2.1. Study sites

Haut Glacier d’Arolla (referred to here as Arolla) and Gornergletscher (referred to here as Gorner) differ in terms of topographical characteristics, areal extent and altitudinal range (see Table 1), but are located in an area of similar general climatic features.

2.2. Meteorological and topographical data

For each glacier, we used meteorological data from one off-and one on-glacier AWS (Fig. 1). On Haut Glacier d’Arolla, we used data from the so-called lowest AWS of the stations described in detail by Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005). Records are available for the 2001 and 2006 seasons (Table 2). On Gornergletscher, an AWS was set up at about 2600 m a.s.l. in both 2005 and 2006. The period of functioning was shorter in 2005 (Table 2). The glacier AWSs recorded data for a period of 5 min (average of 5 s measurements) of ventilated air temperature (°C), relative humidity (%), incoming and reflected shortwave radiation (W m⁻²), wind speed (m s⁻¹) and direction (°). All sensors were set up on an arm fixed to a tripod that sat on the glacier surface and was allowed to sink with the melting of the glacier surface, thus maintaining a nominal height of 2 m between the surface and sensors. Measurements of incoming and reflected shortwave radiation were therefore made parallel to the surface following Reference Sicart, Ribstein, Wagnon and BrunsteinSicart and others (2001), Reference Greuell, Genthon, Bamber and PayneGreuell and Genthon (2004) and Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005).

A permanent station situated at about 900 m from Arolla terminus (Fig. 1) has been recording hourly values of ventilated air temperature, among other variables, since 2001. A permanent off-glacier AWS measuring ventilated air temperature and relative humidity was in use on Gornergletscher in 2005 and 2006 (Fig. 1). Observations of air temperature recorded at these two off-glacier stations are used together with data from the glacier stations for analysis of cloud-cover variations and development of the cloud transmittance factor parameterization.

Station characteristics and period of functioning are listed in Table 2. Four main datasets are used: Arolla 2001, Arolla 2006, Gorner 2005 and Gorner 2006. The Haut Glacier d’Arolla dataset is described in detail by Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005), while the Gornergletscher measurements are discussed by Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others (2009). All meteorological data were aggregated into hourly values before being used for the analysis.

The solar radiation model described below requires a digital elevation model (DEM) of the glacier and surrounding area for computation of the direct incoming shortwave radiation and interaction of the sunbeam with the local topography, in particular to take into account the shading effect. We used a 10 m DEM of Arolla derived from aerial photographs taken in 1999 (Reference PellicciottiPellicciotti, 2004) and a 25 m DEM of Gornergletscher, also derived from aerial photographs (Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others, 2009).

2.3. Cosmo data

The non-hydrostatic limited-area atmospheric model COSMO-7 (http://www.meteoschweiz.admin.ch/) provides detailed climate forecasts for the entire Alpine region at a grid spacing of 2.2 km. We used data extracted from a reanalysis (i.e. continuous integration with continuous assimilation of observations) that started on 1 January 2000 and ended on 31 December 2002, with lateral boundary conditions taken from the global European Centre for Medium-Range Weather Forecasts (ECMWF) model (http://www.ecmwf.int/). COSMO uses terrain-following coordinates and 45 vertical layers of variable height above sea level, hence a variable resolution of layers (increasing from the surface from 12.6 to 2235.4 m). It employs the radiative transfer scheme by Reference Ritter and GeleynRitter and Geleyn (1992), which is based on the solution of the 6-two stream version of the radiative transfer equation incorporating the effect of scattering, absorption and emission by cloud droplets, aerosols and gases in each part of the spectrum (Reference Ritter and GeleynRitter and Geleyn, 1992). We used the modelled 2 m air temperature, incoming shortwave radiation and total cloud cover for the grid of latitude 45°968′ N and longitude 7°532′ E, covering most of Haut Glacier d’Arolla. Mean elevation of the gridcell is 2906 m a.s.l. (similar to the mean elevation of the glacier tongue) and the grid surface type is ice. The solar radiation is with respect to a horizontal surface, and shading by the surrounding topography is not accounted for.

Figure 2 shows hourly incoming shortwave radiation measured at the lowest AWS on Arolla together with hourly incoming shortwave radiation modelled by COSMO for the period 17–27 July 2001. Radiation from the limited-area climate model is higher than the observed radiation by 49.4 W m⁻² on average over the common period of record, most frequently with overestimation in the central part of the day (Fig. 2). The model is able to correctly reproduce cloudy conditions on the two overcast days 20 and 24 July (Fig. 2).

Fig. 2. Incoming shortwave radiation measured at the lowest AWS on Arolla (solid curve) and modelled by the COSMO limited-area climate model in the gridcell corresponding to Arolla (dotted curve) for the period 17–27 July 2001.

Modelled air temperature agrees quite well with the 2 m air temperature at the lowest station on Arolla, but is too low when compared with the temperature time series at the proglacial station. The diurnal amplitude, however, is more like that of the off-glacier temperature record, so the damping effect of the 0°C glacier surface on the temperature variability does not seem to be completely included in the model.

3.1. Solar radiation model

Clear-sky incoming solar radiation is computed using a non-parametric model based on Reference IqbalIqbal (1983) and further modified by Reference Bird and HulstromBird and Hulstrom (1981) to best reproduce the output of the rigorous solar radiation model SOLTRAN. This was shown to be one of the best available approaches in the comparative study by Reference Niemelä, Raisanen and SavijärviNiemelä and others (2001), outperforming three physically based radiation schemes used in numerical weather prediction models. The model computes the transmittance through the atmosphere after attenuation due to Rayleigh scattering, uniformly mixed gases, water vapour, aerosols and ozone. To derive the terrain parameters and solar position from the DEM and the interaction between the solar radiation and the topography, we use the vectorial algebra approach proposed by Reference CorripioCorripio (2003b). In the following, we report only the main equations and refer the reader to Reference CorripioCorripio (2003b) for details.

The fraction of light intercepted by the inclined surface is proportional to the cosine of the angle between the normal to the surface and the sun’s rays. Hillshading is computed by checking the projection of a gridcell over the plane perpendicular to the solar rays, following the direction of the sun (Reference CorripioCorripio, 2003b). The sky-view factor, Ψ, defined as the hemispherical fraction of unobstructed sky visible from any point, is important in the evaluation of incoming diffuse radiation (I _d). Ψ is computed as the ratio of the projected surface of the visible part of the hemisphere to the area of the whole hemisphere of unitary radius. Clear-sky potential incoming shortwave radiation I _POT is computed as

where I _n is the direct normal irradiance and Ψ is the angle between the normal to the surface and the sun’s rays.

I _n is calculated as

where the factor 0.9751 is the ratio of the extraterrestrial irradiance in the spectral interval 0.3–3.0 μm (used by the SOLTRAN model) to the solar constant l _sc (Reference Bird and HulstromBird and Hulstrom, 1981; Reference CorripioCorripio, 2003a);

is the extraterrestrial solar radiation or solar constant corrected for the eccentricity of the Earth’s orbit, where R ₀ is the actual Sun–Earth distance and R is the mean Sun–Earth distance; and the τ functions are transmittance functions for Rayleigh scattering (τ _r) and transmittance due to ozone (τ _o), uniformly mixed gases (τ _g), water vapour (τ _w) and aerosols (τ _a). The total transmittance of the atmosphere is thus accounted for as the product of single transmittance. β(z) is a correction term for altitude, introduced following Reference BintanjaBintanja (1996) to take into account the increase of transmittance with altitude:

where z is altitude. While β(z) is strongly linear up to 3000 m, it is fairly constant and equal to 2:2 × 10⁻⁵ m⁻¹ after 3000 m. The computation of the individual transmit tance follows Reference Bird and HulstromBird and Hulstrom (1981) and requires knowledge of the relative optical air mass, m _r. Reference IqbalIqbal (1983) suggested a formula which approximates values from a transmittance model:

where θ _d is the zenith angle in degrees, or θ(180/2π). This approximation is for standard pressure, and can be corrected for local pressure p_z :

where p_z is computed as explained by Reference CorripioCorripio (2003a).

Transmittance by ozone is calculated as

where l is the vertical ozone layer thickness (cm), assumed to be 0.35 cm according to data from the Total Ozone Mapping Spectrometer Earth Probe (TOMS-EP 2001). Transmittance by uniformly mixed gases is computed as

Transmittance by water vapour is:

where w _p is precipitable water (cm). This is calculated using an empirical equation proposed by Reference PrataPrata (1996):

where T _a is the screen-level air temperature (K) and e ₀ is the actual vapour pressure, defined as e ₀ = e ^* × RH where RH is relative humidity within the range 0.1–1.0 and e ^* is saturated vapour pressure computed using Reference LoweLowe’s (1977) polynomials.

The transmittance by aerosols is not easy to determine when direct information on atmospheric turbidity is not available (Reference CorripioCorripio, 2003a). Here it is computed as a function of visibility v (km) which was estimated.

The diffuse component of solar radiation is also computed following Reference IqbalIqbal (1983). The total diffuse radiation l _d is the sum of the Rayleigh-scattered diffuse irradiance l _dr, the aerosol-scattered diffuse irradiance l _da and the diffuse irradiance from multiple reflections between the Earth and the atmosphere l _dm (Reference CorripioCorripio, 2003a):

The Rayleigh-scattered diffuse irradiance is given by

where τ _aa is the transmittance of direct radiation due to aerosol absorptance. The aerosol-scattered diffuse irradiance is

where τ _as = τ _a/τ _aa and F_c is the fraction of forward scattering to total scattering (assumed 0.84 after Reference IqbalIqbal (1983) as no information on aerosol was available in this study).

The diffuse irradiance from multiple reflections between the Earth and the atmosphere is calculated as:

where α _g is the albedo of the ground which can be estimated from knowledge of the average snow cover around the point considered, as explained by Reference Greuell, Knap and SmeetsGreuell and others (1997). Since this information is not available in our study, we assume a mean value of 0.4. α^’ _α is the atmospheric albedo, computed after Reference IqbalIqbal (1983) as

Finally, incoming shortwave radiation is obtained from I _POT (Equation (1)) as

where cf is a bulk transmittance due to clouds, referred to as the cloud transmittance factor in this paper.

3.2. Computation of cloud transmittance factors

Cloud transmittance factors within the range 0–1 reduce the potential incoming solar radiation of a quantity that corresponds to the amount of clouds present and type of clouds (Reference Zhang, Stamnes and BowlingZhang and others, 1996; Reference Garen and MarksGaren and Marks, 2005; Reference Anslow, Hostetler, Bidlake and ClarkAnslow and others, 2008). A value of cf = 1 corresponds to a clear sky with no clouds, whereas cf = 0 means that no shortwave radiation reaches the surface at all. In general, however, an overcast sky will have cf > 0 (Reference Garen and MarksGaren and Marks, 2005; Reference Anslow, Hostetler, Bidlake and ClarkAnslow and others, 2008), with different values > 0 for different cloud types. Reference Anslow, Hostetler, Bidlake and ClarkAnslow and others (2008), for instance, used cf = 0.4 for completely cloudy conditions for South Cascade Glacier, Washington, USA.

Cloud transmittance factors are computed from hourly clear-sky radiation obtained with the model described above and from the measurements at the AWSs. Following Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005), we first derive hourly cloud transmittance factors cf _H as the ratio of hourly measured (I _MEAS in W m⁻²) to the hourly modelled incoming shortwave radiation (I _POT, also in W m⁻²):

during the day, i.e. when the sun is above the horizon. We compute cf _H only when measured radiation is above a threshold of 120 W m⁻², in order to exclude the low values of the early and late hours of the day, when measurements might be affected by errors and small differences between measured and modelled radiation can result in anomalous values of the cloud transmittance factors. During the night, when by definition the cloud transmittance factor cannot be computed, it is assumed to be constant and equal to the mean cloud transmittance factor of the previous afternoon.

We calculate hourly cloud transmittance factors because they permit derivation of weighted daily cloud transmittance factors. These are calculated by weighting each hourly cf _H according to the importance of the individual hourly cf _Hs to the total daily shortwave radiation flux.

The weighting factors are computed as the ratio of the incoming shortwave radiation in that hour to the daily total incoming shortwave radiation (Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others, 2005). In this way, each hourly cf _H is weighted according to the average theoretical incoming shortwave flux in that hour. In comparison to the arithmetic daily cloud transmittance factor, weighted daily cloud transmittance factors are much more dependent on conditions during the middle part of the day (which dominate the total shortwave receipts on a given day (Reference Mölg, Cullen and KaserMölg and others, 2009)). This implies that the early and late parts of the day, which contribute little to shortwave flux, have little impact. In fact, if the early and late hours of the day are cloudy, this does not affect the total shortwave radiation input for that day very much. In the arithmetic mean daily cloud transmittance factor, all hourly cfs are weighted equally, which seems wrong. We therefore suggest that the weighted cloud transmittance factor is a better indicator of the effects of clouds on shortwave radiation receipts.

3.3. Cloud transmittance factor parameterization

Following Reference Bristow and CampbellBristow and Campbell (1984), Reference PellicciottiPellicciotti (2004) developed a cloud transmittance factor parameterization based on temperature variations during the day. This approach is based on wide evidence that clouds primarily affect the diurnal variations of 2 m air temperature (Reference Dai, Trenberth and KarlDai and others, 1999) and that the daily temperature range can therefore be used as a predictor of changes in cloud radiative forcing (Reference Bristow and CampbellBristow and Campbell, 1984; Reference TangbornTangborn, 1984).

Reference PellicciottiPellicciotti (2004) proposed the following parameterization:

where cf is the daily cloud transmittance factor discussed above, ΔT is the daily temperature range and a and b are two empirical coefficients computed from the data (a = 1 and b = 0.1452). Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005) suggested an alternative linear parameterization,

noticing that differences in the predictive skills of the two functions were very low. The optimized empirical coefficients of Equation (18) are a = 0.0600946 and b = 0.3097. In both cases, ΔT is computed from air-temperature data measured outside the glacier boundary layer, which are more representative of the synoptic air-temperature conditions (Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others, 2005). Temperature measured on the glacier is in fact affected by both the presence of a surface at melting point and by the glacier wind, which contribute to lower air temperature and smooth its diurnal fluctuations. This has already been demonstrated by Reference PellicciottiPellicciotti (2004) and Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005), and was confirmed by our analysis.

For each site and season, we recalibrated Equations (17) and ( 18) with the cloud transmittance factors computed from measured and modelled incoming shortwave radiation. However, to identify alternative parameterizations, we analyze the relationship of computed cloud transmittance factors to all the meteorological variables available.

3.4. Melt models

Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005) used the parameterization described in section 3.3 in an enhanced temperature-index (ETI) model (Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others, 2005, Reference Pellicciotti2008; Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others, 2009) for simulations of hourly melt rates. These are computed as a function of air temperature and the shortwave radiation balance, i.e.

where M is the hourly melt rate (mm w.e. h⁻¹ ), T is air temperature (°C), α is albedo and I is incoming shortwave radiation (W m⁻²). The temperature factor TF (mm h⁻¹ °C⁻¹) and the shortwave radiation factor SRF (m² mm W⁻¹ h⁻¹) are empirically determined and were calibrated against hourly energy-balance melt rates (Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others, 2005). The threshold temperature for melt to occur, T _T, is 1°C (Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others, 2005). We do not recalibrate the model parameters, because they were calibrated at one of the locations investigated in this work and because they have been shown to be robust for Swiss Alpine glaciers (Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others, 2009).

An energy-balance model provides hourly values of computed ablation rate that we use to evaluate the melt rates simulated by the ETI model. The details of the energy-balance model can be found in Reference PellicciottiPellicciotti and others (2008) and Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others (2009); they are not described here because the model has been extensively tested for Alpine sites (Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others, 2009; Reference Pellicciotti, Carenzo, Helbing, Rimkus and BurlandoPellicciotti and others, 2009).

4.1. General meteorological conditions

Figure 3 shows the daily mean values of air temperature and relative humidity at Arolla’s lowest AWS in 2001 and 2006. Air temperature on the glacier is on average lower than at the off-glacier AWS because of the combined effect of the katabatic wind (Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others, 2009) and of the 0°C melting surface. There is clear anticorrelation between low air temperature and high relative humidity. The colder season (2001) shows the highest record of precipitation (Table 3). The main difference between Arolla and Gorner is the amount of precipitation, which is much higher on Arolla (Table 3). Air temperature at Gorner AWS is higher than at Arolla (Table 3), but this is at least partly explained by the lower elevation of the Gorner AWS (Table 2).

Fig. 3. Daily mean air temperature (a, b) and relative humidity (c, d) at Arolla 2001 (a, c) and Arolla 2006 (b, d) for the entire season. Blue indicates measurements on the glacier and red off the glacier.

4.2. Solar radiation model

Agreement between clear-sky modelled and measured hourly incoming shortwave radiation is high on average (Fig. 4). Comparison between modelled and measured radiation can be made only for clear-sky conditions. Differences in the simulated and observed time series also exist for clear-sky conditions, however, as indicated by the values of the efficiency criterion R ² (Reference Nash and SutcliffeNash and Sutcliffe, 1970; Table 4), and are likely attributable to topographic effects.

Fig. 4. Mean hourly incoming shortwave radiation I, modelled by the parametric clear-sky solar radiation model (blue) and measured (black) at Arolla lowest station on 15 and 16 June 2006.

Figure 5 shows the scatter plot of measured and modelled hourly incoming shortwave radiation at the two sites for the period of record for clear-sky conditions only. As expected, differences are higher early in the morning and later in the afternoon (blue and red dots). These are probably due to inaccuracies in the DEM and related uncertainty in the modelling of the interaction with the surrounding topography and of shading in particular. Discrepancies between modelled and measured radiation are relatively small at midday, when solar radiation is highest (Fig. 5). The mean difference over the period 1000–1600 h is 34 W m⁻² for Arolla 2001, 6 W m⁻² for Arolla 2006, −20 W m⁻² for Gorner 2005 (a mean negative value indicating that measured radiation is higher than the modelled) and 29 W m⁻² for Gorner 2006.

Fig. 5. Measured (I _MEAS) versus modelled (I _MOD) hourly incoming shortwave radiation for clear-sky days only over the period of record at the lowest AWS on (a) Arolla 2001 and 2006 and (b) Gorner 2005 and 2006. Colour codes indicate the hours between 800 and 1000 h (blue), 1200 and 1400 h (black) and 1600 and 1800 h (red). Also indicated is the 1 : 1 line.

At the sites considered in this study, the occurrence of cloudy conditions varies between 36.4% for Gorner 2005 and 47.6% for Arolla (Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others, 2009), representing a large percentage of the total period of record and therefore affecting the accuracy of the solar radiation model. The values above and those plotted in Figure 5 are for clearsky days only. These were identified using an algorithm from Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others (2009). Given the large percentage of overcast conditions, however, the algorithm might have failed on a few days, especially those with low cloud cover concentrated over a few hours of the day. This might explain the two high peaks in the maximum difference, which are visible in both the Arolla and Gorner records.

Differences between measurements and estimated clearsky solar irradiance can be caused by low zenith angle, as pointed out by Reference Pfister, McKenzie, Liley, Thomas, Forgan and LongPfister and others (2003). They restricted solar zenith angles to 75° or less in the calculation of the ratio of measured to estimated clear-sky values, because the quality of the fit between the two quantities diminishes for low-elevation sun angles.

4.3. Cloud transmittance factor: regression analysis

Figure 6 shows the computed daily cloud transmittance factors cf together with incoming solar radiation and relative humidity (at the glacier AWS) and the daily temperature range ΔT (at the off-glacier AWS) for Arolla 2006. As expected, cloud transmittance factors are clearly correlated with incoming shortwave radiation (Table 5).

Fig. 6. Daily cloud transmittance factors, together with daily mean solar radiation I (glacier station), relative humidity RH (glacier station) and temperature range ΔT (off-glacier station) for Arolla 2006.

Very low cloud transmittance factors, as on 1, 6 and 19 August, correspond at all sites to high relative humidity (Fig. 6), whereas the correspondence is less clear for higher cf. Correlation with the relative humidity record varies between −0.366 for Arolla 2001 and −0.640 for Gorner 2005. It is clear that the two variables are negatively correlated, since low cloud transmittance factors (corresponding to overcast conditions) are associated with high relative humidity. With the exception of Gorner 2005, correlation with ΔT is stronger (Table 5).

On average, daily cloud transmittance factors are higher on Gornergletscher (Table 6). The low average value for Arolla 2001 is associated with stronger variations over the seasons than at the other sites (standard deviation is higher; Table 6). Besides this, however, higher cloud transmittance factors on Gornergletscher reflect the drier conditions that are typical of Gornergletscher and the surrounding area, despite the fact that the site is only tens of kilometres away from Arolla (Fig. 1).

For the cloud transmittance factors computed above, we examined correlation with the following variables at all sites: minimum, maximum and mean air temperature; minimum, maximum and mean relative humidity; and daily precipitation. For air temperature and temperature range, we correlated the daily cloud transmittance factor with both on-and off-glacier values. Correlation is stronger with the daily temperature range at all sites (Fig. 7). Also relatively strong is the relationship with minimum relative humidity, in the sense that low values of minimum daily relative humidity seem to be good indicators of clear-sky conditions while the scatter in the relationship increases for low cf. Correlation with temperature measured at the glacier AWS is weaker, as shown already by Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005) and discussed in detail by Reference PellicciottiPellicciotti (2004), because of the role of the glacier boundary layer in dampening diurnal temperature fluctuations (Reference Greuell and BöhmGreuell and Böhm, 1998; Reference Klok, Nolan and van den BroekeKlok and others, 2005; Reference PellicciottiPellicciotti and others, 2008). Interestingly, temperature alone (i.e. the minimum, maximum or mean values alone) are no indicator of cloud cover and correlation is very poor with time series both on- and off-glacier. Daily precipitation shows inverse correlation with cf, but the relationship has a rather large scatter. These relationships are consistent across sites and seasons, with only small variations in the shape of the clouds of points. The relationship of cf to precipitation is anomalous on Gornergletscher in 2006, but this is likely explained by the very small amount of precipitation that fell in that season (mean value of 78 mm w.e. as compared to 275 mm w.e. in 2005 and 474 and 342 mm w.e. on Arolla in 2001 and 2006, respectively).

Fig. 7. Daily weighted cloud transmittance factors against daily temperature range at (a) Arolla 2001, (b) Arolla 2006, (c) Gorner 2005 and (d) Gorner 2006. Red dots indicate clear-sky days and blue dots cloudy days.

Similarly, large scatter is visible in the relationship with minimum relative humidity at Arolla in 2001. This might be explained by the large variation in the relative humidity data at the proglacial station (the dataset shows the largest variance of all datasets).

4.4. Cloud transmittance factors: parameterizations

Following Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005), we tested the following regression equations: linear, polynomial, exponential and Gaussian (Table 7). Given the results in section 4.3 above, we performed a regression analysis using as independent variables both the daily temperature range and minimum relative humidity and used the coefficient of determination as an indication of the strength of the regression.

Figure 8 shows the best-fit regression functions for the pool of the four observation series together with the original data. Table 7 lists the coefficients of determination for the regression functions with daily temperature range as an independent variable for the four single datasets. The coefficients of determination for the regression with relative humidity are lower for all five models and are not reported here. The goodness-of-fit varies between 0.327 (Gornergletscher 2005, linear function) and 0.539 (Gornergletscher 2006, Gaussian function) (Table 7), and at all sites it is higher for the Gaussian and polynomial regression (as expected given the larger number of parameters; Table 7). This means that the regression models can explain a percentage variance in the computed daily cloud transmittance factors that ranges between 32.7% and 53.9%, depending on the model. A noteworthy difference is that the exponential functions yield lower cloud transmittance factor values at small ΔT (1–3°C). This might be a limitation of this specific function, and other functions such as the Gaussian might be best suited to reproduce outcomes at the limits of the range if such values are important.

Fig. 8. Scatter plots of daily cloud transmittance factors cf against daily temperature range for all four datasets pooled together. Also shown are the linear (red), polynomial (blue), Gaussian (pink) and exponential 1 and 2 (green and orange, respectively) regression functions that best fit the data.

Since no simple curve can replicate all the variability displayed by the daily cloud transmittance factors, we analyzed the errors of the regression models. Figure 9 shows the distribution of errors for the four case studies, where it is clear that parameterized cloud transmittance factors underestimate the measured cloud transmittance factors (the highest frequency of errors is between −0.1 and 0). From Figure 10 it is also clear that the underestimation occurs for high cloud transmittance factors (corresponding to clear-sky conditions) and this in turn results in an underestimation of incoming solar radiation on clear-sky days (in particular for Gorner 2005). This issue is discussed in more depth in section 4.5. This limitation, however, could be at least partly overcome by employing different functions for various temperature ranges through the use of stacked functions. It should also be noticed that the shape of the function and its ability to reproduce correctly high cf is influenced by outliers, so they should be considered with care before performing the regression.

Fig. 9. Distribution of errors between ‘measured’ (i.e. computed from measurements of incoming solar radiation) and parameterized cloud transmittance factors at (a) Arolla 2001, (b) Arolla 2006, (c) Gorner 2005 and (d) Gorner 2006. Differences are computed as parameterized minus measured cloud transmittance factor.

Fig. 10. Distribution of cloud transmittance factors: measured (i.e. computed from measured and modelled shortwave radiation) and parameterized with five different regression functions (linear, polynomial, Gaussian and exponential 1 and 2) at (a) Arolla 2001, (b) Arolla 2006, (c) Gorner 2005 and (d) Gorner 2006.

4.5. Influence of parameterizations on melt modelling

In order to test the influence of the various parameterizations on modelled melt rate, we run the ETI model (Equation (19)) with the parameterizations described above for the Arolla 2006 (26 May to 30 September) and Gorner 2005 (4 June to 15 September) datasets. We implemented the linear and exponential 2 parameterization of Table 7, given that differences between all parameterizations are small and that these require only one or two coefficients. We applied both parameterizations with: (1) the parameters optimized for the single location; (2) the mean parameters for the four test sites (Table 7); and finally (3) the parameters calibrated by Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005). We also report results of melt simulations with measured incoming shortwave radiation, which correspond to the best model performance. As in Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005), we use energy-balance simulations as reference melt (Reference PellicciottiPellicciotti and others, 2008; Reference Carenzo, Pellicciotti, Rimkus and BurlandoCarenzo and others, 2009) and use the Reference Nash and SutcliffeNash and Sutcliffe (1970) R ² to evaluate the model performance.

Table 8 lists the R ² for hourly melt computed for the entire season with the ETI model forced by measured incoming solar radiation and by the ETI model with the exponential 2 and linear parameterizations. The main result is that there is hardly any difference in model performance associated with varying parameterization or parameters set. The ETI model performance decreases (from 0.943 and 0.935 to values in the range 0.815–0.824 and 0.721–0.765 for Arolla 2006 and Gorner 2005, respectively) when modelled incoming shortwave radiation and parameterized cloud transmittance factors are used for melt computation, depending on the parameterization. The decrease is stronger for Gornergletscher 2005, with the lowest model performance associated for both sites with the original parameters of Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others (2005).

The decrease in model performance indicated by the R ² is due to the underestimation of hourly melt rates by the ETI model on clear-sky days, especially in the hours around noon (Fig. 11). Interestingly enough, the model works very well on overcast days such as 16, 17 and 23 June 2006 (Fig. 11) when agreement with the energy-balance model is very good. This is despite the fact that the model cannot reproduce the sub-daily variations due to passing clouds (e.g. on 16 and 17 June). The underestimation of hourly melt rates on clear-sky days leads to an underestimation of total melt ranging between 7% and 13% of total energy-balance ablation (Table 9) and between 2% and 7% of total model D (ETI model with measured incoming shortwave radiation) melt for Arolla 2006. Values are higher for Gorner 2005 despite the fact that agreement between energy balance and model D total melt is almost perfect, and higher than on Arolla 2006 (0.9% compared to −7.2%). If the ETI model is run with the cloud transmittance factors from the regression functions, however, the difference from the reference melt rates is higher (Table 9).

Fig. 11. Hourly melt rates computed with the energy-balance model (reference), the ETI model with measured input data and the ETI model with various forms of the cloud transmittance factor parameterization (see text) for the period 14–24 June 2006 on Arolla lowest station. Notice the cloudy days on 16, 17 and 23 June.

Assessment of the impact of the cloud parameterization on the performance of the temperature-index melt model has to take into account the fact that the ETI model slightly underestimates hourly ablation rates (Reference Pellicciotti, Brock, Strasser, Burlando, Funk and CorripioPellicciotti and others, 2005). The ETI model empirical parameters have not been recalibrated for the single sites and seasons, which explains some of the difference between the ETI and measured data and the energy-balance model.

4.6. Modelling incoming shortwave radiation and melt using COSMO

Output from weather prediction models offers an alternative to the approach described above in two ways. We can use simulations of hourly air temperature to derive the daily temperature range used in regression analysis to predict the daily cloud transmittance factor (similar to the approach based on in situ air-temperature observations) or we can use incoming shortwave radiation simulated by the climate model directly into the melt model. Both outputs are for a gridcell of 2.2 km resolution, i.e. encompassing the whole of Arolla.

We have tried both approaches in our search for an alternative to the method explored at length in this paper. We first applied COSMO gridded air temperature to derive a time series of daily temperature range and then regressed this against computed daily cloud transmittance factor. The main result is that there is no clear advantage in using air temperature from the regional climate model (but also no clear disadvantage). We obtain the same type of functional relationship between the dependent variable daily cloud transmittance factor and the independent variables ΔT and RH. Maximum, mean and minimum temperature alone, wind speed or precipitation do not show any clear relationship to the air-temperature range, as was evident for the in situ ground measurements.

The strength of the regression (for both types of regression equation) with temperature range is lower for COSMO data as compared to the ground measurements (with coefficients of determination ranging from 0.247 to 0.326), and results in a slightly lower predictive ability. Minimum relative humidity is, after the air-temperature range, the best predictor of variations in the daily cloud transmittance factor. In this case, however, the regression function explains a lower percentage of the variability in the cloud transmittance factor than do the ground data.

The second way in which the outputs of the regional model can offer an alternative to the cloud transmittance factor parameterization based on ground measurements is to directly use the incoming shortwave radiation simulated by COSMO as input to the ETI model. We tried this and found that the R ² is higher than when modelled clear-sky solar radiation is used in combination with parameterizations of the cloud transmittance factor (whether based on ground temperature measurements or COSMO temperature outputs). Total melt at the end of the season also shows much better agreement with the energy-balance reference total melt, with <1% difference. This good performance, however, is due to a compensation of errors, as is clear from comparison of measured and COSMO incoming shortwave radiation (Fig. 2): solar radiation simulated by COSMO overestimates measured incoming radiation by on average about 50 W m⁻².