Introduction
An important tool for understanding the effects of glaciers on stream flow is the measurement of mass balance which has been started on several glaciers in West Greenland, using methods freely adapted from Reference Østrem and StanleyØstrem and Stanley (1969). A basic reality of mass-balance measurements in Greenland, however, is the fact that the glaciers involved are often relatively large, e.g. a few hundred square kilo-metres in area, as well as being poorly delineated and located in inaccessible places. It is therefore impossible to measure specific mass balances within a stake network which would be judged adequate by international standards, e.g. with at least a few stakes per square kilometre rather than a few square kilometres per stake, which is the best that can be achieved on large glaciers in Greenland.
The purpose of the present paper is to discuss whether useful information about mass-balance variations can be obtained from such sparse stake networks. Data from Qamanârssûp sermia in West Greenland are used as an example.
Theory
Suppose the specific annual balance yjt is observed at J stakes distributed over an area S for T years. The area-averaged mean balance of the area S covered by the J stakes will be given by
where sj is the area represented by the jth stake such that
It is generally accepted that a fairly large number of stakes is required to calculate accurate area-averaged balances because of the need to sample all parts of the area S, i.e. that sj should be small.
It is assumed after the simplest case in Reference LliboutryLliboutry (1974) that the annual balance at the stake j and for the year t is a random variable given by the following linear model
where aj and bt express the stake- and time-variations of the balance, while ejt is a centred-random error whose stake-and time-averages are both zero. The term bt fluctuates randomly from year to year with a time average of zero. The model error ejt will reflect the effects of (i) errors in the data whereby observed balances are not true balances, and (ii) violations of the model whereby true balances do not behave exactly according to the postulated model.
Substitution of Equation (3) into Equation (1) gives
where Y 0 is the area average of aj given by
and Et is the area-averaged error ejt given by
This term need not be zero, even if the simple average of the error ejt is zero by assumption. Y 0 is also the time average of Yt , and by re-arrangement of Equation (4), the deviation of Yt from this time-average Y 0 is given by
According to Equation (7), the deviation of the area-averaged balance from its time average is given approximately by the index bt as long as the area-averaged error Et is not too large. This will be satisfied if e jt is small compared with other terms. The error Et is not, of course, the absolute error in the area-averaged balance but gives a measure of how well the stake network allows the errors at single stakes to cancel each other out.
The linear model is useful for isolating time variations in mass-balance data, so that they can be compared with climatological variables. For example,Reference MartinMartin (1975) correlated mass-balance deviations from Reference LliboutryLliboutry (1974) with summer temperatures and annual precipitation. Alterna-tively, such deviations can be compared with time variations in run-off from glacierized basins to study the effects of glaciers on stream flow, e.g. as done by Reference BraithwaiteBraithwaite (1985[b]). Reference LliboutryLliboutry (1974) has suggested that the linear model should be introduced into routine work but there is little sign of this yet. No doubt many potential users have been frightened off by the rigour and detail of Lliboutry’s treatment.
Results
Qamanȧrssûp sermia is an outlet glacier from the inland ice located to the east of Nuuk (Godthȧb), the capital of Greenland. The part of the glacier lying between 80 and 1500 m a.s.l., which coincides approximately with the ablation area, is estimated to have an area of 146.3 km2 (Reference OlesenOlesen, 1981), whilst the delineation of the accumulation area is still unknown. The field programme on Qamanârssûp sermia was started by O.B. Olesen in 1979 and has been led by the author since the late summer of 1981. A summary of climatological and glaciological statistics for the five measurement years 1979/80 to 1983/84 has been given by Reference BraithwaiteBraithwaite (1985[a]), while the stake locations are for shown in Figure 1.
Fig. 1. Sketch map of the lower part of Qamanârssûp sermia, West Greenland, showing the locations of stakes.
The field data
The data analysed in the present study refer to specific annual balances measured at the 13 “centre-line” stakes for Fig. 1 the 4 years 1980/81 to 1983/84, together with 5 years of data from the three “751” stakes which are located within a few metres of each other on the margin of the glacier (at 790 m a.s.l.) near the base camp.
Data for the 13 “centre-line” stakes and the three “751” stakes are given in Tables I and II, respectively. The stake network is sparse as the stake density in the area of 106 km2 between 60 and 1300 m a.s.l. is only 0.12 stakes/ km2.
Table I. Annual Specific Balances at 13 “Centre-Line” Stakes for 4 Years, Qamanârssûp Sermia, West Greenland. Units are m Water Equivalent and the Measurement Year is 1 September to 31 August
Table II. Annual Specific Balances at the Three “751” Stakes for 5 Years, Qamanârssûp Sermia, West Greenland. Units are m Water Equivalent and the Measurement Year is 1 September to 31 August
Stakes up to and including stake 008 (790 m a.s.l) usually have little or no winter snow cover and their specific balances mainly reflect ice ablation, of which about 80–90% occurs in the period June–August. From stake 009 (910 m a.s.l.) there is a fairly substantial winter snow cover which starts to melt in mid May or early June. The “751” stakes were read almost every day during the summer whilst readings of the other stakes were more sporadic, for example, the more inaccessible stakes from 009 to 013 may have been read only two or three times during the summer. The last stake readings are made as close as possible to 31 August and then the data are adjusted with the help of data from the “751” stakes so that annual balances can be calculated for a measurement year 1 September–31 August.
Although the data in Table I refer to repeated measurements on the same stakes, it must not be assumed that conditions at each stake are constant. The “centre-line” stakes move with speeds of up to 250 m a−1, whilst the main features on the glacier, e.g. ice falls, crevasse zone, and smoother areas, appear to remain fixed. This means that each stake passes through a variety of conditions with respect to local topography, so that the 4 year series at each stake are not homogeneous, but are sampled over trajectories of up to 1 km. By contrast, the total displacement of the “751” stakes is only of the order of 100 m so that the data in Table II are more homogeneous.
Testing the linear model
The model in Equation (3) is actually the simplest one of several discussed by Reference LliboutryLliboutry (1974) and is chosen for that reason. The present sample sizes, e.g. a 13 × 4 matrix for the “centre-line” stakes, are probably too small to examine the more subtle effects of autocorrelation which Lliboutry considered. A simplification is gained by only considering complete data matrices rather than using more complicated procedures to take account of missing data. For this reason, incomplete data for 1979/80 at the “centre-line” stakes are excluded, as are incomplete data for some other stakes above stake 013. The reader is referred to Reference BraithwaiteBraithwaite (1985[a]) for the most complete summary for the 5 years of measurement.
The value of aj for each stake is the mean value of each row (stake) in Tables I and II, respectively, whilst the value of bt for each year is the mean value for each column (year) minus the mean of all stakes (Reference LliboutryLliboutry, 1974, p. 382). The sources of variation within the data matrices in Tables I and II were computed by the “two-way analysis of variance” programme on a Hewlett-Packard HP-97 desk calculator. The results are summarized in Table III for the two data sets. A statistical test (Reference KreyzigKreyszig, 1970, p. 277) shows that both stake and year effects are significant at the 5% level for the “centre-line” stakes. On the other hand, at the “751” stakes the year effect is statistically significant at the 5% level whilst the stake effect is not.
Table III. Sources of Variation Among Annual Specific Balances at the 13 “Centre–line” Stakes and at the Three “751” Stakes Qamanârssûp Sermia, West Greenland
For the “centre-line” stakes, the stake effect is the largest source of variance, comprising 89% of the total variance. This illustrates the fact that differences between stakes may be very large compared with differences between years, i.e. the same stakes must be used for different years. The error in the linear model for the “centre-line” stakes is only 4% of the total variance, corresponding to a standard deviation of ±0.34 m equivalent.
The effect of errors in the measurements
In the case of the “751” stakes, the stake effect must be regarded as an additional error to the error effect, as both would be zero if readings on the three stakes were identical. The two effects have a combined standard deviation of ±0.23 m water equivalent. This gives a measure of the accuracy of single stake readings. However, the readings of the “751” stakes should be more accurate than those at the more remote “centre-line” stakes because they are read almost every day which allows the detection of gross errors or mistakes. By contrast, the “centre-line” stakes are only read a few times during the season and the data often suffer from the fact that the observers are tired, and a long way from home, when they make the readings. There is little doubt that there are some gross errors in the data for the “centre-line” stakes shown in Table I, e.g. the value for stake 008 in the year 1981/82 appears especially suspicious and is probably an error. The measurement errors at the “centre-line” stakes could therefore be greater than those at the “751” stakes, i.e. larger than ±0.23 m water equivalent. Measurement errors would then account for more than 45% of the total error variance, e.g. about one-half, whilst the rest would be caused by real violations of the linear model.
Variations of annual balance
The annual deviations of mass balance for the 13 “centre-line” stakes, given at the bottom of Table III, have a standard deviation of ±0.52 m water equivalent. The standard deviation of the corresponding error effect is ±0.34 m water equivalent, which is reduced to only ±0.05 m water equivalent by stake averaging according to Equation (6). This latter figure does not, of course, represent the absolute error, i.e. the difference between the data collected in the present sparse network and in a hypothetical dense network. However, it does indicate a reasonable degree of averaging out of errors in the present network.
The differences between the mass-balance deviations for the “centre-line” and “751” stakes have a standard deviation of only ±0.24 m water equivalent, which is of the same order as the error in the data for the “751” stakes. This represents reasonable agreement between the two sets of stakes. However, they cannot be expected to give identical results as their responses to climate control will not be the same, especially because winter snow cover tends to be absent from the “751” stakes. The discrepancy between the 1983/84 deviations, i.e. +0.41 and +0.07 m water equivalent, respectively, may reflect the fact that there was generally more winter snow at the “centre-line” stakes but not at the “751” stakes. On the other hand, the extreme positive deviations for both data sets for 1982/83 reflect the fact that the 1983 summer was exceptionally cool with reduced ablation at all stakes.
Discussion and Outlook
Statisticians can object that the samples analysed here are too small to draw general conclusions. This is correct and further analyses will be made as more data become available. However, it is a hard fact of life that the applied glaciologist cannot afford to collect data over a statistically adequate number of years before attempting to draw any conclusions at all. The present conclusion is that useful information can be obtained from sparse stake networks such that they should be continued in the future, rather than being discontinued; the alternative of making stake networks in Greenland more dense cannot be considered at present.
The present preoccupation is with the assessment of mass–balance variations on a scale of tens to hundreds of square kilometres. There is, however, the broader perspective of the whole Greenland ice sheet and its possible changes in response to climate. It would be quite impossible to measure the mass balance of the whole ice sheet within an adequate stake network. On the other hand, it might be possible to monitor time variations of the mass balance by a relatively small number of strategically spaced stakes, i.e. a sparse stake network on a scale of hundreds of thousands of square kilometres. Considering the global consequences of any major change in the mass balance of the ice sheet, the development of such a large-scale, albeit sparse, stake network should be considered seriously within the next few years.
Acknowledgements
This paper is published by permission of the Director, The Geological Survey of Greenland. The project at Qamanârssûp sermia was set up by O.B. Olesen. Many student assistants helped to collect data, especially J.-O. Andreasen and O. Bendixen from Aarhus University, and J. Andsbjerg and C.E. Bøggild from Copenhagen University.
