The Clouds and the Earth's Radiant Energy System (CERES) instruments are used to estimate the amount of reflected shortwave (SW) flux and emitted longwave (LW) flux at the top of the atmosphere (TOA) . These fluxes are widely used in studies of Earth's energy budget. However, CERES measures radiances (in Wm-2 sr-1) not flux (in Wm-2). In order to derive the flux from the radiance we use scene-dependent angular distribution models (ADMs) . ADMs relate the reflected radiance at a given satellite viewing geometry to the total reflected flux. The viewing geometry is described in terms of satellite viewing zenith angle, θv, the solar zenith angle, θ0 and the relative azimuth angle, ϕ=ϕv-ϕ0, where ϕ0 and ϕv are the solar and viewing azimuths (see Fig. 1). The CERES next-generation ADMs are described in , which improved upon the ADMs provided by . In this paper, we focus on details pertaining to the development and testing of the ADMs over clear Antarctic scenes that were briefly described in .

This study is conducted with the CERES Edition 4 Single Scanner Footprint (SSF) and the CERES Single Scanner Footprint–MISR (SSFM) data sets. The SSF data set consists of CERES radiances, fluxes, viewing geometry and geo-location information combined with MODIS radiances and cloud information derived from those radiances , matched to the CERES field of view (FOV) and convoluted using the CERES PSF (Point Spread Function). The SSFM data set consists of the same SSF with the addition of co-located, and PSF-weighted, MISR radiance measurements. The MISR instrument is also flying on NASA's Terra satellite. It consists of push-broom-style cameras and provides near-simultaneous measured radiances of the same footprint from nine different viewing zenith angles (Df/a =±70.5∘, Cf/a =±60.0∘, Bf/a = ±45.6∘, Af/a =±26.1∘, and An =0.0∘) in the along-track direction. MISR measures at four different wavelengths (blue – 446 nm, green – 558 nm, red – 672 nm, and near-infrared – 867 nm), with resolution ranging from 275 m at nadir to 1.1 km for the off-nadir cameras. The SSFM data set is discussed in more detail in . Additionally, we use the Modern-Era Retrospective Analysis For Research and Applications (MERRA) re-analysis data set to determine wind speed and direction .

As noted above, sastrugi can strongly affect the BRDF of snow, with the effect depending on their orientation relative to the sun. When they are aligned parallel to the sun the reflectance in the forward-scattering direction is relatively higher, and the backscattering reflectance relatively lower than when the sastrugi are aligned perpendicular to the sun. This can be demonstrated using MISR measurements and the MERRA wind directions. If we assume that the sastrugi are aligned parallel to the mean wind field from MERRA, then we can sort the MISR measurements by the relative sastrugi angle, ϕsas=ϕwind-ϕ0. This is shown in Fig. . Here we have used the MISR near-infrared (NIR, 0.86 µm) reflectance (ρnir): ρnir=πInirF0,nircos(θ0)E02, where Inir is the measured radiances, F0,nir is the incoming solar flux in the MISR NIR band, θ0 is the solar zenith angle and E0 is the Earth–Sun distance. We calculate the mean reflectance for each MISR camera, separating the scenes into three ϕsas bins: 0∘≤ϕsas<30∘ (sastrugi mostly parallel to the sun, green line), 30∘≤ϕsas<60∘ (sastrugi neither parallel nor perpendicular to the sun, blue), and 60∘≤ϕsas≤90∘ (sastrugi mostly perpendicular to the sun, red). The Figure shows that in the forward direction (positive viewing zenith angles) the reflectance is highest when the sastrugi are aligned parallel to the sun and decreases as the sastrugi become more perpendicular to the sun. The opposite occurs in the backscatter direction. The highest reflectance occurs when the sastrugi are aligned perpendicularly, decreasing as the sastrugi become parallel. This is what we would expect to happen and demonstrates that MISR is picking up a sastrugi signal in its measurements.

One of the underlying assumptions about ADMs is that the anisotropy does not change as the radiance changes. It is assumed that the radiances are positively correlated at all viewing geometries. This allows us to then assume that if a measurement at one viewing angle is greater than the mean, then the radiances at all the angles that we cannot simultaneously see are also greater than the mean. The anisotropy is then the same and the albedo scales accordingly. For single-viewing geometry instruments such as CERES there is no easy way to test this assumption. By using a multi-angle instrument such as MISR, which essentially gives us a slice of the instantaneous BRDF, we can test this. The results in Fig.  suggest that this assumption is being violated and the presence of sastrugi causes the reflectance to become anti-correlated. This behavior is caused by the highly consistent orientation of the sastrugi and is not present for other scene types that do not have surface features with such a preferred azimuth (such as trees and clouds). As outlined briefly in , we examine this further using the joint probability distributions of standard scores between NIR reflectance (ρnir) measurements from any two MISR cameras. The standard score is calculated as znir(θ0,θm,ϕ)=ρnir(θ0,θm,ϕ)-ρ‾nir(θ0,θm,ϕ)σρnir(θ0,θm,ϕ), where the averages (ρ‾nir) and standard deviation (σρnir) are calculated in 5∘ bins for all available θ0 and ϕ at each MISR viewing angle, θm. The absolute value of z represents the departure of a measurement from the mean in units of the standard deviation, where negative z indicates the measurement is below the mean and positive z is above the mean. We use the NIR band of the MISR instrument as these radiances have the highest correlation with the CERES SW radiances for different solar zenith angles. We examine this by matching MISR and CERES radiances over clear-sky Antarctic scenes to within an angular separation of 5∘, separating these matched radiances into 10∘ solar zenith bins and calculating the correlation. For all solar zenith angles, the NIR has the highest correlation, ranging from r2=0.95 for 50∘≤θ0<60∘ to r2=0.98 for 80∘≤θ0<90∘. The MISR red band has the second-highest correlations, ranging from r2=0.92 for 50∘≤θ0<60∘ to r2=0.98 for 80∘≤θ0<90∘. The blue band has the worst correlations: from r2=0.69 for 50∘≤θ0<60∘ to r2=0.95 for 80∘≤θ0<90∘. The green band correlations lie between the blue and red bands. Based on these results we chose the NIR band to create the adjustments as it is most similar to the broadband.

Figure  shows the joint distributions of znir between the forward viewing Df (θm=70.5∘) camera and the remaining cameras. Distinct correlations can be seen in the figures, with the slope of the relationship changing as the angular distance between the cameras changes. There is a strong positive correlation at the adjacent camera, the correlation remains positive but decreases in strength as the viewing angle becomes less oblique. Once the viewing angle becomes negative, the correlation also becomes negative, increasing as the angles become more oblique. This indicates that the sastrugi are increasing the reflectance in one direction, while at the same time decreasing the reflectance in the opposite direction. We use these relationships to help us develop ADMs for CERES that take account of this observed behavior.

For other surfaces that might have a similar azimuthal alignment to that of sastrugi, namely desert sand dunes and ocean waves, the standard scores showed positive correlations between all the cameras. This indicates that these surfaces are not exhibiting sastrugi-like behavior. For sand dunes this may be due to the MISR relative azimuth sampling angles. As an example, for one desert area with large sand dunes known as the “Empty Quarter” (44–56∘ E, 16–23∘ N) the forward-scatter-viewing mean-relative-azimuth ranges from 45∘ (for 40∘<θ0<45∘) to 68∘ (for 10∘<θ0<15∘). As these are closer to 90∘ than to the principal plane, this limits our ability to detect any effect the sand dunes might have. A study by found that the presence of sand dunes in the Taklamakan Desert in China altered the MISR red band reflectance shape compared to a nearby dune-free area. However, when we applied the above approach to this same area (77–88∘ E, 36–41∘ N) we found positive correlations between the standard score for all cameras. The reason for this discrepancy is not clear, it may be the result of the SSFM data set having reduced spatial and temporal sampling over desert areas compared to Antarctica, or it may be that this approach is not suitable for deserts which are less reflective than snow. For ocean scenes, the BRDF is largely dependent on the wind speed, which has been shown to affect the size and intensity of the glint region . As the glint region is the dominant feature of the ocean BRDF, any changes in this due to wave orientation would likely be second-order to changes due to the wind speed, whereas it appears that the sastrugi-alignment is a first-order effect for sastrugi on snow.

Figure a shows the mean CERES clear-sky reflectance (red line) for the region 88–89∘ S, 76–83∘ W, averaged over all Decembers from 2000 to 2004 as a function of the solar azimuth angle. At this latitude, sampling is available at all solar azimuth angles and a limited viewing zenith angle (45∘<θ<65∘) range at a fixed relative azimuth angle (65∘<ϕ<75∘). The reflectance shows a clear sinusoidal signal as a function of the solar azimuth angle. This suggests that there is a surface feature that periodically brightens and then darkens the reflectance with azimuth, i.e sastrugi. The mean anisotropic values used to convert the reflectance to albedo are also shown in dashed lines. The anisotropic factors from (hereafter KL05) (Fig. a dashed blue) are fairly constant across the solar azimuth range, whereas the new anisotropic factors (Fig. a, dashed green) follow the shape of the reflectance curve more closely. The corresponding mean albedos are shown in Fig. b with albedos inverted using KL05 ADMs in blue and using the ADMs described in this paper (hereafter CS15) in green. The KL05 albedo also follows a sinusoidal shape with solar azimuth. Although sastrugi are expected to alter the albedo slightly with solar azimuth angle, the change is much larger than expected. Also, as the relative azimuth is about 70∘, the change in albedo is not in the correct phase with respect to the solar azimuth (the maximum albedo would occur when the sun is aligned perpendicular to the sastrugi, which is an offset of about 70∘). This is a result of the anisotropic factors being relatively constant across the solar azimuth range. The CS15 albedos show a slight variation with solar azimuth; however, the dependence is greatly reduced. This indicates that the CS15 albedo retrievals behave more realistically with the solar azimuth than the KL05 ADMs, and that the anisotropic factors are more accurately compensating for the change in reflectance. The standard deviation is greatly reduced too, suggesting the retrievals are more consistent.

Figure a shows the reflectance values against viewing zenith angle for two separate 1∘ × 1∘ grid boxes over Antarctica; circles (81∘ S, 163∘ W, A), squares (81∘ S, 35∘ E, B). These measurements were acquired when one of the CERES instruments was scanning in along-track mode, so it sees the same area from different viewing zenith angles but with a fairly constant relative azimuth angle close to the principal plane (ϕ≈20∘). As can be seen, the reflectance curves from the two regions have quite different shapes. Grid box B (purple squares) has a much higher backward peak than grid box A (green circles) which has a higher forward peak. The difference in shape can be understood by looking at the viewing geometry angles shown in Fig. b and c. The green line shows the viewing azimuth with respect to north (0∘), the red line shows the solar azimuth. The blue line shows our estimate of the sastrugi direction, based upon the mean wind direction from the MERRA data set over the period 2000–2004. The viewing geometry is fairly similar in both cases, with the viewing azimuth close to 270∘ and the solar azimuth around 300∘. The main difference between the two is the wind direction or sastrugi angle. In the first region, the sastrugi angle is more parallel to the solar azimuth angle and the viewing azimuth is about 30∘ off the sastrugi angle. In the second region, the sastrugi angle is approximately north–south, resulting in a viewing angle almost perpendicular to the sastrugi. This is consistent with the higher backward peak and lower forward peak seen in the reflectance. The corresponding albedo retrievals against viewing zenith angle for the two regions are shown in Fig. d and e. Ideally, the albedo at a given location should be independent of viewing zenith angle. However, we see that the KL05 ADMs (shown in green) return a very systematic dependence of albedo on viewing zenith angle. This is especially true for the region B where albedo varies from 0.78 to 0.60 as the viewing zenith goes from -70∘ (i.e., backward viewing) to 60∘ (forward viewing). While the FOVs from the different angles are not all looking at exactly the same area, there is considerable overlap and it is hard to believe that the albedo is actually varying by this much over such a small area, indicating that the retrievals are probably erroneous. For region A, the dependence is not as large, yet we do see an increase in the KL05 albedo from the backward viewing to the forward viewing angles. In both regions the albedos retrieved using the CS15 ADMs are much flatter with viewing zenith angle. This indicates that these ADMs are better at accounting for the variation of anisotropy caused by sastrugi.

To estimate the monthly grid box level bias over Antarctica due to sastrugi we first calculate the 24 h equivalent flux for each flux measurement using the method described in . Briefly, this method uses the instantaneous albedo and the diurnal variation of the KL05 ADM albedos to estimate what the reflected SW flux would be at any time over a 24 h period for a given location, assuming there are no changes in the atmospheric or surface properties. The average of these values is then the estimate of the 24 h mean reflected SW flux, F^24h. Equation 3 of describes this mathematically as F^24h=A^[θ0(t0)]αk[θ0(t0)]1Nt∑l=1Ntαk[θ0(tl)]μ0(tl)F0, where A^ is the estimated instantaneous albedo determined using Eq. (3), t0 is the time of observation, αk[θ0(tl)] is the scene-type-dependent (k) ADM albedo at the solar zenith angle corresponding to local time tl, l is the time step increment, Nt is the number of time steps used to compute the 24 h flux and μ0(tl) is the cosine of the solar zenith angle at tl. F0 is, as above, the TOA solar insolation. To improve calculation speed the (1/Nt)∑l=1Ntαk[θ0(tl)]μ0(tl)F0 term is pre-calculated for each Julian day at a resolution of 1∘ in latitude and 1 min in time and stored in a look-up table.

We then calculate two estimates of the F^24h for each 1∘ × 1∘ grid box. The first estimate, F^24h, is calculated using measurements from all viewing zenith angles. The second, F^24hn, from measurements where the viewing zenith angle is close to nadir (θ<20∘). We then define the bias as the difference between the flux calculated from all angles and the flux from nadir angles: ΔF^24h=F^24h-F^24hn. This is the procedure used in , and it allows us to estimate what effect sastrugi have on the mean TOA flux over Antarctica. The idea is that the sastrugi will have the most impact on the ADMs at more oblique angles than at the nadir angles; thus, the difference between the total flux and near-nadir flux will provide an estimate of the bias in the averaged monthly level 3 CERES data products. We performed this analysis for the austral summer months (October–March) for 4 years of Edition 4 SSF data from 2001 to 2004, for both the CS15 and KL05 ADMs. In this analysis we consider values in the upper and lower 0.5 percentiles to be outliers and exclude them, this allows us to make a more consistent comparison between the two ADMs. Figure a shows the 0.5–99.5 percentile range (whiskers) and 25th–75th percentile range (box, also referred to as the interquartile range) of the ΔF^24h for each month listed. The dots show values of the outliers and the stars indicate that there are values outside the ±15 Wm-2 axis limits. The blue box/whiskers/dots are the CS15 ADM results and the red are the KL05 ADM results. The green line within the box shows the median value and the grey line the mean value (not area-weighted). The box/whisker values here show that the CS15 ΔF^24h tend to have a narrower range than the KL05 ΔF^24h. The month with the largest KL05 ΔF^24h range is December 2002 (DEC 02), from -12 to 7.5 Wm-2. The range of CS15 ΔF^24h for December 2002 is -5 to 4 Wm-2, the mean and median have shifted closer to 0, and the interquartile range has also decreased. The interquartile range has decreased for all months, indicating a significant decrease in the spread of the ΔF^24h when using the CS15 ADMs instead of the KL05 ADMs. The maximum positive bias for the CS15 ADMS is 7 Wm-2 (December 2004), down slightly from 7.5 Wm-2 for the KL05 ADMs (December 2003). The maximum negative CS15 bias is -5 Wm-2 (December 2004), down significantly from -12 Wm-2 for the KL05 ADMs (December 2002). An interesting aspect of this plot is the seasonal dependence of the ΔF^24h values. The biases tend to be higher in December and lower in October and March. Part of this is because Antarctica receives more sunlight in December, so the 24 h bias will be higher, but part of it is likely due to the sastrugi decreasing in size over the summer. observed this decrease in sastrugi over the summer months at the south pole due to erosion an sublimation. If we compare months that have similar sunlight hours (i.e., October/March, November/February, and December/January, though there are some differences in incoming solar) we see that the months that occur earlier in the summer have higher biases than the late-summer months. This is especially true for the KL05 ADMs, the CS15 ADMs do show this but with a reduced magnitude, further indicating they are accounting for the sastrugi effects. Note that the tests we perform here are designed to capture biases caused by sastrugi, it is possible that other biases may exist in the CERES fluxes (e.g., scene identification errors) but these would require additional tests to be determined.

Figure b–g show the maps of ΔF^24h for November 2002 (b, c), December 2002 (d, e), and January 2003 (f, g) for both KL05 (b, d, f) and CS15 (c, e, g). The largest sastrugi bias effects can be seen in the KL05 ADMs over the east Antarctic Plateau area, especially on the eastward slope. The CS15 ADMs show that the bias has decreased over most of Antarctica. The distinct pattern of the bias over the east Antarctic Plateau can still be seen; however, its magnitude has been reduced, with both the large positive and negative biases smaller in the CS15 plots than in the KL05 plots. There are two regions where the bias seems to have actually increased when using the CS15 ADMs. One of these is approximately located between 135 and 180∘ W and north of 80∘ S. The other region is approximately between 30 and 60∘ W and north of 80∘ S. Interestingly, these correspond, respectively, to the Ross and Ronne ice shelves and the relatively steep slopes that lead to them. It suggests that they may have a slightly different anisotropy compared to the rest of Antarctica, and may need to be accounted for in future ADMs. These positive biases may explain why the range of ΔF^24h for CS15 decreased more on the negative side than on the positive side in Fig. a.

Generally, the area-weighted mean bias and area-weighted standard deviation of the ΔF^24h values have both decreased for all of the months we looked at when using the CS15 ADMs. The largest absolute change in the mean bias occurred for October 2002 where the mean ΔF^24h decreased from -0.87 to -0.23 Wm-2 and the largest relative change was in February 2004 where the mean ΔF^24h decreased from -0.53 to 0.03 Wm-2, a relative decrease of 95 % (using absolute values to calculate the difference). Only 1 month showed an increase in mean bias, November 2001, where the mean ΔF^24h increased from -0.77 to -0.97 Wm-2. Otherwise the changes in the mean ΔF^24h ranged from -0.02 to -0.64 Wm-2 or, using relative changes, from -2 to -95 %. The standard deviation of the ΔF^24h decreased for all of the months we looked at, with relative decreases between -24 % (January 2004) and -56 % (November 2002). In absolute values these decreases were from 2.02 to 1.54 Wm-2 and from 2.89 to 1.26 Wm-2, respectively. As this is clear-sky only, and as Antarctica occupies a relatively small portion of the globe, these biases (and the reduction in bias from the CS15 ADMs) have a negligible effect on the global monthly reflected SW flux value.

The change in the mean 24 h flux between the KL05 and CS15 ADMs is shown in Fig.  for the months October–March for 2002. The changes are generally quite small at the grid box level, within 20 Wm-2, and a maximum 2σ range of 4.5 % of the mean 24 h flux (equivalent to 7.4 Wm-2 in October), or 12.2 Wm-2 (3.6 % of the mean 24 h flux in December). The Antarctic-wide mean changes range from -1.9 Wm-2 in November to 1.4 Wm-2 in January. Qualitatively, the changes in flux show a similar pattern to the KL05 bias maps, with the sign reversed. This is especially noticeable over eastern Antarctica, where the positive biases over the ridge of the plateau have become negative flux changes and the positive biases along the slopes at 90 and 135∘ E are associated with negative flux changes. The negative flux biases located approximately between 0 and 45∘ E, north of 75∘ S, are associated with positive flux changes, as are the negative flux biases between 90 and 135∘ E. The flux changes also capture to some extent the alternating biases located between 45 and 90∘ E. The changes are as we expect, negative biases, suggesting we were underestimating the flux, are associated with positive flux changes. Positive flux biases, indicating overestimation of the flux, are associated with negative flux changes. The correlation between flux changes and bias is shown in Fig. , with Fig. a showing the correlation for all of Antarctica and Fig. b showing the correlation for the eastern part of Antarctica between 0 and 180∘ E. The correlation coefficients are -0.33 and -0.48, respectively. The improved correlation for the eastern part seems to stem from excluding the large flux increases over the Ross and Ronne ice sheets in the western part of Antarctica in December and January. These large flux increases are not associated with a negative flux bias, suggesting that the new ADMs might be overestimating the fluxes in these areas. However, in general, the relationship between biases and flux changes is as we would expect it to be and provides further support for this new method of creating ADMs.

Due to the along-track sampling of the MISR instrument, and Terra's polar orbit, the SSFM data used here are only available close to the principal plane. This means that the relationships we derive between the cameras are only really valid for that region. We assume here that the relationships hold as the relative azimuth changes. Unfortunately, there is no way to test this assumption as it is not possible to get simultaneous measurements of the same location from different viewing zeniths at the relative azimuths sampled by CERES at high latitudes, approximately 70–80∘ and 100–110∘ while in cross-track mode. However, the standard deviation used to calculate the ADMs is generally lower at these angles than at the principal plane angles. This means the change in albedo possible from these angles is much less than from the principal plane angles which have higher standard deviations. We note that some of the above results can be used as a form of validation of this assumption. Especially the results in Fig. , which test the ADMs from cross-track angles with a wide range of reflectance values. The consistent results achieved regardless of the reflectance values suggests that this assumption is valid.

Another limitation is that the absolute value of the CERES reflectance and the standard deviation is used to determine which ADM to use. By doing this we are assuming that the mean reflectance and standard deviation calculated for the period 2000–2005 will remain unchanged for the rest of the record (a length of time as of yet undetermined). This is not a completely unrealistic assumption, as the expected changes in Antarctic snow surface properties (mainly snow grain size) that would affect the absolute brightness are very minimal . As we also use an interpolation scheme between standard deviation bins, this limits the effect that any shifts in the distribution might have on the anisotropic factor and, hence, the final albedo.

A third source of potential error is the use of the MISR ellipsoid-projected radiances in the SSFM data set. The ellipsoid-projected radiances are projected onto the WGS (World Geodetic System) 84 surface ellipsoid. This means that over the high Antarctic Plateau the radiances, registered to the same point on the ellipsoid, will be offset slightly at the true surface. Based upon the height of the Antarctic Plateau (≈ 3 km) above the ellipsoid and the height of Terra's orbit (705 km), this offset will be ≈ 10 km between the Da and Df cameras. However, this offset is still within the size of the CERES FOV (20 km) at nadir, so at least within the CERES footprint the cameras will be viewing portions of the same scene. Because of the CERES footprint size and the large-scale homogeneity of the Antarctic Plateau, this is unlikely to cause any significant errors; however, it probably contributes to the noise between the Df and Da cameras in Fig. .

Sastrugi are an interesting problem for radiance-to-flux inversion. They have the ability to significantly alter the BRDF, without changing the albedo. Capturing this in a consistent and physically plausible way is important to accurately estimate the albedo and irradiance over clear-sky Antarctica. In our attempt to find a solution to this issue we have incorporated the multi-angle information from MISR into the creation of the CERES clear-sky Antarctic snow ADMs. By using angular relationships derived from MISR we are able to create a set of ADMs that successfully removes most of the bias caused by the sastrugi. To examine the bias caused by sastrugi on the reflected SW flux, we compare measurements taken from nadir angles only, which are less sensitive to sastrugi, with those taken from all viewing zenith angles. We find that the regional, monthly mean biases between the nadir-only fluxes and the all-angle fluxes have decreased from -12 to 7.5 Wm-2, using the existing ADMs, to between -5 and 7 Wm-2 when using the ADMs described in this paper. The Antarctic-wide mean biases from sastrugi have decreased for all but one of the months we tested, with the decreases ranging from -0.02 to -0.64 Wm-2. These results are encouraging and suggest this novel method for creating ADMs is successful in accounting for the effects of sastrugi on anisotropy.