Albedo

α was measured with a Kipp and Zonen CM7B albedometer sensitive to radiation in the wavelength range 0.3–2.8 μm. The instrument was calibrated by comparison with a reference pyranometer (model: Kipp and Zonen CM6B) using a 1000 W tungsten–halogen lamp. To record spatial variations and seasonal trends in a, rather than α variations caused by changes in cloud cover and θ, the following measurement procedures were used.

1. 1. α was measured under cloud-free conditions within 3 hours of local solar noon, when pyranometer accuracy is ±2%. Point α was found to vary by < ±0.02 within this period, irrespective of surface type. Therefore, the influence of varying θ can be ignored in the a measurements. Similarly, the impact of diurnal freeze–melt cycles was avoided since the glacier surface was always melting when measurements were made, except during periods of cold weather when the surface remained frozen throughout the day.

2. 2. The albedometer was mounted on a camera tripod to ensure that α readings were made in a surface-parallel plane. This effectively maintained consistent viewing geometry between different sites and avoided the serious errors which may occur when α is measured in a horizontal plane over a sloping surface (Reference MannsteinMannstein, 1985).

3. 3. Measurements were made at 1 m above the surface, at which height 90% of the total irradiance recorded by the downfacing pyranometer is received from a circular area with 3 m radius directly beneath it (Reference SchwerdtfegerSchwerdtfeger, 1976).

Surface properties and meteorological variables

To investigate the influence of surface conditions on α variation, the following variables were measured at each snow-covered point, based on the mean of three samples: snow depth (error = ±1 cm); surface snow density (error = ±5 kg m⁻³); surface snow grain-size, defined as sphere diameter (error = ±0.1 mm). At 36 sample points the solid impurity content of the surface snow was determined by filtering 500 mL of melted snow, sampled from directly beneath the albedometer, through a 0.7 μm millipore filter paper. The weight of solid impurities contained in the snow sample was calculated as the difference in filter paper weight before and after filtering (error = ±2 parts per million (ppm) by weight). On ice, the percentage cover by debris and finegrained material, on the surface and within cryoconite holes, was assessed with the aid of a 0.5 m² quadrat.

To assess the effects of meteorological conditions on α and to develop α parameterizations, meteorological variables were recorded at an automatic weather station located at a proglacial site 200 m from the glacier snout at 2547 m a.s.l., which operated continuously throughout the fieldwork period (LMS in Fig. 1). An identical meteorological station (UMS in Fig. 1) was located on the glacier at 2884 m a.s.l. in July and August 1993 and 1994 to provide the temperature lapse rate. Air temperature (T) was measured at 2 m height using thermistors housed in unaspirated radiation shields. As with all T measurements, there is the possibility of artificial heating under strong incident radiation conditions. However, due to katabatic effects wind speed tended to be high (2–5 m s⁻¹) when global radiation was > 700 W m⁻², and the likely T overestimate is <0.5 K (manufacturer’s figure). To determine the relationship of α to accumulated melt, regular measurements of surface lowering were made at ablation stakes at 16 sample points along the glacier centre line.

Sampling strategy

To determine glacier-wide variations in a and surface conditions, 68 sample points, ranging in elevation from 2572 to 3002 m a.s.l., were established (Fig. 1). Above 3000 m a.s.l., measurements could not be made safely due to the hazards of steep slopes and crevasses. No measurements were made on the western margin of the glacier tongue, which is entirely debris-covered. The spacing of sample points was varied from about 200 m in the upper basin to about 50 m on the snout to accommodate the greater a variability at low elevations. Sample point locations were surveyed onto the Swiss grid using a Geodimeter 400 total station.

The entire network was sampled at 2–3 week intervals throughout the 1993 ablation season. All points were normally sampled within a 2–3 day snowfall-free period, producing an almost instantaneous picture of the distribution of α across the glacier surface (Table 1). The proportion of sample points monitored increased during the ablation season as the variability in surface conditions increased. To enable the broad patterns of α variation during 1993 to be compared with those during a second ablation season, with different meteorological and surface conditions, two glacier-wide surveys were also conducted during the 1994 ablation season (Table 1). Prior to glacier surveys 2 and 4, measurements were also made between sample points, to assess small-scale α variability. To study the impact of new snowfalls on α, additional point measurements were made on the days following summer snowfalls on 3 September 1992 and 21 May, 3 and 13 June, and 28 August 1993.

Relationship of albedo to physical control variables

α _s is inversely correlated with snow grain-size (γ) and impurity concentration (I) (Table 3; Fig. 6a and b) (r ² = 0.46 and 0.74, respectively). Hence, changes in γ and I cause most of the observed α _s variability. α _i is inversely correlated with debris cover (D) (Table 4; Fig. 7a) (r ² = 0.41). This is expected since rock material, dust, carbon soot and organic matter all have lower α than pure ice. Values of α _i > 0.30 occurred only when D was < 5%.

Fig. 6. Relationships between snow albedo and (a) snow grain-size, (b) snow impurity content, (c) snow density, (d) snow depth, (e) accumulated days since snowfall, (f) accumulated daily maximum temperatures since snowfall and (g) accumulated melt since snowfall.

Fig. 7. Relationships between ice albedo and (a)% debris cover, (b) elevation, (c) accumulated days since ice exposure and (d) accumulated melt since ice exposure.

Parameterization of albedo

Previously published forms of the relationship between α and surrogate variables are shown in Table 5 (referred to as parameterizations A–F hereafter). In these parameterizations, accumulated days (t _a), accumulated daily maximum temperatures >0°C (T _a) and accumulated melt (M _a) (each of which accumulates from a value of zero at the time of the most recent snowfall), snow density (ρ) and snow depth (d) are assumed to be surrogates for increases in γ and I as the snowpack ages. Hence, all parameterizations except A attempt to calculate temporal α _s variability. Parameterizations C–F also aim to calculate spatial a_s variability through spatial variations in d, ρ, M _a and T _a. Parameterizations C and E assume that the influence of the underlying ice or debris will increase with decreasing snow-cover thickness and hence α _s will approach the underlying α _i as d decreases.

In parameterization E, α _i decreases as a function of accumulated melt since exposure of the ice surface (M _a), assumed to be a surrogate for increases in D during the ablation season. Similarly, a decrease in α _i over time was calculated as a function of accumulated days since exposure of the ice surface (t _a) using parameterization B (Reference Oerlemans and HoogendoornOerlemans and Hoogendoorn, 1989). Parameterization E also calculates spatial α _i variability as a function of elevation (E), on the assumption that α _i will increase up-glacier with decreasing D. The remaining parameterizations do not calculate spatial α_i variability. Only parameterization D attempts to account for α variability at time-scales of <1 day.

The validity of the assumptions made by the earlier parameterizations can be tested with the field data. On snow, all surrogate variables used by the parameterizations are significantly correlated with α _s, γ and I, except for d, which is significantly correlated with α _s and I, but not γ (Table 3). The correlations with ρ, T _a and t _a are particularly strong taking sample size into account. Hence, parameterization forms B, D and F have a strong physical basis. However, the occurrence of low α _s (<0.40), causes scatter in the relationships of α _s with surrogate variables (Fig. 6d–g) (snow samples taken for the measurement of p required a snow thickness of ≥0.1 m, so few values < 0.40 appear in Figure 6c). The assumption that α _s approaches α _i at low values of d (parameterizations C and E) is unsupported (Fig. 6d). As d decreases, α _s becomes increasingly scattered, with a range from < 0.20 to 0.80 occurring when d < 0.1 m, compared with the total α _i range of 0.07–0.39. This suggests that variability in γ and I, rather than in α _i, causes most of the scatter in α _s at low values of d.

To determine whether the value of α _s is dependent on the underlying ice or debris albedo (α _u) at very low values of d, correlations between α _s and α _u, for a sample of 54 albedo measurements over shallow snow covers (d ≤ 2 cm w.e. depth) at which α _u is known, are shown in Table 6. All of the depth ranges in Table 6 include both light deposits of fresh snow and the last remnants of older snow overlying the ice. Measurements of γ and I are not available for most of these points. Instead, correlations of α _s with T _a, α _s proxy for snow metamorphism and accumulation of impurities, are shown. Snow covers of < 0.5 cm w.e. depth were discontinuous, with bare patches of the underlying ice or debris exposed within the field of view of the albedometer.

The most striking result is that α _s is significantly correlated with α _u only on snow <0.5 cm w.e. deep (Table 6). α _s and α _u are not correlated on deeper snow, despite the large range in α _u. Therefore, snow metamorphism and accumulation of impurities control α _s variability on snow > 0.5 cm w.e. deep, as demonstrated by the very strong inverse correlation between α _s and T _a and by the very large range in α _s (Table 6). Furthermore, α _s is as strongly correlated with T _a as with α _u on snow covers <0.5 cm w.e. deep, where the α _s range is 0.20–0.68. This implies that a light dusting of fresh snow on an ice or debris surface will dramatically increase the albedo to as high as 0.68, indicating that α _s is virtually independent of α _u. By contrast, the last remnants of an old snowpack lying on top of the ice will have an α _s value similar to that of the underlying ice or debris, due partly to the low α _s of the snow itself and partly to the influence of α _u. An important implication is that α _s will not necessarily approach α _u with decreasing d. Rather, for fresh snow the transition from high α _s to low α _u will occur extremely rapidly when d decreases below 0.5 cm w.e. On old snow, however, the transition to the value of α _u will occur largely as a result of increases in γ and I during snow metamorphism, with α _u an additional factor playing a role only briefly before the snow melts away completely.

On ice, the assumption that t _a and M _a are surrogates for increases in D over the ablation season (parameterizations B and E) has some support from the correlation analysis, but t _a and M _a are weakly correlated with α _i (Table 4; Fig. 7c and d). Of the surrogate variables, E is most strongly correlated with α _i (Table 4; Fig. 7b), although this cannot be explained through a longitudinal trend in D, as assumed in parameterization E, since E and D are not significantly correlated (Table 4). This may be because of longitudinal variability in air-bubble and crack density, which were not recorded.

Development of new parameterizations: snow albedo

A new α _s parameterization is proposed which differs from previous schemes in form and in the treatment of shallow snow covers.

1. 1. In previous work, α _s is calculated as an exponential function of t _a , T _a or d (Table 5). Hence, following snowfall, the calculated α _s decays rapidly to a constant and is invariable on snow > 10 days old (e.g. Reference TangbornTangborn, 1984) (although parameterization E allows for further decrease in α _s with increasing M _a). In reality, however, α _s decreases gradually for several weeks after snowfall (Fig. 6e-g). This may be attributed to increasing I in the snowpack. Therefore, it is more realistic to calculate α _s from a logarithmic relationship in which the initial rapid α _s decrease following snowfall becomes more gradual over time, without reaching a constant minimum value:

   (1)

   

   where p ₁ is the fresh snow α, p ₂ is a parameter and S is a surrogate variable.

2. 2. In contrast to earlier work, which relates α _s to α _u as a function of decreasing d, the relationship of α _s to α _u on shallow snow covers (d < 0.5 cm we.) must be parameterized using surrogate variables which correlate with γ and I, to calculate high values of α _s when the snow is fresh and a transition to α _u as the snow metamorphoses.

Taking the above points into account, the best α _s parameterization was identified from least-squares regression analyses between α _s and all the surrogate variables, using linear, logarithmic and exponential models. The analysis was conducted first for deep snow ≥0.5 cm w.e. deep (α _ds), and second for shallow snow <0.5 cm w.e. deep (α _ss). The largest amount of α _ds variability is explained by a logarithmic relationship to T _a. Additional surrogate variables either violate colinearity or result in no significant increase in r ² (Table 7):

snow depth ≥0.5 cm we.:

(The standard error of each regression coefficient is shown in parentheses.)

In Equation (2), α _ds decreases from 0.71 when T _a = l to 0. 50 when T _a = 80 (Fig. 8a), but further decay in α _ds is very gradual. For example, α _ds = 0.40 is reached only at T _a = 623. Values > 0.71 are calculated when T _a is 0.0–1.0.

Fig. 8. (a) Variation in measured albedo over deep snow (symbols) and albedo calculated from Equation (2) (solid line). (b) Variation in measured albedo over shallow snow (symbols) and albedo calculated from Equation (3) (solid line). (c) Comparison of measured snow albedo and albedo calculated from Equation (4). The straight line indicates a 1:1 relationship.

The largest amount of α _ss variability is explained through an exponential relationship to T _a (Table 7):

snow depth < 0.5 cm we.:

When T _a = 1, calculated α _ss is equal to α _u plus 0.42, but α _ss decays rapidly to α _u as the snow metamorphoses, and is equal to α _u + 0.01 when T _a = 65 (Fig. 8b).

There is discontinuity between Equations (2) and (3) for a snowpack melting down on top of the ice. For example, at T _a = 100, α _s will jump from 0.49 (Equation (2)) to α _u (Equation (3)) when d < 0.5 cm w.e. This is overcome through a transition from α _ds to α _ss as a function of decreasing d:

where d* is a scaling length for d, which was found by optimization to be equal to 2.4 cm w.e. Hence, for d > 5 cm w.e., α _ss has negligible influence on α _s. When d = 1.7 cm w.e. α _ds and α _ss make equal contributions to α _s, and when d < 1.0 cm w.e. α _s is dominated by α _ss. Equation (4) results in a successful parameterization of α _s variability (Table 7). In general, calculated α _s matches measured α _s well (Fig. 8c). The main problems are a tendency to underestimate the high α _s and to overestimate low α _s.

Generally, α _s values calculated from earlier parameterizations must be constrained to ensure that unrealistic values are not calculated when applied outside of the calibration range. Similarly, when T _a ≤ 0, no solution to Equation (2) can be found and α _s values >1.0 may be calculated by Equation (3). Additionally, α _s >1.0 will be calculated by Equation (2) when T _a is very small (e.g. 0.001). Therefore, α _s values calculated from Equations (2–4) should be constrained below an upper limit of 0.85 (the highest α _s recorded), which can be achieved simply in a numerical model.

Development of new parameterizations: ice albedo

The best α _i parameterization was identified through regression between α _i and the surrogate variables using a stepwise least-squares procedure, applying linear and polynomial models. Neither t _a nor M _a accounted for a significant amount of α _i variability. α _i is best parameterized through a second-order polynomial based on elevation (E) (Fig. 9a):

Fig. 9. (a) Variation in measured ice albedo (symbols) and ice albedo calculated from Equation (5) (line) with elevation. (b) Comparison of measured and calculated ice albedo. The line represents a 1:1 relationship.

The correspondence between measured and calculated α _i is moderate, as only longitudinal α _i variation, not crossglacier or temporal variation, is calculated (Table 7; Fig. 9b).

Comparison between albedo parameterizations

The performances of the new α parameterizations (Equations (4) and (5), referred to as parameterization G in this subsection) and previous schemes (Table 5) were compared using a common dataset of α _s and α _i measurements. Half of these data were selected randomly for use as a calibration dataset, from which optimal parameter values for each parameterization were obtained. Parameterization performance was assessed on the basis of the accuracy (as expressed by the r ² and rms error) with which the remaining α values were calculated (the test dataset). Separate calibration and test datasets ensure an impartial comparison of the different schemes. On snow, a dataset of 60 α _s measurements which contained values of all surrogate variables required by the different parameterizations was used initially (Table 8a). However, this dataset contained no measurements on shallow snow, as ρ and M _a (required by parameterizations D and E; Table 5) were generally measured only for deep snow covers. Therefore, the other parameterizations (A, B, C, F and G) were also tested using a larger dataset of 276 α _s measurements from both deep and shallow snow covers, which contained only the required values of t _a, d and T _a (Table 8b). The α _u values required by parameterizations C and G were calculated from Equation (5). On ice, the parameterizations were calibrated and tested using a dataset of 252 α _i measurements, although parameterization E was calibrated and tested using a subset of 90 α _i measurements for which the required M _a values were available (Table 8c). Although not measured specifically, the terms relating to m, θ and n in parameterization D are effectively constants in the Arolla dataset, hence only the term which calculates α _s as a function of ρ was tested.

Results of the comparison: deep snow

All of the parameterizations except A explain a significant amount of α _s variability and have a small rms error (Table 8a). However, parameterization D’s performance is relatively poor, suggesting that ρ does not make a good basis for α _s parameterization. Parameterization E, the most complex scheme tested, performs quite well, but the extra computational effort required is hardly worthwhile when greater accuracy (based on r ² and rms values) is obtained from simple relationships to T _a (parameterizations F and G) or t _a (parameterizations B and C). Parameterization G has the highest r ² and lowest rms error of all, and its calculated α _s range corresponds most closely with the measured α _s range. The α _s range calculated by parameterization F is more restricted. This supports the proposal that α _s varies as a logarithmic (i.e. Equation (2)) rather than exponential function (parameterization F) of T _a.

Results of the comparison: all snow depths

The performance of the new scheme (parameterization G) is a dramatic improvement over that of previous schemes (Table 8b). The r ², rms and α _s range statistics demonstrate that Equation (4) successfully calculates both the α _s variation on deep snow and changes associated with the transition to a thin, discontinuous snow cover. By comparison, the performance of the remaining parameterizations is disappointing. The problem of exponential schemes is highlighted by parameterizations B and F. These produce lower α _s limits of 0.49 and 0.50, respectively, leading to an overestimate of low α _s values. Parameterization C, meanwhile, produces a higher rms error than the assumption of a constant mean α _s (parameterization A), and greatly overestimates the mean α _s. These problems occur because this scheme assumes that α _s approaches α _u primarily as a function of d, rather than due to changes to the snowpack during snow metamorphism.

Results of the comparison: ice

Parameterization G is the only one to explain a significant amount of α _i variation and it has the smallest rms error (Table 8c). Parameterization E explains a small amount of α _i variation, but tends to overestimate α _i. Although not specifically developed to calculate α _i variability, the low r ² values for parameterizations B and F demonstrate the difficulty of calculating temporal α _i variations in terms of t _a and T _a, respectively.

Physical basis of the new albedo parameterizations

Patterns of α _s variation are dominated by increases in γ and I following snowfall. When T > 0°C this is due mainly to snowmelt, which increases γ through the formation of grain clusters and increases I through snowpack depletion. Thus, T _a is a proxy for the amount of melt, and accounts for both temporal α _s variations and increases in α _s up-glacier, associated with lower melt rates. The logarithmic form of the T _a−α _s relationship (Fig. 8a) results from the combined effects of γ and I. The rapid α _s decay of fresh snow can be attributed to γ growth in approximately the range 10–100 μm, while the accumulation of absorptive impurities causes a more gradual and sustained α _s decrease in older snow as the influence of γ growth diminishes.

The finding that α _s varies independently of α _u on snow covers > 0.5 cm w.e. deep is surprising. This occurs because the influence of α _u on α _s is overwhelmed by the impact of snow metamorphism and the accumulation of impurities. According to theory, the α _s of fresh snow will be independent of α _u, even when d is very small, due to efficient light reflection at the snow surface, but the influence of α _u on α _s will strengthen with the increased forward scattering and penetration of light into the snowpack caused by γ growth (Reference Wiscombe and WarrenWiscombe and Warren, 1980). However, it is probable that old snow, which has high concentrations of impurities, is less effective at transmitting light to the underlying ice or debris than theoretical models of pure snow suggest. This explains why α _s and α _u converge on old snow only when d < 0.5 cm w.e. depth. However, the scaling depth, d*, at which the α _s calculation transfers from Equation (2) (independent of α _u) to Equation (3) (dependent on α _u) suggests that α _u does have a small influence in reducing α _s on snow covers up to about 5 cm w.e. deep. The separate parameterization of shallow snow covers enables a much greater amount of α _s variability to be explained, compared with schemes which treat all snow as independent of α _u. Furthermore, the majority of summer snowfall events, which play a significant role in the surface energy balance, have depths of only a few cm w.e.

Calculation of glacier-wide patterns of α _s using a parameterization based on T _a is dependent on the extrapolation of 2 m T measurements. Normally, this is achieved by assuming a linear lapse rate of 2 m T from a weather station located at a proglacial site, or between two weather stations at differing altitudes. However, the distribution of 2 m temperatures along a glacier is complex, due to the thermal influence of the glacier itself and due to locally strong adiabatic heating of the katabatic wind (e.g. at the base of an icefall). The along-glacier profile of 2 m T is better estimated using a wind model, based on glacier length and slope (Reference Greuell and BöhmGreuell and Bohm, 1998). However, it is useful to consider the error in α _s values calculated from Equation (4) if a standard lapse rate (i.e. 6.5 K km⁻¹) is used to extrapolate 2 m T along-glacier from a proglacial site, as is common procedure in surface energy-balance models. A standard lapse rate may overestimate the true 2 m T by as much as 5 K in warm weather (e.g. a 2 m T in the proglacial area of 288 K) (Reference Van den BroekeVan den Broeke, 1997). In this case, the resulting α _s underestimate on deep snow (Equation (2)) is only 0.02 (0.58 instead of 0.60) on the day following a fresh snowfall, and remains 0.02 if 2 m T is overestimated by 5 K on subsequent days. On shallow snow (Equation (3)) the α _s underestimate is 0.06 (α _u + 0.39 instead of α _u + 0.45) on the day following a fresh snowfall, but reduces to 0.03 if 2 m T is overestimated by 5 K for several days. Therefore, even very large 2 m T errors have a minor impact on α _s estimated using Equation (4). Furthermore, the high temperatures which may result in a large 2 m T overestimate, and a large underestimate of fresh snow α _s in Equation (3), are unlikely to occur immediately after a snowfall.

Comparison of different parameterizations showed that α _s variability may be successfully calculated as a function of t _a. M _a should also be a good predictor of a_s since increases in γ and I are caused by melt. However, M _a correlates with α _s and γ less strongly than T _a does (Table 3). This might be due to errors in converting measurements of surface lowering at ablation stakes into M _a, caused by ρ variation with depth. In any case, M _a makes a poor basis for α _s parameterization in numerical models. First, the accuracy with which M _a values may be estimated across a glacier is subject to errors in the extrapolation of several meteorological and surface (e.g. aerodynamic roughness) variables. Second, the calculation of M _a in an energy-balance model depends on prior knowledge of α _s. Thus, a positive feedback loop may be generated in which an error in the initial α _s estimate would cause an error in the calculation of M _a, which in turn would result in an enlarged α _s error and so on.

A more general problem for surface melt models is the accurate estimation of α in the vicinity of the transient snowline, assuming model gridcells cover areas of > 100 m². While, for a given cell, calculated d might be a few cm, in reality the area covered by the cell would be a mosaic of snow patches and bare ice. Hence, α would tend to be overestimated. A possible way to treat this problem is to allow d to vary around the mean from cell to cell to simulate patchy snow cover.

On ice, D accounts for >40% α _i variability. The remainder is probably caused by variability in air-bubble content and crack density, which are difficult to quantify, α _i was generally <0.30, lower than the Greenland ice-sheet margin (Reference Konzelmann and BraithwaiteKonzelmann and Braithwaite, 1995) and Morteratschgletscher, Greenland (personal communication from W. H. Knap, 1999), but similar to Hintereisferner, Austria (Reference Van de Wal, Oerlemans and van der HageVan de Wal and others, 1992), and Peyto Glacier, Canada (Reference Cutler and MunroCutler and Munro, 1996). This implies that D, air-bubble content and crack density vary between glaciers.

The α _i parameterization, based on E, explains slightly more than a quarter of total α _i variability. Albedo measurements derived from 1993 Landsat imagery of Haut Glacier d’Arolla also demonstrate a weak dependency of α _i on E during August, but the longitudinal α _i variability is less erratic than in Figure 4b and c (Reference Knap, Brock, Oerlemans and WillisKnap and others, 1999). Satellite measurements do not resolve small-scale spatial variability in α _i, but represent the mean α _i over large (e.g. 30 m × 30 m) areas more accurately. The satellite measurements are also able to delineate longitudinal features such as medial moraines and debris bands. As a result, cross-glacier variability appears to be even stronger in the Landsat-derived α _ii measurements than in Figure 2d and e. It may therefore be questioned whether Equation (5) is worth the extra computational effort compared with using a constant mean value for α _i, which will suffer from similar magnitude errors.

Parameterization assumptions

Most earlier parameterizations make implicit assumptions about the physical controls on α variation. Some of these assumptions are found to be unsubstantiated when tested with the field measurements collected in this study.

1. 1. The common assumption that α _s systematically approaches α _u with decreasing snow thickness is not substantiated by the field measurements or theory. The effects of metamorphism and accumulation of impurities in the snow dominate over the influence of α _u until the snow cover has all but melted away. Furthermore, d is not significantly correlated with γ, and its correlation with I is weaker than other surrogate variables. Hence, there is little basis for calculating α _s as a function of d, and, when tested with field data, parameterizations based primarily on d are relatively poor at estimating α _s variability.

2. 2. Many previous α _s parameterizations have an exponential form reflecting the behaviour of a pure, deep snowpack. However, this ignores the impact of absorptive impurities, with the result that exponential parameterizations overestimate α _s on old snow. Exponential forms can be “forced” to calculated lower α _s values, but this seriously diminishes the overall parameterization accuracy.

3. 3. Trends of decreasing α _i over an ablation season are generally assumed to be associated with increasing concentrations of dust and debris on the exposed ice surface (e.g. Reference OerlemansOerlemans, 1993). The α _i increase observed between July and August 1994 demonstrates that such trends are not universal. Clearly, there must be additional physical mechanisms which can cause α _i to increase over time, most likely through removal of surface debris by rainfall and runoff. Parameterizations making the assumption that α _i decreases continuously throughout the ablation season will overestimate α _i following an event in which fine debris is washed from the glacier surface.

4. 4. Our measurements do not show the strong dependence of α _i on E which is often assumed. Given the probable differences in D, air-bubble content and crack density between glaciers, universal relationships between a_i and E are unlikely to exist.