The 1959, 1980 and 1989 ice surfaces

Ice surface topography for Rabots Glaciär is available in maps (Fig. 3a-c) produced from aerial photographs for the years 1959 (unpublished), 1980 (Reference Holmlund and SchyttHolmlund and Schytt, 1987) and 1989 (unpublished). Data from a 1981 survey of a stake network (n = 23) used for velocity measurements indicate that the mean difference between measured ice surface elevations and those shown on the 1980 map is -1.2 ± 1.8 m. Considering that changes brought about by ablation (typically ∼0.9 m ice a^-1 ) or accumulation (typically ∼1.3 m ice a^-1) had occurred in the interim, the uncertainty in the ice surface depicted in the 1980 map is estimated to be ±1 m. This value is consistent with estimates of the uncertainty of topographic maps reported in other studies (e.g. Reference HolmlundHolmlund, 1987; Reference Reynolds and YoungReynolds and Young, 1997; Reference AndreassenAndreassen, 1999; Reference Hagen, Melvold, Eiken, Isaksson and LefauconnierHagen and others, 1999). For the unpublished 1959 and 1989 maps, uncertainties could not be assessed independently. The 1959 map exists in draft form and was constructed in the 1980s. The accuracy of the map is, however, considered to be similar to that of the 1980 map (personal communication from P. Holmlund, 1986), so we assign the same uncertainty (±1 m) to it. The 1989 map (also a draft) is more problematic in that there is a slight distortion of the glacier’s projection with respect to known bedrock fixed points (personal communication from P. Holmlund, 1991). We therefore assign an uncertainty of ± 2 m to the 1989 ice surface.

Fig. 3. Ice surfaces for Rabots Glaciär in 1959 (a), 1980 (b), 1989 (c) and 2003 (d). Coordinates shown are in the Swedish RT-90, 2.5 gon system. Contour interval is 25 m. Circled points are known fixed points. Small crosses in (d) show the locations of most survey points, but some are obscured by contour labels.

The 2003 ice surface

In July 2003 a detailed resurvey of the surface of Rabots Glaciär was carried out using differential global positioning system (GPS) techniques. The use of GPS surveys in glaciological studies is well known (e.g. Reference Hulbe and WhillansHulbe and Whillans, 1994; Reference VaughanVaughan, 1994; Reference Eiken, Hagen and MelvoldEiken and others, 1997; Reference Theakstone, Jacobsen and KnudsenTheakstone and others, 1999). In the interest of brevity, therefore, the theory is not reviewed here and only a brief description of certain aspects of our survey is given.

The base station receiver (Trimble 4600LS) was located on a moraine crest ∼0.5 km from the glacier’s terminus and was linked via radio to the roving receiver, permitting realtime kinematic (RTK) surveying. The RTK technique also eliminated the need for post-processing of the data. After initialization, ice-marginal positions and surface elevations were measured using preset quality control parameters. In most instances, measurements were automatically recorded only when horizontal and vertical precision were both ≤0.05 m and position dilution of precision (PDOP) values were low. Typically occupation times for data acquisition were on the order of several seconds. For some points measured in the upper reaches of the glacier, the criterion for acceptable horizontal and vertical precision was relaxed to ≤0.1 m in order to reduce the time required for data acquisition. In these areas, up to 5min was required to achieve the better precision, if achieved at all. This was done to maximize the number of points measured, as the ‘window’ available for maintaining initialization during RTK surveying was severely limited by both the constellation of satellites and the steep, high valley walls that surround much of the glacier. The location and density of the nearly 1000 points (Fig. 3d) surveyed varied according to irregularities in local surface topography (or ice margin) and our expectations of where more significant changes in the ice surface should have occurred. Thus point density is greater at the terminus and over the undulating surface of the lower reach of the glacier than it is over the more planar surfaces up-glacier. We were unable to survey points in the heavily crevassed ice of the southern tributary cirques and the uppermost reach of the trunk valley. Consequently the survey data were augmented with ice surface elevations taken from the 1989 map. Measured elevations suggest that virtually no appreciable changes have occurred in these areas in the intervening 14years (Fig. 3d).

Several fixed points were surveyed in order both to facilitate georeferencing in conjunction with existing maps and to provide a tie to other fixed points used for velocity measurements in the 1980s. Differences between the GPS-determined elevations of these fixed points (n = 13) and those indicated on the 1980 map are similar, having a mean of 16.1 ± 2.2 m. No systematic trends are evident in these differences. Thus this value was uniformly used to make elevations on the 1959, 1980 and 1989 maps consistent with those measured in the 2003 survey.

Changes in the longitudinal profile

The recent evolution of the longitudinal profile of Rabots Glaciär is shown in Figure 4. Not surprisingly, the most pronounced changes have occurred at and near the terminus in connection with its retreat. Above elevations of ∼1500 m, profile changes are relatively small not only for each interval, but also over the entire 44 years for which profiles can be constructed. It is also clear that the most significant adjustments in glacier geometry were made during the interval 1959-80.

Fig. 4. Longitudinal profile changes for Rabots Glaciär during the interval 1959–2003. Profile is along the line A-A’ shown in Figure 1 . Dotted portions of the bed profile are areas where the glacier’s depth is less certain.

Volume changes

Maps of 1959, 1980 and 1989 ice surfaces were digitized using the software package Didger® 3 (Golden Software). The 1959 and 1980 maps were digitized following their 5 m contour interval resolution, and the 1989 map following its 10m contour interval resolution. Data from the 2003 survey were used in their raw form and supplemented with digitized data from the 1980 surface in areas above 1500 m. We assume minimal changes have occurred in these areas based on a few measurements and extrapolations of the glacier’s 2003 longitudinal and transverse profiles. The appropriateness of this assumption is discussed subsequently. All maps were georeferenced using the fixed points established during the 2003 survey. Irregularly spaced digitized or survey data were transformed into regular grid files using a linear kriging interpolation method within the software Surfer® 8 (Golden Software). Grid spacing was specified to be 10 m. Changes in ice volume (hence mass) between the periods 1959-80, 1980-89 and 1989-2003 were calculated using three algorithms (Trapezoidal Rule, Simpson’s Rule and Simpson’s 3/8 Rule) within Surfer®. The use of all three methods provides a measure of the accuracy of volume calculations. In all cases, values determined by the three different methods of calculating volume differed by <0.1%.

Changes in the glacier’s surface elevation between the periods 1959-80, 1980-89 and 1989-2003 are shown in Figure 5. Between 1959 and 1980, the volume of Rabots Glaciär decreased by about (51.1 ± 6) x 106m³ , equivalent to ∼2.4 × 106m³ a^-1 . The net volume change corresponds to a mean change in surface elevation (averaged over the entire former surface of the glacier) of -12.4 ± 1.5 m and a change in area Δ A of -3.0 × 10⁵ m².

Fig. 5. Changes in the ice surface of Rabots Glaciär between the intervals 1959–80 (a), 1980–89 (b) and 1989–2003 (c). Note that the glacier margins differ slightly in some areas, as a result of inability to distinguish the ice margin from marginal snow banks on aerial photos during the map-making process or during measurement in the field. These differences were not deemed significant in the analysis presented here. Minimum mean surface change indicated in (b) and the areas indicated by questions marks in (b) and (c) are discussed in the text.

Figure 5 indicates thickening of a few areas of the glacier, but the total positive volume change calculated for the interval 1959-80 is only 1.8 × 106m³ . Given that most of the thickening is indicated where there are differences in glacier boundaries and that this positive volume is small (well within the uncertainties), we believe it is either not real or not significant.

After 1980, the glacier’s volume decreased at a rate of ∼1.1 × 10⁶ m³ a^-1 , so that by 1989 the total volume loss was (10.4± 7) x 106m³ and its area reduced by 7.1 × 1 0 ⁴m² . The large uncertainty in the volume change results from both that in surface elevations and the brevity of this interval. Of greater concern is the thickening (mean of 4.7 m) in the tributary cirques and the upper reach of the main valley (Fig. 5b). We believe this to be largely an artifact of the rotation in the 1989 map that was not wholly removed by warping during georeferencing, the effect of which would be to introduce a false thickening in these areas, especially in the tributary cirques. If in reality no significant thickening occurred in these areas, the magnitude of volume loss for the 1980-89 interval could be as much as 15.3 × 1 0 ⁶m³ , or more if thinning had occurred.

To investigate this problem further, we used the mean specific net mass balance for 1981-89 (about −0.30 m w.e.) as that for 1980-89. (Systematic mass-balance measurements on Rabots Glaciär began in 1981.) Total mass loss estimated in this way is ∼10.4 × 10⁶ m³ w.e. To convert from mass to volume we follow Reference Sapiano, Harrison and EchelmeyerSapiano and others (1998) and assume a mean density for the surface layer of 795 kgm–³, weighted proportionally to the areal distribution of snow (typically 550 kg m–³) and ice (900 kg m–³) given by the mean equilibrium-line altitude (ELA; ∼1480m). This yields a volume loss of ∼13.1 × 10⁶ m³.

For the 1980-89 interval, was estimated to be -2.7 ± 1.9 m on the basis of changes in mapped elevations. However, based on the foregoing mass-balance approach, we believe this is a minimum estimate and could have been as much as -3.5 m for this interval.

Between 1989 and 2003, an additional (14.4 ± 7) × 106m³ was lost at a rate of 1.0 × 106m3a–¹. Again the rather large uncertainty is associated principally with those in surface elevations. Changes in the glacier during this interval were such that was − 3 .9 ± 1.9 m and ΔA about -5.4 × 104m² . We evaluate our assumption that little change has occurred in the areas unsurveyed in 2003 using mass balance in the manner described above. The mean specific net balance for 1989-2001 is -0.20 m w.e. (mass-balance data for the balance years 2001/02 and 2002/ 03 are unavailable at this time). Assuming again that this closely approximates the mean specific net balance for 1989-2003, mass loss for this interval is estimated to be ∼10.4 × 106m³ w.e. The mean ELA of ∼1360m implies a mean density of 720 kg m–³, from which a net volume loss of 14.5 × 106m³ is calculated. Based on the similarity of the two estimates of volume change, we conclude that no significant changes have occurred in the tributary cirques and the upper reach of the main valley between 1989 and 2003.

An estimate for volume changes, 1910-59

The change in the volume of Rabots Glaciär from 1910 to 1959 was estimated using an empirical relationship between changes in the glacier’s surface area, ΔA, and those in volume, ΔV. The extent of the glacier in 1910 and its associated ice margins are well documented by a series of photographs taken by F. Enquist. The Enquist photos provide multiple views of the entire glacier, and in sufficient detail to determine former ice limits to within a few meters using known landmarks (e.g. bedrock outcrops and features, uniquely shaped boulders) and comparisons with photos taken in the years 1982-84. On this basis, the aerial extent of the glacier in 1910 is estimated to be ∼4.57 × 10⁶ m².

The correlation between known Δ A and ΔV for Rabots Glaciär is shown in Figure 6a. Although the dataset is limited to three points, the correlation is significant at the 95% confidence level. To assess the robustness of such a relationship, a similar correlation (Fig. 6a) was performed for Storglaciären using known values of ΔV (explained subsequently) and Δ A (Reference HolmlundHolmlund, 1987; unpublished reports) (Table 1). Again the statistical significance of the regression is very high (n = 5; r² = 0.92; p< 0.001). This can be seen better in Figure 6b where values of ΔV for Storglaciären, predicted using ΔA, are compared with ‘observed’ variations in glacier volume. The latter are taken from Reference HolmlundHolmlund (1987) who calculated cumulative mass (hence volume) variations using a measured mass-balance series for the interval 1946-86 and a reconstructed mass-balance series prior to 1946. The record was extended to 2002 by including more recent mass-balance measurements. (It bears mentioning that based on a map analysis similar to that presented here, Reference HolmlundHolmlund (1987) concluded that the cumulative mass balance accurately represents mass changes of Storglaciären.) Volume changes based on ΔA agree reasonably well with the observed record of volume changes.

Fig. 6. (a) Correlation of volume changes with changes in glacier length and area for Rabots Glaciär and Storglaciären. (b) Observed volume changes for Storglaciären compared with those predicted using the relationships shown in (a).

Table 1. Changes in glacier area and volume for Rabots Glaciär and Storglaciären and derived changes in mean thickness

Given this success, the volume change on Rabots Glaciär between 1910 and 1959 is estimated to be 77.3 × 106m3, based on a value for Δ A of -4.5×105m³ . The rate of volume change over this interval would have then been -1.6 × 106m3a–¹. The glacier would have also thinned by an average of 16.3 m. (Only slightly different results are obtained when substituting the volume change suggested by the mass-balance analysis for the 1980-89 interval.)

Ice retreat

The retreat of Rabots Glaciär and that of Storglaciären are compared in Figure 2. While the behavior of a glacier’s terminus might be the most obvious manifestation of its reaction to climate change, factors other than the glacier’s dynamic response can influence the observed behavior of the terminus. These include local energy and mass balances, catastrophic collapse of the ice margin, changes in local bed slope, and non-uniform retreat (particularly because of the accumulation of debris). Moreover, the dynamic response of a glacier’s terminus is particularly sensitive to local ice velocities (Reference NyeNye, 1960; Reference Jóhannesson, Raymond and WaddingtonJόhannesson and others, 1989). Thus, direct detailed comparisons of terminus behaviors can be complicated by the inability to deconvolve those behaviors into precise longer-term dynamic and shorter-term nondynamic components.

In light of the foregoing, some caution must be exercised when inferring the dynamic response of Rabots Glaciär and Storglaciären based on their records of retreat. Nevertheless, the very disparate behaviors of the glaciers’termini shown in Figure 2a suggest that: (1) Storglaciären reached at least a transient equilibrium with the existing climate (with due consideration of more recent warming and/or other climate trends) by the 1980s, while Rabots Glaciär remains significantly out of equilibrium and continues to retreat in response to the early-20th-century warming; and therefore (2) the response time of Storglaciären might be inferred to be shorter than that of Rabots Glaciär.

Storglaciären’s initial rate of retreat and the quickness with which it achieved its maximum rate of retreat also indicates its greater climatic sensitivity. Figure 2b shows that the maximum rate of retreat for Storglaciären occurred during the period ∼1945-50. The anomalous 1933-46 peak notwithstanding (it is evident in the Storglaciären record as well), maximum rates of retreat for Rabots Glaciär were not attained until decades later. Finally, we note the ‘peakedness’ in the time distribution of retreat rates for Storglaciären, as opposed to the broader ‘plateau’ for Rabots Glaciär.

Thickness changes

The magnitude, timing and rate of ice-thickness changes on the two glaciers have also been different (Fig. 7). The calculated mean change in thickness on Storglaciären from 1910 to 1959 is about -29.1 m (Table 1), about twice that estimated for Rabots Glaciär. In general terms, the ablation area of Storglaciären thinned between 1959 and 1980, while the surface of its accumulation area remained unchanged (Reference Holmlund, Karlén and GruddHolmlund and others, 1996). There was a net thinning, however, on the order of 7.5 m, but this was considerably less than that experienced by Rabots Glaciär (Table 1). In the following decade, Storglaciären’s ablation area experienced little change, but more significantly, parts of the accumulation area thickened by as much as 10m (Reference Holmlund, Karlén and GruddHolmlund and others, 1996). As a result, the glacier thickened an average of about 1 m during this time, whereas the net thinning on Rabots Glaciär was 2.7 m or more (Table 1). After 1990 there was a general increase in ice thickness in the ablation area of Storglaciären as well (Reference Holmlund, Karlén and GruddHolmlund and others, 1996). Consequently, between 1989 and 2001 the glacier thickened by another ∼1 m. During the same time the surface of Rabots Glaciär lowered by ∼4m. In fact, we cannot say with any certainty that any significant thickening occurred on Rabots Glaciär for any of the intervals under investigation.

Fig. 7. Mean surface elevation (thickness) change and rate of change of glacier surfaces.

The calculated mean decrease in ice thickness on Storglaciären for the period 1910-2001 is on the order of 35 m. Over essentially the same period, the mean lowering of the surface of Rabots Glaciär was at least 35 m and will likely increase substantially as the glacier continues to retreat. Inferred variations in the time rate of change in over this interval (Fig. 7) are also very different for the two glaciers. For Storglaciären, a consistent transition from a rapid rate of thinning to slight thickening is obvious. In contrast, thinning rates have been more variable on Rabots Glaciär. Most notable is the pronounced increase in the rate of thinning centered on the 1959-80 interval.

Volume changes

Volume changes on Rabots Glaciär for the period 1910-2003 are compared with those on Storglaciären in Figure 8. The general trends on Storglaciären mirror those in its retreat, in that there is a systematic decrease in volume (and, by inference, in the rate at which this decrease occurs) until about the late 1970s. By that time, Storglaciären had reached quasi-equilibrium, at least with respect to mass changes, as was noted by Reference HolmlundHolmlund (1987, Reference Holmlund1988). Increased winter accumulation over the last decade has led to a small increase in glacier volume. As of 2001 the total volume change from the 1910 configuration is estimated to be about -126.9 × 106m3.

Fig. 8. Volume changes for Rabots Glaciär and Storglaciären, 1910– 2003. Also shown are volume changes for Rabots Glaciär predicted using its dependence on area changes shown in Figure 6a. Uncertainties in the 1980, 1989 and 2003 data points reflect uncertainties in individual glacier surfaces.

Conversely, Rabots Glaciär has consistently lost volume over the last century. Moreover, this loss appears to have occurred less systematically (Fig. 8). While the period of largest rate of volume loss on Storglaciären was 1910-59 (more specifically 1933-43 (Reference HolmlundHolmlund, 1987), also coinciding with very warm temperatures), on Rabots Glaciär it was 1959-80. This delay might again suggest Storglaciären’s greater sensitivity and therefore quicker reaction to a change in climate. Since 1980, volume loss on Rabots Glaciär has continued at a more-or-less constant rate (with due consideration of uncertainties) of ∼1.0×106m3a^–1. Total volume change on Rabots Glaciär during the period 1910-2003 is estimated to be about -153.2 × 10⁶ m³.