles, reanalyzed from Reference Adams,Adams (1966). n, number of stakes in profile; σb, standard deviation of cross-profile balances
The stake network on White Glacier is uneven both in space and in time. Stake density is greater on the lower tongue than elsewhere, exceeding two (and in some years ten) stakes per km2, while stake densities are usually 0.5 stakes per km2 or less above about 1000 m. Therefore, uncertainty in the band-averaged balance b(h) is probably greater at higher elevations. In recent years gaps have developed in the coverage, such that there are no stakes at all between 400 and 600 m or between 800 and 900 m.
The stake network does not cover the entire vertical range of the glacier. Until 1983 the uppermost 10–15% of the glacier had never been sampled, the highest long-lived stake being at about 1420 m. Since 1983 the highest stake has been at nearly 1600 m, so that extrapolation is required over only about 1% of the glacier. However, extending the stake network upwards has had minimal effect: for 1983–91, when B is recalculated, the mean difference (balance with new stakes minus balance without) is 2 mm, and the root-mean-square (rms) difference is 8 mm.
Variation in stake density from year to year introduces two kinds of uncertainty. First, when the number of stakes is large the uncertainty ought to decrease; this point is complicated by correlation between stakes, and is discussed below. Secondly, there will be a “missing-stake bias” if a stake cannot be read because of melt-out or burial. Typical accumulation is less than the length of a typical stake (3.5 m), so burial is not a problem on White Glacier. Stakes in the ablation zone do melt out occasionally; we do not have complete records, but for 1983–91 only three readings out of well over 100 have been lost in this way. (The most common explanation for losing a reading is that the stake is inaccessible for safety-related or other reasons; this does not introduce a bias.)
The areas of elevation bands, and of the glacier as a whole, have been assumed constant at values based on surveys dating from 1959–60. The glacier has shrunk since then. The snout is retreating at 3–5 m a−1 and has a mass balance of about −2 m a−1 (Table 2). Thus, in any cross-section through the snout a (triangular) area of about ½ × 4 × (−2)m2 is lost each year but is not accounted for in calculations. Volumetrically, with the snout width taken as 1000 m, this is equivalent to a spurious trend of −0.1 mm a−2 in the calculated balance series, or to a balance in 1991 which is 3.2 mm a−1 too low, and is negligible even if we double or treble it to allow crudely for shrinkage along the sides of the glacier tongue. We also neglect the additional error due to the decrease in total glacier area, which is perhaps 0.3 km2 or 0.8% over the 1960–91 period.
Table 2. Mass balance β(h) (mm) by elevation band and year. Elevations are band midpoints. 1750 and 50 m bands each occupy less than 0.2% of area, and are omitted; they contain no stakes, and their estimates are extrapolated by setting them equal to the highest/lowest band with stakes. The rightmost column is affected by round-off error
Finally, the incorrect recording and processing of field measurements is an important source of error. As noted by Reference Braithwaite, and Olesen,Braithwaite (1986), mass-balance measurements are usually made by tired observers working in harsh and sometimes dangerous conditions, and so errors are to be expected in field notebooks. There is less excuse for errors made in the office, but they seem to occur nevertheless. These errors are very difficult to find, let alone quantify, but their impact can be reduced greatly by careful double-checking.
Although this discussion is not a rigorous analysis of errors, the conclusion we draw from it is that measurements of b should be considered to have a standard error of about ± (200–250) mm. In the accumulation zone the errors are greater, by at least a few tens of millimetres, than in the ablation zone, because the measurements are more sparse, and because in places the actual accumulation is either overestimated due to overestimated densities assumed for superimposed ice, or underestimated due to losses by percolation (i.e., internal accumulation).
Errors in whole-glacier mass balance
Measurements at different stakes are in general highly correlated, so an increase in the number of stakes does not necessarily increase significantly our confidence in the estimate of B. If all pairs of stakes give perfectly correlated sequences of results, then our confidence in B is no better than our confidence in each b, while at the other extreme perfectly uncorrelated stakes will reduce the uncertainty towards zero as their number grows large. In fact, the White Glacier network exhibits very high correlation between stakes (Fig. 3), such that it is prudent to take the uncertainty in single-stake b as a good estimate of the uncertainty in whole-glacier B. There must be some reduction in uncertainty as the number of stakes increases, but the reduction is unlikely to be large and is not investigated further here; it is the subject of work in progress. Nor do we try to infer errors from the reduction of variance achieved by fitting the curve β(h). The variation we are trying to infer is not adequately sampled by the observations, so that the good fit of the curves β(h) to the data b(h) is not relevant here.
Fig. 3. Balance time-series correlations of 2676 stake pains vs height difference. Each dot represents two stakes with at least 5 years of common record. Triangles: average correlations within 100 m wide bins on horizontal axis. Three-quarters of stake pairs differ in height by less than 610 m; four-fifths have correlations exceeding +0.6.
The time series of annual mass balances of While Glacier has a lag-1 autocorrelation of 0. For estimating the balance normal 〈B〉 this means that we can assume that each B is an independent, random sample of 〈B〉. Thus the standard error of this long-term average will be inversely proportional to the square root of Ny , the number of years of record. With Ny = 29 the error in 〈B〉 is less, by a factor of five or more, than the error in B, i.e. about ± (40–50) mm rather than ± (200–250) mm.
Results
Revised series
Figure 4 shows b as a function of elevation for all years, with simple linear interpolation between stakes. These are not graphs of b(h) or β(h); the tangle of lines illustrates the observed long-term variability of stake mass balance. The largest measured net accumulations are about 500–600 mm and the largest net losses are as great as −5000 mm, although typically ablation does not exceed −3000 mm. This observed range is considerably smaller than that found on many mid-latitude glaciers, especially in maritime climates.
Fig. 4. Stake mass balances for all years. 1960–91, with linear interpolation between stakes. All stakes that survived our reassessment are shown. These graphs show all of the raw measurements on which the mass-balance calculations of this paper are based.
Figure 5 illustrates the polynomial β(h) for a small sample of balance years, and Table 2 tabulates β(h) for all years, together with flanking data on a(h) and the time average of β(h). Table 3 summarizes B, h0 and AAR for each year. The balance normal 〈B〉, i.e. the average of the 29 values of B in Tables 2 and 3, is −100 mm a−1, while the average h0 is 974 m and the average AAR is 0.65. The estímate of h0 is representative of values estimated by simpler, visual methods for the locality, and the AAR is close to what would be expected from observation of mountain valley glaciers in many parts of the world.
Table 3. Mass balance and related properties. n, number of stakes: B, annual whole-glacier mass balance; h0, equilibrium-line altitude; AAR: accumulation area ratio (area of accumulation zone divided by area of glacier)
Fig. 5. β(h) in four different years: least (1978) and greatest (1962) numbers of stakes, and most positive (1964), most negative (1962) and second most negative (1987) mass balances. Thick lines: third-order polynomial β(h). Thin lines: a(h), the hypsometric curve of the glacier. Small open circles: measured stake mass balances. Large solid circles: b(h) for 100 m elevation hands. Standard deviation of measurements within each band is shown as an error bar; in bands with 0 or 1 measurement there is no bar, but even with several measurements the error bar is sometimes invisible because it is narrower than the symbol for b(h).
Alternative estimates of mass balance
The expense and difficulty of fieldwork make more economical methods of mass-balance measurement attractive. This subsection deals with some previous estimates of the mass balance of White Glacier, and with the calibration of some alternative methods. The alternative mass-balance time series discussed here may be found in Table 4.
Table 4. Estimates of the mass balance (mm) of White Glacier by different methods
Previously published estimates
Mass-balance data for White Glacier were first described in B.S. Reference Müller.Müller (1961) for 1960 and 1961, and F. Mülle, and others (1963) for 1960–62, with further reports by Reference Adams,Adams (1966), Reference Young. and Müller.Young (1972), Reference Arnold,Arnold (1981) and Reference Weiss,Weiss (1984).
The record is published in Fluctuations of Glaciers (Reference Müller,F. Müller, 1977; Reference Harberli,Haeberli, 1985; Reference Harberli. and HoelzleHaeberli and Hoelzle, 1993), but with an unfortunate recording error: one series reported from Trent University was displaced by 1 year, such that the published balances for 1984–89 are actually those for 1983–88 (i.e., for the balance years 1982–83 to 1987–88). The rms difference between the published series (as corrected: Table 4) and the new series reported here is 39 mm, attributable to the correction of stake data during reassessment and to the substitution of automated for visual curve fitting in the estimation of β(h). This is well within the standard error for stake measurements b.
Estimates from ELA
Many authors have noted that glacier mass balance is will correlated with elevation of the equilibrium line (e.g. Reference Koerner,Koerner. 1970: Reference Østrem,Østrem, 1975; Reference Braithwaite,Braithwaite, 1984; Reference Hagen. and Liestøl.Hagen and Liestøl. 1990; Reference Kulkarni,Kulkarni, 1992). The relationship is to be expected from the success of Reference Lliboutry,Lliboutry’s (1974) model, which decomposes stake data sets into a stake-dependent function (i.e. an algebraic representation of the shape β(h)), a time-dependent function representing the translation of the shape, and an error term.
The result of a simple first-order regression for White Glacier is
where h0 is in m a.s.1., and B in mm. The standard error of the estimate of B is 126 mm, and the explained variance r 2 is 77%, Model performance is illustrated in Figure 6 and Table 4. The model is simply a rearrangement of existing information, h0 having been derived from the same data as were used to estimate B. But independent measurements of h0 are possible, and the relationship Suggests that they are likely to he useful.
Fig. 6. Whole-glacier balance series as observed (“This Work”) and as predicted from h0 (“Modelled”; Equation (3)). a. Time series; b. scatter graph.
Whether such a model is useful in practice depends on whether h0 can be determined economically, for example from satellite imagery. Independent estimates of h0 are likely to have errors uncorrelated with those in B. Therefore the error in the regression may he added in quadrature (i.e., we add the variances) to the error of ± (200–250) mm in the measured B, in order to estimate the total error, about ± (235–280) mm, in the hypothetical space-based estimate of B. The extra cost-an added uncertainty of (35–50) mm — seems attractive when set against the saving achieved on fieldwork expenses (partly offset by the cost of imagery).
Reference Jung-Rothenhäuslcr., Cogley, and Adams,Jung-Rothenhäusler and others (1992) and Reference Jung-Rothenhäusler,Jung-Rothenhäusler (1993) discuss measurement of h0 from space, and illustrate some of the reasons why early expectations (e.g. Reference Østrem,Østrem, 1975) have not been realized. For example, there is only a handful of suitable end-of-season images of White Glacier from the entire Landsat series of satellites. Moreover, much of the surface texture which helps the human observer to tell snow apart from glacier ice is fine-scaled, and is lost in satellite pixels having dimensions of 10–100 m. A detailed investigation of one end-of-season image showed that neither the equilibrium line nor the closely related snowline were detectable. After full correction for atmospheric and topographic effects (Reference Cogley,Cogley, 1992a, Reference Cogley,b), the surface reflectance was found to be clearly bimodal, the darker mode representing the glacier tongue, and the brighter mode parts of the accumulation zone. However, the transition was gradual, over a vertical span of at least 150 m, and much of the accumulation zone was in the dark, low-reflectance mode. Reference Jung-Rothenhäusler,Jung-Rothenhäusler (1993) adduced two factors to explain this incomplete separation: residual errors, and glazing. Glazing produces strong contrasts in reflectance between snow surfaces, and slopes which face the afternoon sun are more likely to be glazed. A rapid transition from dark ice to bright snow, which is a prerequisite for exploiting Equation (3), simply was not present on Jung-Rothenhäusler’s Landsat image.
This result seems at odds with the spectacular images often shown as indications of the potential for satellite remote sensing of glacier mass balance. We suspect that such images have either been enhanced, using oversimplified algorithms, or simply do not represent the annual snowline; rather, they are images of transient, bright high-altitude snow cover dating from shortly before the time of the image. The distribution of this transient snow will not in general be related to the position of the annual snowline. Jung-Rothenhäusler’s image was unusual in dating from the very end of an ablation season, showing a mature snowpack with reflectance only moderately greater than that of exposed ice.
Thus space-based estimates of h0 for While Glacier do not seem to be practicable, at least in the visible spectrum and until understanding of glacier surface reflectance has advanced considerably.
Estimates from reduced networks
Savings could be realized if we could exploit high inter-stake correlations (Fig. 3) to infer B from measurements on a few stakes near the snout. This idea is the same as that underlying advocacy of h0 as an estimator B.
We examined the series of b at each stake, selecting stakes with records of at least 6 years which correlated with concurrent estimates of B at r ≥ 0.9, or r ≥ 0.707 (r2 = 0.5) for stakes below 650 m a.s.l. We then chose subsets of five to ten stakes located close to the snont. Linear regressions of B on Btongue, the (unweighted) average of b at the subset stakes, were computed over as many years as possible (up to nine). Standard errors were always between ±100 and ±120 mm. Figure 7 and Table 4 show the predictive power of a typical subset.
Fig. 7. Whole-glacier balance series as observed (“This Work”) and as predicted from five tongue stakes with 9 years of record (“Modelled”). a. Time series; b. scatter graph.
There are statistical objections to this procedure: Btongue and B are not independent. Nevertheless a reduced stake network could yield useful estimates of mass balance, but with greater uncertainly than the equilibrium-line relationship discussed above, Because B and Btongue are so highly correlated, the standard error of the regression should be added linearly to the standard error of B, giving a total error of about ±(300–370) mm. The extra uncertainty is a serious penalty, because the only saving is in the time and difficulty of visiting more stakes higher up the glacier. But we conclude that a mass-balance estimate need not be forfeited for lack of time, or because of inability to move up-glacier for other reasons.
Estimates from meteorological data
Reference Glenday,Glenday (1989) modelled the mass balance of While Glacier as a function of climatological data. The advantage of such models is that the glacier need not be visited at all; the cost is negligible.
Glenday’s best relationship explains 71% of the variance in the mass-balance record (Fig. 8; Table 4):
where temperature T, precipitation P and global solar radiation K* are all measured at Eareka (Fig. l). The standard error of the estimate of B is ±119 mm, to be added in quadrature to the nominal error of ±(200–250) mm for a total error of ±(230–280) mm. Thus. Equation (4) is about as powerful as Equation (3), and has considerable appeal. Of all the alternatives discussed here it is the only one that requires no contact with the glacier, remote or otherwise.
Fig. 8. Whole-glacier balance series as observed (“This Work”) and as predicted in terms of weather data from Eureka (“Modelled”; Equation (4)). a. Time series; b. scatter graph, projected onto B-T plane. T, mean summer (June-August) temperature (°C) at 1000 mbar (about 112 m above surface); P, annual precipitation (mm); K*, summer average global solar radiation (Wm−2). Coefficients yield estimate of B in mm a−1.
Equation (4) could probably be improved upon. For example Reference Lefauconnier, and Hagen,Lefauconnier and Hagen (1990) found higher correlations between the mass balance of a glacier in Spitsbergen and climatological quantities from a nearby weather station.
Mass balance for 1980–82
Stakes were measured on White Glacier in 1981 by J. Weiss. We have readings from 1983 for three of Weiss’s 1981 stakes, all of which have remained in use. Their time series of b are well correlated with the time series of B (the correlations being 0.822, 0.866 and 0.882, with Ny = 9 in each case), so we can estimate B for the 2 year interval 1980–82 (recalling that measurements in spring refer to the end of the previous balance year). By linear regression, we find the relationship
between whole-glacier balance B and the (unweighted) average, Btongur, of the balances for the three stakes over the nine years 1983–91; r is 0.95, and the standard error is ± 105 mm a−1. For the same three stakes, Btongue for 1980–82 is −1755 mm a−1, and hence, from Equation (5), B = −61 mm a−1. This estimate has an uncertainly of ±(300–350) mm a−1, but it is consistent with meteorological estimates from Equation (4) (Table 4), which average −148 mm a−1. Apparently the mass balance during the gap was close to normal.
Discussion
The balance normal
The standard error in any single B is ±(200–250) mm. To estimate the uncertainly in the balance normal 〈B〉, which is simply the average of all the annual balances, we may take as a starting-point either this estimate or an independent estimate based on the variability in the time series. Mainly because ±(200–250) mm is an informal rather than a rigorous estimate of the standard error of B. we choose the latter course.
If we assume (e.g. Reference Lyons,Lyons, 1991) that the Ny
estimates of B are random, independent samples from a population with finite variance, then the best estimate of the standard error, se (〈B〉), of the normal is equal to the standard deviation, 258 mm, divided by 
. 〈B〉 for 1960–91 may thus be quoted as −100 ± 48 mm a−1. se (〈B〉) defines a ±68% confidence region for (〈B〉); twice se (〈B〉); defines a ±95% confidence region. The normal differs from zero by more than two standard errors, so, if our assumptions are justified, we may be 95% Confident that it is negative.
To judge whether the Bs are independent of each other, we have computed the lag-I autocorrelation r (i.e. the correlation of the balance series with a copy of itself offset forwards by one year). If r is not zero, then (Reference Zwiers, and Storch.Zwiers and von Storch, 1995) the effective sample size might be less than Ny, and thus the estimated standard error might be too small. However, for White Glacier r = 0.02. which is indistinguishable from zero.
The coefficient of variation (standard deviation divided by the magnitude of the mean) is 2.6. Thus the mass-balance record is rather noisy, due to natural variability and measurement error.
The balance trend
Detecting a trend is more difficult than determining the sign of B. Whether an estimate of dB/df is consistent with the true dB/dt being zero follows from inspection of the standard error. We risk wrongly accepting the hypothesis that there is a trend, but the risk is readily quantifiable. The hard part is avoiding the error of wrongly concluding that dB/dt = 0, i.e. that there is no trend. This risk is enlarged upon below.
The trend and its standard error sedB(dB/dt), in mm a−2, are both derived from a linear regression of the series B on the time t, measured in years since t0 = 1960:
How should we expect glaciers to respond to changes in forcing? In going from one equilibrium to another, a glacier must evolve from 〈B〉 = 0 back to (〈B〉) = 0 via an excursion. Figure 9 shows the mass balance of a glacier in equilibrium with the climate until time t0, when there is a step change in forcing. There is an immediate change in B: the glacier is now either too long and thick or too short and thin, so B becomes negative or positive accordingly. (We assume that β(h) varies so that a glacier which is loo long will grow shorter, and one which is too short will grow longer. Note that B is a Specific quantity, expressed per unit area over the changing total extent of the glacier.) B remains non-zero until the glacier attains the area and thickness appropriate to the new climate, but its magnitude must approach zero more and more closely. In other words, there ought to be a non-zero trend dB/dt in the mass balance of a glacier not at equilibrium.
Fig. 9. Schematic response of a glacier to a change in forcing. The climate (“Farcing”) changes at time t0; the balance B, and therefore also the trend dB/dt, respond accordingly. The shape drawn for B is arbitrary, but in general dB/dt is never zero while B is not zero. Only in the limit of a step-function response (i.e. a rectangular shape for B) will dB/dt be zero during adjustment to the altered forcing. Note that we are not regarding the balance as “forcing” for changes in glacier length, as is sometimes done.
For White Glacier, dB/dt is estimated as −2 ± 5 mm a−2. Thus the conclusion that dB/dt = 0 seems fairly safe. But it may be a type II error, that of wrongly accepting a false hypothesis. We cannot quantify the risk of such an error, because, of the total stock of information in the data, that on deviations of the observed B from the fitted line has already been “used up” (cf. Reference Lyons,Lyons, 1991). We need this information on residuals if we are to estimate the probability that dB/dt is not 0 (i.e., the risk of a type II error). We must not use it twice. But we were obliged to draw on it when estimating the standard error of the trend, se (dB/dt), because we do not have independent estimates of the error in each B, only the nominal error of ± (200–250) mm derived earlier.
To grasp the practical importance of these uncertainties, consider White Glacier’s response time τ, which we may estimate crudely as ∼102a because the glacier is ∼104a long and flows at ∼102m a−1. Reference Jóhannesson., Raymond and WaddingtonJóhannesson and others (1989), who reanalyze the kinematic-wave model of glacier terminus fluctuations (Reference Hutter,Hutter, 1983; Reference Lliboutry,Lliboutry, 1987). suggest calcinating τ as — H/bsnout, where H is a typical thickness, and bsnout the mass balance at the snout. Appropriate values, H = 200 m and b>snout = −2 m a−1, again suggest τ ≃ 100 years. If dB/dt is indeed −2 mm a−2, and if this trend continued for τ years, the balance normal would become −300 mm a−1 - surely a significant response. But is it physically likely or realistic? From Equation (4), the temperature sensitivity of the balance ∂B/∂T is −74 mm a−1 K−1, roughly consistent with estimates by Reference Kuhn,, Warrick,, Barrow, and WigleyKuhn (1993) based on energy-balance modelling. With predicted warming for the next century of ∂B/∂T ∼0.01−0.03Ка−1 (Reference Houghton,, Jenkins, and Ephraums,Houghton and others, 1990, Reference Houghton,, Callander, and Varney,1992), a (transient) balance trend dB/dt of about −1 or −2 mm a−2 should thus be expected in the near future, and perhaps now. This is just what has already been measured, and rejected because of its large uncertainty.
We cannot measure such subtle trends. Natural variability may defeat our best efforts, but it seems that a way must be found to measure mass balances and balance trends with about ten-fold greater accuracy. With the data available now, se (dB/dt) = ± 5 mm a−2 is far too large, but with se (dB/dt) = ± 0.5 mm a−2 we might already have a firm detection of a climatically plausible trend for White Glacier. To repeat an earlier conclusion: because its mass balance is not zero, there ought to be such a trend.
One line of inquiry which does not depend on improved accuracy is to explore how uncertainties evolve as the available record grows. The result (Fig. 10) shows a slow increase, proportional to 
, in our confidence in estimates of 〈B〉 and dB/dt. Given only the First half of the record. We might draw quite misleading conclusions. Two of the most negative balances in the scries were measured in its first and third years; presumably this placement is due to chance, but, whatever the reason, it explains why the estimates evolve rapidly in die first half of the record. Only after 25 years can we claim with 95% confidence that 〈B〉 is not zero, and a very much longer record will be needed before we can detect with the same confidence that dB/dt is not zero.
Fig. 10. Evolution of estimates of the balance normal and trend. We step through the record one rear at a time, recalculating the normal and trend with the record available to date. (Except that the information became available in this chronological order, there is no particular reason for stepping forwards through the record, or for starting at the beginning.) Thick lines: estimated normal (a) and trend (b). Thin lines: 95% confidence regions (i.e. ±2 × standard error of estimate of normal or trend, assuming no serial correlation in the time series). 1980–82 gap filled with estimates from Equation (4).
Are trends detectable in the climatic forcing? Surface temperatures in the Canadian Arctic appear, from the large-scale analyses of Reference Hansen, and LebedeffHansen and Lebedeff (1987), Reference Jones,Jones (1988) and Reference Folland,, Karl, Vinnikov., Houghton,, Jenkins, and Ephraums,Folland and others (1991), to have cooled moderately in recent decades, but the trends are not strong. Reference Walsh, and Chapman,Walsh and Chapman (1990) analyzed station temperature data; none of the Canadian Arctic stations showed significant trends. Reference Kahl,, Serreze,, Stone,, Shiotani,, Kisley and Schnell,Kahl and others (1993) show that the tropospheric temperature in the Canadian Arctic during 1958–86 exhibited moderately significant summer cooling, at levels up to 300 mbar. It would be interesting to know whether this represents an increase in lapse rate. But as it stands, this study is consistent with the surface studies: no evidence for regional warming in recent decades.
For precipitation, the most recent large-scale compilation (Reference Vinnikov,, Groisman and Lugina,Vinnikov and others, 1990) excludes the Canadian Arctic, so we have used our own data set (Reference Briggs, and Cogley,Briggs and Cogley, 1990). Precipitation can be highly variable spatially, and in particular altitudinally (Reference Ohmura, and Müller,Ohmura and Müller, 1977). Moreover, it is difficult to measure because gauges, especially snow gauges, tend to undercatch (e.g. Reference Woo., Heron, Marsh and SteerWoo and others, 1983). Thus the evidence (Table 5) is of doubtful quality, and is also inconclusive. At three stations there appears to be no trend, while ai two there are marginally significant increases. A significant decrease in accumulation was observed by Reference Müller, and Müller,F. Müller (1963d) at a site not far from White Glacier. This record derives from an earlier interval, and the information is described as preliminary, but it adds to a strong impression that no trend in precipitation can be identified with confidence.
Table 5. Annual precipitation at High Arctic weather stations
Since climatic trends are not evident, it is perhaps unsurprising that trends are not detectable in glacier response to climate. But then (Fig. 9), why is White Glacier’s mass balance negative? This paradox illustrates the most important conclusion from the analysis above: with the present technology of mass-balance measurement, the trends which we should expect now and in coming decades will be almost impossible to detect.
The regional balance normal
We have gathered data for other High Arctic glaciers from literature sources. For each glacier, Table 6 includes basic geographical information as well as the balance normal and, if the record is long enough, the trend. The time-span given may include one or more years without measurements. For three glaciers we present additional multi-year balances based on stake readings more than 2 years apart; in these cases Ny is simply the number of years separating the two sets of readings.
Table 6. Mass-balance series for High Arctic glaciers (Fig. 1). Sources: 1, This work; 2, R.M. Koener (personal communication, 1994); 3, Reference Blatter. and KappenbergerBlatter and Kappenberger (1988); 4, Reference Bradley, and Serreze,Bradley and Serreze (1987); 5, Reference Kasser.Kasser (1973); 6, Reference Alean, and Müller.Alean and Müller (1977); 7, Reference Ommanney,Ommanney (unpublished a); 8, Reference Ommanney,Ommanney (unpublished b); 9, Reference Sagar,Sagar (1964), Reference Arnold,Arnold (1968)
Unfortunately the spatial distribution of the glaciers is unfavourable for mapping. To estimate the regional balance normal 〈BR〉, it seems more realistic to settle for a simple regional average. The average of the normals of the five longest records (Table 6), weighted by record length Ny, is 〈BR〉, = −82 mm a−1. Various weighting schemes such as not weighting at all, including the shorter records, and weighting by extent, all give estimates within 30 mm a−1 of −82 mm a−1. The error in 〈BR〉, is not easy to evaluate, because of spatial auto-correlation and extreme undersampling, but it must be of the order of the uncertainty in the measured normals 〈B〉, namely about ±50 mm a−1.
This estimate of 〈BR〉, and Reference Meier,Meier’s (1984) value of 108 600 km2 for the glacierized area of the Canadian High Arctic, give a contribution to sea-level rise of 0.024 ± 0.015 mm a−1. This estimate is compatible with Meier’s 0.019 mm a−1, but being based on fewer assumptions it is more robust. Thus the Canadian High Arctic, although it contains one-fifth of the world area covered by small glaciers, supplies only one-twentieth of the total small-glacier contribution to rising sea level, estimated by Meier as 0.46 ± 0.26 mm a−1.
Summary
Errors in stake mass balance b are about ± (200–250) mm, due largely to undersampling. Errors in whole-glacier mass balance B are of the same order as single-stake errors. However, the lag-1 autocorrelation of the time series of B is zero, so the error in the long-term average of В is noticeably less. A number of recording and processing errors were detected during reassessment of the record, but we do not think that our work has been less careful than that of other glaciologists. Rather, the errors are a consequence of the complexity and longevity of the enterprise.
The balance normal 〈B〉 is −100 ± 48 mm a−1, derived from 29 annual measurements of B. The reassessed balance series agrees closely with previously published values.
Among less accurate alternatives to the present stake-based programme, a reduced stake network is recommended only for salvaging an estimate in difficult field conditions. Hydrometeorological modelling is a practical solution if it is impossible even to visit the glacier. Remote sensing of mass balance is not presently practical, and visible-band imagery may never become a routine tool in the estimation of mass balance.
White Glacier is representative of the Canadian High Arctic, and the dispersion of single glaciers about a regional balance normal of about −80 mm a−1 is not very great. The evidence for trends is weak, and in particular While Glacier seems to have a constant mass balance, consistent with a similar absence of evidence for trends in climatic forcing. However, present methods of balance measurement are inadequate for detecting trends of the magnitude to be expected from generalized climatological reasoning.
This is a serious problem for glaciology. Large errors are attached to our estimates, errors which we suspect to be typical of many other mass-balance estimates. A foundation of measurements, how ever sparse, is essential in monitoring climatic change. But the measurements reported here are rather inaccurate, and all of the cheaper alternatives are even more inaccurate. Research on more accurate ways of estimating mass balance deserves high priority.
Acknowledgements
We gratefully acknowledge the work of all those who have made or assisted with mass-balance measurements on White Glacier. We thank the following organizations that have supported mass-balance work on White Glacier: Polar Continental Shelf Project, Energy Mines and Resources Canada; National Hydrology Research Institute. Environment Canada, Saskatoon; Geographisches Institut, ΕΤΗ, Zürich; Trent University, Canada; the McGill Station, Axel Heiberg Island; the Jacobsen-McGill Arctic Research Expeditions; Science Institute, Government of the Northwest Territories. We are also grateful to R.M. Koerner for valuable comments. Finally, we owe a special debt to the late F. Müller, who initiated the programme of glaciological research on Axel Heiberg Island in 1959 and kept it going energetically and insightfully for 20 years until his death in 1980.
