3.1. The available images

In view of the possibility of substantial changes in the surface albedo from day to day, we limited the comparison of the helicopter and satellite albedos to satellite data acquired on the days of the experiments. This requirement was fulfilled by some AVHRR and MODIS images. Unfortunately other, possibly interesting, satellite data such as TM, MISR and Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) images, all of which have a coverage of less than once per day, were not available for the days and locations of the experiments.

Table 2 lists the acquisition times of the AVHRR and MODIS images used in the present study, as well as the solar view geometry (see Fig. 5) and resolution of the region of the helicopter experiments on these satellite images (note that the resolution varies across AVHRR and MODIS images). There is considerable variation in the view zenith angle and the relative azimuth angle whereas the range in solar zenith angles is rather limited (42-56° for Vatnajokull and 45-49° for the K-transect).

Fig. 5. Coordinate system used to describe the solar-view geometry. The system consists of the solar zenith angle (θ _s), the view zenith angle (θ _v) and the relative azimuth angle (ϕ). The latter is defined relative to the solar principal plane, i.e. the Sun is at ϕ = 0°. Consequently, forward scattering corresponds to ϕ = 180° and back scattering to ϕ = 0°.

Table 2. Time of acquisition, solar-view geometry and approximate resolution of the AVHRR (across-track-along-track) and MODIS data used for comparison with the helicopter albedos

Three AVHRR images were available for each experiment (six images in total), but in four (two for both experiments) of these images, the area of the experiment was viewed from relatively large zenith angles (>54°). The original AVHRR data were resampled to a resolution of 1.0 km, which is slightly finer than the resolution of the AVHRR data for nadir view (1.1 km), so that no information was lost. The AVHRR data were all obtained from the NOAA-14 satellite and delivered as uncalibrated counts.

Because MODIS was launched after the experiment in Iceland, MODIS images were only available for the Greenland experiment. We used ‘calibrated, geolocated radiances’ (level 1B data), a product valid for the top of the atmosphere. The acquired data were ‘Version 4 data’, which according to NASA is a ‘validated product, ready for use in scientific publications, although improved versions may appear later on’. Our MODIS data were resampled to a nominal resolution of 1.0 km. The real resolution at the view angles under which the K-transect was observed is somewhat coarser.

3.2. The retrieval method

To obtain surface albedos, we processed the raw satellite data as follows:

1. 1. The AVHRR images were geolocated manually, both with ground-control points and by matching the ice-cap (or ice-sheet) margin on the images with the position of the margin in DEMs. Differences between these two methods do not exceed 1 km, which is therefore considered the accuracy of the geolocation of the AVHRR data. According to the requirements, MODIS geolocation should be accurate to 0.1 pixel (~0.1 km).

2. 2. In agreement with our own observations in the field, clouds were almost absent on the satellite images within the region of the helicopter experiments. The only notable exception was the AVHRR image of 1735 GMT for Greenland. At this time, cirrus clouds coming from the southeast covered the easternmost part of the flight legs. We excluded the corresponding satellite data from the comparison.

3. 3. The AVHRR counts were converted into effective narrowband reflectances by means of the calibration coefficients, where ‘reflectance’ refers to the fraction of the incoming radiation reflected in the direction of the satellite (note the difference to the albedo, which is the hemispherically averaged reflectance). The AVHRR calibration coefficients were taken from Reference Rao and ChenRao and Chen (1999) and are time-dependent in order to account for degradation of the instruments. The MODIS Science Team calibrated the MODIS data before delivery.

4. 4. The effective narrowband reflectances (r _eff) were converted into planetary narrowband reflectances (r _p) by taking into account the effects of the solar zenith angle with respect to a horizontal surface (θ) and the varying distance between the Sun and Earth (d in astronomical units):

   (1)

   

   Here, ‘planetary’ means that the reflectances are valid for the top of the atmosphere.

5. 5. We made corrections for the atmosphere and for anisotropic reflection by the surface. The directional distribution of the reflected radiation is described by bidirectional reflectance distribution functions (BRDFs). The atmospheric and BRDF corrections must be made simultaneously because the atmosphere also has an effect on the directional distribution of the signal (Reference RahmanRahman, 1996). This effect is taken into account by the radiative transfer model (6S) that we used for the calculation of the atmospheric correction. This model allows the use of custom-prescribed surface BRDFs.

   For each image a relationship of the following form was established between the known planetary narrowband reflectance and the unknown surface narrowband albedo (α_s):

   (2)

   

   where the c_i are coefficients and z is surface elevation. The coefficients depend on the solar-view geometry, atmospheric conditions and the surface type. The latter dependence is due to the fact that different surface types have different types of BRDFs. This approach was only feasible because the solar-view geometry varied little within the regions of the experiments (none of the angles characterizing the solar-view geometry varied by >3°).

   We specified the atmospheric conditions as follows:

   atmospheric profiles of pressure, temperature and water vapour from European Centre for Medium-Range Weather Forecasts analyses;

   a mixture of 15% (by volume) dust-like aerosols, 70% (by volume) water-soluble aerosols and 15% (by volume) oceanic aerosols; no soot;

   an optical thickness for aerosols equal to 0.065 at 550 nm (as measured by Reference Stroeve, Nolin and SteffenStroeve and others, 1997, at the ETH Camp), which corresponds to a visibility of 109 km;

   total ozone determined with space-borne Total Ozone Mapping Spectrometers.

   We made calculations, and therefore established different relationships, for three different types of BRDF:

   BRDFs of melting ice according to the parameterization proposed by Reference Greuell and de Ruyter de WildtGreuell and De Ruyter de Wildt (1999);

   BRDFs of melting snow according to the parameterization proposed by Reference KoksKoks (2001);

   isotropic reflection.

   We used the BRDFs measured in TM band 2 for AVHRR band 1 and MODIS bands 1 and 4, and the BRDFs measured in TM band 4 for AVHRR band 2 and MODIS band 2 (see Fig. 2; Table 1).

   For each situation (a specific image with its solar-view geometry and atmospheric conditions, satellite band and BRDF), we ran the radiative transfer model for various values of a _s ranging between 0.1 and 1.0, and various values of z (varying roughly over the range of elevations within the region of the experiment). After completing these calculations, a least-squares fit was made to Equation (2). Residual standard deviations were all ͬ0.01. Inclusion of other terms on the righthand side did not improve the performance of Equation (2). Note that the residual standard deviation is only a measure of the quality of the fit of discrete values of a _s(r_p), computed with the radiative transfer model, to Equation (2). It is not an estimate of the total accuracy of the atmospheric and BRDF correction since it includes neither inadequacies in the 6S radiative transfer model nor uncertainties in atmospheric conditions and BRDFs.

6. 6. Finally, a correction for the effect of the sloping surface on the incoming radiation was carried out as by Reference Klok, Greuell and OerlemansKlok and others (2003):

   (3)

   

   where α _s and α’_s are the corrected and uncorrected surface narrowband albedos, f _diff and f _dir the diffuse and direct fractions of the incoming shortwave radiation (computed with the 6S model) and θ _s and θ the solar zenith angles with respect to the real and a horizontal surface. We computed surface orientation and tilt from DEMs with a grid size of 0.5 × 0.5 km for Vatnajokull and ~2 × 2 km for Greenland (Reference EkholmEkholm, 1996).

3.3. Accuracy of the satellite-derived albedos

In this subsection, the causes of uncertainty in the satellite-derived albedos are discussed so that possible differences with the ‘ground-truth data’ can be interpreted more easily.

1. 1. Equations for time-dependent calibration coefficients for the AVHRR band 1 and 2 sensors have been proposed in various papers. They may differ substantially. For example, the coefficients for the sensors on board NOaA-14 as derived by Reference Tahnk and CoakleyTahnk and Coakley (2001) differ by up to 12% from the calibration coefficients that we used (those given by Reference Rao and ChenRao and Chen, 1999). Both Reference Rao and ChenRao and Chen (1999) and Reference Tahnk and CoakleyTahnk and Coakley (2001) take the calibration coefficients of the AVHRR sensors on board NOAA-9 as reference. Therefore, the absolute accuracy of the NOAA-9 sensors must be added to the relative accuracy of the NOAA-14 sensors, but we were unable to find this information. In conclusion, the absolute accuracy of the AVHRR coefficients is of the order of 10% or more. Compared to the AVHRR sensors, the absolute uncertainty in the calibration coefficients of the MODIS sensors is much better, namely ͬ2% (Reference JusticeJustice and others, 2002).

2. 2. For individual pixels, errors in the surface albedo due to errors in the surface slope and orientation can be considerable. However, the slopes of the observed surfaces in Iceland and Greenland were generally relatively small. More importantly, for the comparisons (section 5) we considered averages of satellite-derived and helicopter albedos over distances larger than 10 km, a scale on which topographic errors cancel out because it is much larger than the grid size of the DEMs. In conclusion, errors due to insufficient knowledge of the topography are negligible (ͬ0.01).

3. 3. We calculated the uncertainty in the atmospheric correction due to uncertainties in the composition of the atmosphere by performing sensitivity runs with the radiative transfer model. We did this for all the relevant AVHRR and MODIS bands and for the solar-view geometries of the satellite data used for the comparison. In the sensitivity runs, the water-vapour content of the atmosphere was reduced to 80% of its original value, the mixture of aerosol types was varied within reasonable limits, the aerosol optical thickness was halved and the ozone amount was reduced to 95% of its original value. For all the satellite bands and solar-view geometries, the effect on the surface albedo of different amounts of water vapour and ozone and of different aerosol types was <0.001, with the exceptions of water vapour in AVHRR band 2 and the soot fraction of the aerosol, both of which had an effect on the surface albedo >0.001 but <0.01. Variation in aerosol optical depth had an effect <0.01, but for the images with view zenith angles >50°, the effect could become somewhat larger than 0.01, depending on the type of BRDF chosen. Note that in regions where the atmosphere contains more water vapour and aerosols than the relatively dry and clean polar atmosphere over Iceland and Greenland, the atmospheric correction is more sensitive to uncertainties in atmospheric composition.

4. 4. Reference LyapustinLyapustin (2002) assessed errors due to shortcomings in the treatment of radiative transfer in the 6S model, given perfect knowledge of atmospheric and surface conditions. He did so by comparing 6S calculations with calculations made with SHARM-3D, which is a more rigorous radiative transfer model than 6S. He showed that the differences increase with the solar and view zenith angles and he claims that ‘in cases of high surface anisotropy’, which is the case for glacier ice, ‘the error can be as high as 9-10%’. We cannot derive error estimates for our Sun-satellite target geometries and BRDFs from the examples provided by Reference LyapustinLyapustin (2002).

5. 5. It is not possible to make a meaningful estimate of the uncertainty caused by inadequacies in the applied BRDFs. The BRDFs for melting ice as employed in this study are parameterizations based on measurements at three different sites, with varying amounts of near-surface meltwater, all on Morteratschgletscher, Switzerland. These BRDFs have a maximum in the forward direction which becomes more pronounced with increasing solar zenith angle. Certainly, varying amounts of dust and debris as they occur especially on Vatnajokull have a completely different effect on the BRDFs than varying amounts of meltwater. Moreover, along the K-transect, the ice within 10-20 km of the ice-sheet margin was characterized by bumps and valleys of considerable steepness (typically 10-20°) and height (up to 10m). For a surface that is an isotropic reflector on a scale smaller than that of the surface roughness elements, such elements cause maximum reflection in the backward direction and minimum reflection in the forward direction. Therefore, the effect of surface roughness at any scale larger than the length scaleof the surface over which the BRDFs were measured (≈10 × 10 cm) is to reduce the amount of anisotropy found in the measured BRDFs (see also Reference Warren, Brandt and O’Rawe HintonWarren and others, 1998). Under large solar zenith angles, very rough snow and ice surfaces should even have a maximum in the backward direction, instead of the maximum in the forward direction for horizontal terrain. The problem is that the AVHRR and MODIS data used in this study do not contain any information about the surface characteristics in terms of amounts of meltwater, concentrations of dust and debris, or surface geometry. In addition, BRDFs of ice surfaces with variable concentrations of dust and debris and variable surface geometry have not been established. Similar arguments apply to the uncertainty in snow BRDFs. For snow, the variable characteristics that affect the BRDFs are probably grain- size, water content, dust content and surface geometry. In conclusion, quantification of BRDF-related errors is currently impossible, but the error can be of the order of magnitude of 0.10 or larger.

In summary, errors in the satellite-derived surface albedo due to inaccuracy in the description of the surface topography and in the assumed composition of the atmosphere are small (<0.01). At view zenith angles >50°, unknown aerosol loads can cause errors slightly more than 0.01. The largest error sources are probably inaccuracy of the calibration coefficients of the satellite sensors (only for AVHRR: 10% or more) and insufficient knowledge of surface BRDFs (0.10 or more). Errors due to shortcomings in the radiative transfer model (6S) could be up to 10%, but will certainly be less for our images obtained at small view zenith angles.

3.4. Ground truth for MODIS sensors

We did not perform helicopter measurements in MODIS bands, but Figure 2 shows that MODIS bands 4, 1 and 2 are situated more or less in the middle of TM band 2, and AVHRR bands 1 and 2, respectively (see also Table 1). The question is, therefore, whether the helicopter measurements in the AVHRR and TM bands can be used as ground truth for surface albedos derived from satellite data in the corresponding MODIS bands. The answer is positive if the corresponding narrowband albedos (e.g. MODIS 4 and TM 2) have ratios that are independent of the type of snow and ice. To check this, we generated 70 spectral albedo curves with a simple radiative transfer model (Reference Reijmer, Bintanja and GreuellReijmer and others, 2001). These curves represent snow ranging from fine- to coarse-grained, ice ranging from clean to very dirty, and blue ice. Figure 2 shows two examples. The generated curves match measured curves (Reference Reijmer, Bintanja and GreuellReijmer and others, 2001) very well. Next, we computed narrowband albedos (α_nb) for these 70 fictitious snow and ice surfaces, using the following equation:

(4)

where α(λ), R(λ) and G(λ) are the spectral albedo, the spectral response of the sensors (Fig. 2) and the spectral incoming radiative flux, respectively. Table 3 gives statistics of the ratios Averages are close to 1 (1.001, 0.999 and 1.006, respectively) and, more importantly, the standard deviations and ranges are much smaller than the differences between the helicopter and the satellite albedos (see section 5). This is especially true for because within these bands the spectral albedo is almost constant. Therefore, we used the albedos in TM 2 and AVHRR 1 measured from the helicopter as ground truth for the satellite albedos in MODIS bands 4 and 1, respectively. The variation in is larger than the variation in the other two ratios because the spectral albedos of snow and ice vary dramatically within AVHRR 2. Nevertheless, we used the helicopter albedos for AVHRR 2 to validate the satellite-derived albedo in MODIS 2 after dividing the satellite albedos by the mean ratio of 1.006. The associated uncertainty (0.015) should be kept in mind.

Table 3. Statistics of ratios of narrowband albedos for different types of snow and glacier ice. The ratios were computed for 70 types of snow and ice (see text)

5.1. Introduction

In this section we compare the satellite-derived albedos with the helicopter albedos. Figures 7-10 show albedos as a function of the distance along the flight-lines. The albedo curves, for both Vatnajokull and the K-transect, tend to show two levels, one for ice and one for snow. Therefore, we have computed and compared the averages of the satellite- derived and the helicopter albedos over entire flight portions above ice and snow (see Table 4).

Fig. 7. Comparison of the surface albedo along the K-transect as measured from a helicopter with surface albedos derived from satellite data acquired on the day of the helicopter measurements (7 July 2000). The upper two panels are for AVHRR band 1 and MODIS band 1. The lower two panels are for AVHRR band 2 and MODIS band 2. The two panels to the left represent the flight before solar noon; the two panels to the right represent the flight after solar noon (see also Fig. 4). The legend gives the times at which the data were acquired.

Fig. 8. Same as lower right panel of Figure 7, but here satellite albedos are derived with two different assumptions about the BRDFs.

Fig. 9. Same as lower right panel of Figure 7, but satellite albedos are derived from three AVHRR images obtained from the view angles given in the legend.

Fig. 10. Comparison of the helicopter-measured surface albedo in AVHRR band 2 over Vatnajökull with surface albedos derived from satellite data in the same band and acquired at 1524 GMT on the day of the helicopter measurements (6 July 1999). The three panels correspond to different flights over the ice sheet (see Fig. 3). Satellite albedos retrieved with and without BRDF correction are shown.

Table 4. Comparison of helicopter-measured surface albedos along the K-transect (7 July 2000) and of Vatnajoökull (6 July 1999) with surface albedos derived from AVHRR and MODIS images acquired on the days of the helicopter measurements. Satellite albedos retrieved with and without BRDF correction are given. The values shown (one for snow and one for ice) are averages over the given flight distances. For the K-transect, we only considered those flight portions that were made within roughly half an hour of the acquisition time of the images. Also, data obtained from a helicopter height of <100m above the surface were omitted

5.2. The Kangerlussuaq transect

Figure 7 shows albedos along the K-transect in AVHRR bands 1 and 2 and MODIS bands 1 and 2. For the computation of the satellite-derived albedos we used the complete retrieval method as described in section 3.2, including the correction for anisotropic reflection at the surface. No satellite-derived albedos are shown for the transitional areas with a mixture of snow and ice, because ice and snow BRDFs are different. For AVHRR, we considered only the image with the smallest, and therefore most favourable, view zenith angle (16°). However, we did not consider this image for the snow-covered area because of the presence of clouds at the time of the satellite overpass. The differences between the various albedos in the ice-covered area can possibly be explained by temporal variations in the real surface albedo (see section 4, Fig. 6) in the following way. Until 1600 GMT, the albedo decreased due to increased accumulation of meltwater. After that time, the albedo increased again. Perhaps the amount of meltwater had already been decreasing between 1600 and 1735 GMT (note that solar noon is at 1525 GMT). Errors in BRDF corrections could also explain the apparent increase in albedo. Real changes in albedo due to an increase in the solar zenith angle are unlikely to play a role at the relatively small solar zenith angles (45° at 1602 GMT and 49° at 1735 GMT). In order to remove the complication of the temporal variations in the albedo, Table 4 contains only averages of the helicopter measurements over those flight portions of the K-transect that were made within roughly half an hour of the acquisition time of the images used in the comparison. The differences are small: satellite-derived albedos corrected with BRDFs differ from the helicopter albedos by 0.01 in both AVHRR bands, by 0.03 (ice and snow) in MODIS band 1, and by 0.00 (ice) and 0.02 (snow) in MODIS band 2. In MODIS band 4, the differences are 0.02 (ice) and 0.01 (snow).

The required accuracy of the surface albedo in climate models and in models simulating the mass balance of glaciers and ice sheets is of the order of 0.05. So, in the light of that requirement, the satellite-derived albedos discussed in the previous paragraph are ‘validated’. This could mean that each of the steps of the retrieval method is accurate enough. However, it is not proof of that, because there could be compensatory effects (e.g. errors in the calibration coefficients that compensate for errors in the BRDF corrections).

It is noteworthy that in the cases discussed so far (all the K-transect comparisons in Table 4), the BRDF correction produced an improvement. This is shown in Figure 8 for AVHRR band 2 and MODIS band 2,where satellite-derived albedos with and without BRDF correction are compared with helicopter albedos.

Figure 9 considers all of the three AVHRR images (band 2) of the K-transect. The albedos derived from the image of 1735 GMT have been discussed above (Figs 7 and 8). Here we focus on the two images acquired at relatively large view zenith angles (68° and 57°). This implies larger footprint sizes (Table 2). Portions of the curves within one footprint size of the ice margin were omitted because the tundra influenced those data. Surface albedos derived from the image of 1415 GMT (view zenith angle = 68°) agree very poorly with the helicopter albedos. The data were acquired from a forward direction (relative azimuth angle = 146°), where, according to the BRDFs that we used, the radiance varies strongly with view direction. Therefore, the uncertainty in the BRDF correction is relatively large in all areas (snow, bare ice and transition from snow to bare ice). In addition, errors in the calibration coefficients of the satellite sensors and in the 6S model might play a significant role. The uncertainty related to the composition of the atmosphere, including aerosol optical thickness, is too small, by one order of magnitude, to explain the discrepancy. In fact, the albedo curve for the image of 1415 GMT suggests strong spatial smoothing, but note that this occurs on a scale that is much larger than the resolution of the image (still only ~3 km in the helicopter flight direction).

The agreement between the helicopter data and the data of the image of 1554 GMT is reasonable for large portions of the flights, but there is a significant mismatch in the bare ice area, which increases towards the margin. This may be an unaccounted BRDF effect caused by bumps and valleys with a steepness that increases towards the margin. We conclude that satellite data obtained at too large view zenith angles are not suitable for retrieval of the surface albedo. The threshold lies at a view zenith angle between 43° (albedos derived from the MODIS image acquired at this angle are in good agreement with the helicopter albedos; see Fig. 7) and 57° (AVHRR image of 1554 GMT).

5.3. Vatnajökull

For Vatnajökull, three AVHRR images were available (but no MODIS images). Figure 10 compares the satellite-derived albedo with the helicopter albedo for the image obtained from a near-nadir view direction and band 2. Table 4 summarizes the comparison for both AVHRR bands and two images, excluding the image of 1706 GMT because its view zenith angle (65°) is larger than the threshold determined from the data of the K-transect. We averaged the helicopter data, measured over snow and ice during the entire day, to obtain single snow and ice values. This is justified by the fact that, in contrast to the situation along the K-transect, the surface albedo of the ice did not change during the helicopter measurements (see section 4). A small decrease in albedo probably did occur over the snow, in analogy with the situation along the K-transect.

We first compare Vatnajokull’s ice albedos. There are four samples (two images x two bands). The differences between the helicopter and satellite albedos corrected for anisotropic reflection at the surface are 0.06, -0.01, 0.03 and -0.05 (Table 4). These differences may seem small in an absolute sense but they are large in a relative sense, with measured albedos of 0.29 for band 1 and 0.22 for band 2. Errors in the calibration coefficients of the satellite sensors and errors in the atmospheric composition can be ruled out in explaining the differences, as errors in the albedo due to these factors are more or less linear in the albedo and the albedo is relatively small. For the same reason, shortcomings in the 6S model probably have only a small effect (Reference LyapustinLyapustin, 2002, gives errors in terms of percentages). This leaves the BRDF corrections as the main factor to explain the discrepancies. These corrections are large (0.05-0.07) because the albedo itself is small and the corrections increase in absolute magnitude with decreasing albedo (see Reference Greuell and de Ruyter de WildtGreuell and De Ruyter de Wildt, 1999). It is striking that in three out of the four cases, the satellite albedo derived with the assumption of isotropic reflection is nearer to the helicopter albedo than the satellite albedo that is corrected with the BRDFs. Figure 10 displays one such case (AVHRR band 2, image 1524 GMT). This suggests that the ice BRDF parameteriza- tions used in this study are not applicable for Vatnajokull. Indeed, the BRDF parameterizations represent surfaces with varying amounts of meltwater. Therefore, they might be applicable to the wet and clean surfaces of the K-transect but not to the dry and dirty ice of Vatnajokull. We conclude that, since the uncertainties in the BRDF corrections are of the same magnitude as the corrections themselves, the differences between the helicopter and the satellite-derived ice albedos could actually be caused entirely by erroneous BRDFs.

We also had four samples of helicopter and satellite- derived albedos to be used in a comparison for the snow of Vatnajokull. They differ by 0.03, 0.06, -0.03 and 0.04 (Table 4). As argued above, uncertainties due to atmospheric composition are ͬ0.01. A sensitivity experiment showed that this was also true for the uncertainty in aerosol optical depth and the image of 1343 GMT, even though the view zenith angle of this image is relatively large (54°). Because the albedos of snow are higher than the albedos of ice, the uncertainties due to the calibration coefficients of the satellite sensors are significant (0.09 in band 1 and 0.07 in band 2 for an error of 10% in the coefficients), and errors due to shortcomings in the 6S model could also be important. The uncertainty due to the use of incorrect BRDFs is difficult to quantify but could be as large as the uncertainty due to the calibration coefficients (~10%). Possibly, errors in the calibration coefficients, in 6S and in the BRDFs all have a significant effect on the satellite- derived snow albedo. With our small number of samples, it is impossible to tell which of these three error sources is the most important in explaining the differences between the helicopter albedos and the satellite-derived albedos.