Glacier fluctuations constitute an important indicator for climate change, both current and past. Glacier mass balance measurements are made to correctly reflect the state of the glacier. Very few studies have been made to study the representability of each point measurement to the average mass balance of a particular glacier, an exercise that requires a large number of measurements. Such studies are rare due to the practical constraints and costs involved in collecting data. On Storglaciären, Sweden, a very dense system of measurements of both distributed winter (∼100 points km^{−2}) and summer (∼15 points km^{−2}) balance allows a spatial analysis of the mass balance components. The results show that local summer balance values are strongly correlated to the average summer balance value of the glacier. Local winter balance values are also generally well correlated to the average winter balance value, but small areas on the glacier exhibit no correlation. These areas correspond to wind-eroded areas of low accumulation on the glacier. The local net balance values are also well correlated to the average net balance value, indicating that the effect of the summer balance is strong and, at least partly, counter-balancing the spatial inhomogeneities in the local spatial winter balance values. These results show that detailed knowledge of both mass balance components and their spatial variability may be necessary to safely use a sparse system of measurements points. On Storglaciären, this is especially true for winter balance measurements since the spatial snow distribution is highly variable and not necessarily representative of the glacier average at each measurement point. The results strictly apply to Storglaciären but similar effects should be present on most glaciers in a similar setting; the results thus serve as an example of conditions that can be expected on a typical mid-latitude to subarctic glacier.

## Introduction

Mass balance measurements on glaciers aim at capturing the climate signal hidden in the net gain or loss of mass or volume over time (Lemke et al., 2007). One problem that faces students of mass balance is how each measurement on the glacier represents or contributes to the average mass balance of the glacier (e.g. Cogley, 1999; Fountain and Vecchia, 1999). The use of a single stake (exemplified by the long Claridenfirn, Switzerland, record, initiated in 1914; Müller-Lemans et al., 1994; Aellen, 1995) providing a measure of the average mass balance relies on the assumption that local mass balance on all parts of the glacier, at least those considered for establishing the stake, is correlated to the glacier average mass balance value. The number of mass balance studies where measurements are made at a sufficiently large number of points to evaluate such relationships is small. On Storglaciären, northern Sweden, measurements with high spatial density (e.g. Holmlund and Jansson, 1999; Jansson, 1999a; Holmlund et al., 2005) have been carried out since the mass balance year 1945/1946 (Fig. 1). In this program, both winter and summer balances are measured, and the net balance calculated as the sum of the two mass balance components (summer balance, mass loss, is a negative quantity). Hence, the Storglaciären data can be used to evaluate mass balance relationships in great detail.

We have used maps of winter, summer, and net balance, based on spatially extensive snow accumulation and melt distribution data (e.g. Jansson, 1999a), to study the spatial correlations between the components on Storglaciären. This provides a detailed view of how the spatial distribution pattern of the different mass balance components relate to the glacier average values. The results from Storglaciären indicate that the local winter balance values may not generally be well correlated to the glacier average winter balance. The result is, however, only a weak effect on the net balance due to the dominance of the summer balance on the net balance. Our results shows that knowledge of the influence of both mass balance parameters on the net balance is necessary to successfully use strongly reduced observation stake nets.

## The Mass Balance of Storglaciären

Storglaciären is a ∼3 km^{2} glacier in northern Sweden with an ongoing mass balance program initiated in 1946 (e.g. Holmlund et al., 2005). A wealth of investigations on Storglaciären and other glaciers in the region (e.g. Wallén, 1948; Holmlund, 1987, 1988, 1993; Holmlund et al., 1996, Hock and Holmgren, 1996; Holmlund and Schneider, 1997; Holmlund and Jansson, 1999; Klingbjer, 2004; de Woul and Hock, 2005; Hock and Holmgren, 2005; Jansson and Linderholm, 2005; Jansson et al., 2007) have set the climatological framework for the Tarfala Research Station mass balance program (Holmlund and Jansson, 1999). The winter balance of Storglaciären is established by traditional snow depth probing and density measurements. Summer balance is measured using standard ablation stake methods. The measurement method used on Storglaciären is applicable to most glaciers in the temperate, subarctic, and arctic regions where a pronounced cold accumulation and warm ablation season is present. The entire mass balance series of Storglaciären is available as isopach maps of the winter and summer balances (in meters of water equivalent, m w.e.). These maps have been constructed from the measured values of winter and summer balance according to the methods given by Østrem and Brugman (1991), and in fact many of the methods described there originate from ideas developed during the first years of measurements on Storglaciären. Common to all measurements is that they have good spatial distribution. Since 1966, snow depth is probed at ∼300 points (∼100 points km^{−2}) in a 100 × 100 m grid, and ∼50 stakes distributed across the entire glacier (∼15 stakes km^{−2}) are used for melt measurements (Fig. 2f; Jansson, 1999a). The number of stakes and snow depth measurements have never been smaller than approximately half the modern values but have varied from year to year, especially before 1966.

## Definitions

In this paper we discuss mass balance parameters on local scale, in terms of their glacier-wide average values, as well as the temporal average of the local values. Anonymous (1969) defined the nomenclature for mass balance measurements and we follow the recommendations. Hence, we can define the following parameters:

Local scale: *b*_{W}, *b*_{S}, and *b*_{N} refer to the mass balance at a single point, e.g. at a stake, of the glacier in terms of its winter (*b*_{W}), summer (*b*_{S}), and net (*b*_{N} = *b*_{W} + *b*_{S}; *b*_{S} is defined as a negative quantity).

Glacier average: *b¯*_{W}, *b¯*_{S} and *b¯*_{N} refer to the glacier-wide average values of each component as established by calculating the mass balance for the glacier based on field measurements according to standard methods outlined by Østrem and Brugman (1991).

Temporal average: According to Anonymous (1969), ⟨*b*_{i}⟩, where *i* is W, S, or N, denotes the temporal averages of mass balance. Consequently, ⟨*b*_{W}⟩, ⟨*b*_{S}⟩, and ⟨*b*_{N}⟩ denote the temporal averages of local balance (e.g. at each node point in a grid).

We will use this nomenclature to avoid misunderstandings when referring to local and glacier wide values and averages.

## Methods

The basic data for our analyses are the digitized, interpolated, and, around the edges, extrapolated maps of distributed winter and summer balance of Storglaciären. Maps rather than raw data were used since only maps and their evaluated average balance values are available for the first 20 years of the series; only the balances from 1994 have been calculated digitally from raw data. The data used here was produced by Holmlund et al. (2005) for their reanalysis of the mass balance data. Their study shows that the data does not contain major inconsistencies arising from the use of sometimes poor topographical support for the mass calculations; however, larger corrections are evident only in the early measurements where map support was also at its worst. We will, therefore, only briefly reiterate the procedure for calculating the gridded mass balance data.

Holmlund et al. (2005) used ordinary kriging (Isaaks and Srivastava, 1989) in the form of a Matlab implementation of the GSLIB (Deutsch and Journel, 1998) implementation to create distributed grids from the digitized water equivalent isopach maps of *b*_{W} and *b*_{S} for the mass balance years 1945/1946 to 2003/2004. Maps are missing for the years 1960, 1963, and 1972, so these have not been included in the analysis. For the Kriging, we used an automatic fitted linear variogram model with no nugget effect. The variogram was isotropic. The resulting grids have 10 m node spacing.

The correlation analysis was made by correlating the *b¯*_{i} given by the mass balance series (Fig. 1) with the corresponding data at each grid point in the interpolated *b*_{i} maps. We have produced contour maps of the temporal average distributions ⟨*b*_{W}⟩, ⟨*b*_{S}⟩, and ⟨*b*_{N}⟩ at each grid point from all available interpolated maps. This also allowed us to produce maps of the corresponding standard deviation, which indicate areas where balances vary from the ⟨*b*_{i}⟩. All calculations were made with a Matlab script using its built-in statistical functions.

## Results

Figure 2 shows the results from correlations between *b¯*_{i} and values of *b*_{i} at each interpolated grid point on the glacier. The *b*_{W} map shows an overall high correlation with *b¯*_{N}. However, there are a few small areas with near zero or even negative correlation between *b*_{W} and *b¯*_{N}. The *b*_{S} map shows a strong agreement with *b¯*_{N}. Figure 3 shows two examples of the correlations between *b¯*_{i} and *b*_{i} at locations A and B in Figure 2, corresponding to areas of maximum and minimum *b¯*_{W}*–b*_{W} correlation. Relating *b*_{N} to *b¯*_{N} provides a similar distribution of correlation coefficients as for *b*_{W}. The correlations between *b¯*_{S} and *b*_{S} and between *b¯*_{W} and *b*_{W} provide slightly better correlations than between *b¯*_{N} and *b*_{S} or *b*_{W}, respectively.

The average distributions ⟨*b*_{W}⟩, ⟨*b*_{S}⟩, ⟨*b*_{N}⟩ are shown in Figures 4a–4c with maximum and minimum values summarized in Table 1. The average patterns show similar features to that visible in distributions from each individual year (see maps in Tarfala Research Station Annual Reports; e.g. Jansson, 1999b). The ⟨*b*_{W}⟩ pattern shows high accumulation values near the head wall and a pattern of highs and lows that are related to wind erosion and deposition on convex and concave areas of the glacier surface, respectively (e.g. Jansson, 1999a). The ⟨*b*_{S}⟩ shows a strong monotonic decrease in melting with elevation consistent with expectation from temperature forcing and its lapse rate (e.g. Jansson, 1999b). The ⟨*b*_{N}⟩ being the sum of ⟨*b*_{W}⟩ and ⟨*b*_{S}⟩ (negative) balance inherits patterns from both seasonal mass balance distributions.

## Table 1

Maximum and minimum values of average and standard deviation in the maps of Figure 4.

The maps of standard deviations relating to the ⟨*b*_{i}⟩ distributions are shown in Figures 4d–4f and Table 1. The ⟨*b*_{W}⟩ generally shows lower standard deviations where accumulation is low and vice versa. The ⟨*b*_{S}⟩ shows very small spatial variability in standard deviation, except for near the terminus and lower margins. The ⟨*b*_{N}⟩ standard deviation distribution closely follows the distribution from the ⟨*b*_{W}⟩, as can be expected.

## Discussion

The spatial distribution of correlation coefficients (Fig. 2) obtained by correlating each of the spatially distributed *b*_{i} values and their corresponding *b¯*_{i} values (represented by Fig. 1) provide a detailed picture of the spatial and temporal statistics of the mass balance measurements.

The *b*_{W} (Fig. 2a) is generally well correlated to the *b¯*_{W} values. However, there are areas with considerably poor correlations, reaching zero and even slightly negative in two areas in the ablation area. Comparing with the average accumulation pattern in Figure 4b, it is evident that the poor correlations occur in areas with generally very low snow thickness values. Obviously areas of low accumulation restrict the magnitude of possible variations relative to areas of high accumulation because the distributions must be skewed since negative accumulation values cannot occur. These are the areas of wind erosion on convex parts of the glacier surface topography. Although Storglaciären has retreated substantially during the course of the mass balance program, the strong bedrock undulations in the subglacial topography (Herzfeld et al., 1993) produce a pattern of convex and concave areas on the glacier surface that has remained pronounced through time (see maps of Storglaciären from 1949 to 1990 reproduced in Albrecht et al., 2000, fig. 1, p. 92). The areas with very low accumulation thus do not have snow depths that correlate with the annual *b¯*_{W}. In these areas the year-to-year variability in snow depth is also smaller. The reason for this, again, is that the *b*_{W} values tend to zero and that snow depth values approach the resolution of the measurements so that the measurement errors (Jansson, 1999a) become significant. Figure 3 shows that *b*_{W} values at B in Figure 2 average around 0.3 m w.e. The estimated error in measurements is around 0.1 m (Jansson, 1999a), i.e. one-third of the typical measurement value at the site. Consequently the climate signal represented by the total value is locally lost.

Figure 2b shows the correlation coefficients for the *b¯*_{S}–*b*_{S} relationship, which are very high across the glacier surface. This is the effect of the strong dependence of *b*_{S} on average summer (June–August) air temperature, which is not influenced by local conditions on the glacier but only on elevation (lapse rates). The correlation map clearly shows that the ablation measured at any point on the glacier will yield a value proportional to *b¯*_{S}. Poorer correlations at the terminus and at the head walls are likely due to effects resulting from extrapolation problems. This is especially true at the head walls since stakes have usually not been established in these areas due to risks associated with e.g. avalanching.

The *b¯*_{N}–*b*_{N} correlations (Fig. 2c) is a combination of the *b¯*_{W}–*b*_{W} and *b¯*_{S}–*b*_{S} correlations. The spatial variability of *b*_{W} can clearly be seen but the *b¯*_{N}–*b*_{N} correlations are significant even in areas where the *b¯*_{W}–*b*_{W} correlation was near zero. The strong *b¯*_{S}–*b*_{S} correlations thus compensate the locally poor *b¯*_{W}–*b*_{W} correlations. Since *b*_{S} increases in importance at lower elevations due to the lapse rate and the areas of low *b¯*_{W}–*b*_{W} correlation occur low on the glacier, the compensating effect would not be universally applicable on the glacier. We can conclude that measuring *b*_{N} at any point on the glacier yields a reasonable proxy for the *b¯*_{N}, but the results from our study cannot be taken as a guarantee that they hold on every glacier. It is therefore important to know local conditions before applying sparse monitoring networks on an unknown glacier. The results from Storglaciären further suggest that *a priori* knowledge of local climatology and climatological forcing on the glacier including energy balance for *b*_{S}, spatial variability of solid precipitation and snow drifting for *b*_{W}, as well as the importance of other effects such as condensation, sublimation, and evaporation are necessary to ensure good measurements.

The patterns of *b*_{W}, *b*_{S}, and consequently the *b*_{N} ( = *b*_{W} + *b*_{S}) (Figs. 4a–4c) are consistent from year to year. The calculated ⟨*b*_{W}⟩ pattern (Fig. 4a) reflects the snow distribution found every year, which is the result of snow drifting, wind scouring, and deposition related to the surface topography of the glacier. The pattern of calculated standard deviations (Fig. 4d) indicates that larger deviations occur where snow thickness is large and vice versa. This is explained by the fact that large variability is more likely where snow thickness is large and that accumulation in areas of wind scour tends toward zero, which also narrows possible variability. The latter relationship leads to the corollary observation that the wind redistribution probably does not vary substantially from year to year since the existence of variable wind distribution effects would be reflected by a more even average distribution of snow on the glacier and hence also larger spatial variability in standard deviations.

The calculated ⟨*b*_{S}⟩ (Fig. 4b) reflects the well-known dependence of *b*_{S} on summer temperature (June–August) (Oerlemans, 2001; Ohmura, 2001) and its lapse rate, the ablation gradient (e.g. Haefeli, 1962). There is some second-order lateral variability, possibly due to shading effects (Hock, 1999), in *b*_{S} maps from individual years, but these largely cancel when considering the average. It is not clear why such effects do not appear each year (according to the measured data), but it is likely that temporal variations in melt observation stake net density and location are part of the explanation. The corresponding distribution of standard deviations shows small spatial variability consistent with largely constant spatial pattern in *b*_{S} from year to year. The larger values seen in standard deviation near the terminus and margins at lower elevations are probably artifacts from strong lowering of the surface in these areas from glacier retreat (e.g. Albrecht et al., 2000, fig. 1). These areas also suffer from extrapolation effects. The overall trend seen in the pattern of standard deviation may also be a result of stronger thinning of the glacier in the ablation area because the glacier has retreated ∼200 m since the initiation of the mass balance measurement program. However, similar to the effects in *b*_{W} variability, smaller variability is expected at higher elevations due to smaller *b*_{S} values in such regions and vice versa for lower regions. It is likely that the effect of thinning, involving tens to several tens of meters, is second-order to the effect of the elevation dependence of temperature when considered over the ∼600 m elevation difference between terminus and head wall area.

The maps for ⟨*b*_{N}⟩ (Figs. 4c and 4f) show the combined effects of ⟨*b*_{W}⟩ and ⟨*b*_{S}⟩ distributions. As can be expected, the pattern in ⟨*b*_{W}⟩ produces a similarly distinct pattern in ⟨*b*_{N}⟩. The resulting distribution of standard deviations follows the same line of reasoning.

In summary, the ⟨*b*_{i}⟩ maps and their standard deviation distributions support long-standing observations by Tarfala personnel and the conclusions of Albrecht et al. (2000) that the patterns observed each year on Storglaciären are consistent from year to year and that the spatial distribution in *b*_{N} is determined primarily from the pattern in *b*_{W} apart from an elevation-dependent modification introduced by the *b*_{S} pattern. The results further show that *b*_{W} is not as easily predicted as *b*_{S}, which shows a simple elevation dependence because of the large variability in *b*_{W}.

## Conclusions

The analysis of distributed *b*_{W}, *b*_{S}, and *b*_{N} versus the *b¯*_{N} for the glacier over a ∼60 year period reveals that *b¯*_{N} correlates well with the *b*_{N} and that each point on the glacier carries the mass balance signal well. This relationship is also extremely well developed for the *b*_{S}. However, *b*_{W} is not well correlated with *b¯*_{W} or *b¯*_{N} in all areas. This shows that the calculated *b¯*_{W} may be unreliable when measurements have been made at only a few points if these were located in unfavorable areas because of the significant redistribution of snow by wind-drifting. The *b*_{N} is influenced by the large spatial variability in *b*_{W}. However, the strong positive correlation between *b¯*_{S} and *b*_{S} values means *b¯*_{N} still correlate well with *b*_{N} and, hence, that *b*_{N} at any point on the glacier can be expected to reflect *b¯*_{N}. This relationship is not generally universal but applies in cases where *b*_{S} is a strong or dominating component in the mass balance. Using very few measurement points or index stake methods thus requires good knowledge of the representativity of the point or points chosen or alternatively an *a priori* knowledge of the distribution of *b*_{W} and *b*_{S} and their influence on the *b*_{N} so that poor correlations do not occur.

## Acknowledgments

We sincerely acknowledge work of the numerous volunteers who have enabled the activities on Storglaciären over the years and without whom the mass balance series may not have existed. Detailed constructive reviews by two anonymous reviewers as well as guidance by the AAAR scientific and chief editors significantly helped us improve the manuscript; however, remaining ambiguities are the responsibility of the authors. We wish to acknowledge Prof. A. Ohmura for initiating our thinking of these relationships.

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