The BBRs deployed during ARISE are modified CM 22 precision pyranometers from Kipp & Zonen (Bucholtz et al., 2010). The BBR included downward-looking and upward-looking sensors. The radiometers were fix-mounted on the aircraft and measured upwelling and downwelling broadband irradiance (unit: Wm-2), that is, the spectrally integrated irradiance from 200 to 3600 nm. To account for the change of sun-sensor geometry due to aircraft attitude (pitch and roll), a software attitude correction (Long et al., 2010) was applied to the BBR data. In addition, a sunshine pyranometer (SPN1) was flown to measure diffuse and global radiative fluxes (Badosa et al., 2014; Long et al., 2010). The SPN1 radiometer was originally intended for ground-based use but is suited for airborne measurements of global and diffuse radiative fluxes because it does not have any moving parts, unlike traditional instruments such as the Multifilter Rotating Shadowband Radiometer (MFRSR). Smith et al. (2017) provide mission-specific details on both instruments. The BBR has a reported uncertainty of 3 % (Smith et al., 2017).

To attribute discrepancies between satellite-derived irradiance and airborne observations to causes such as erroneous water vapor, cloud properties, or three-dimensional radiative transfer effects, spectrally resolved measurements are needed (Schmidt and Pilewskie, 2012). SSFR is a moderate-resolution flux spectrometer built at the Laboratory for Atmospheric and Space Physics (LASP, University of Colorado Boulder). It is an updated version of the heritage spectrometer system originally developed at NASA Ames (Pilewskie et al., 2003). The SSFR radiometer system consists of two spectrometers for each viewing direction (zenith and nadir): (1) a Zeiss grating spectrometer with a silicon linear photodiode detector array covering a wavelength range from 350 to 950 nm and (2) a Zeiss grating spectrometer with an InGaAs linear photodiode detector array covering a wavelength range from 950 to 2150 nm. The spectral resolution of the silicon channels is 6 nm with a sampling of 4 nm. For the InGaAs channels, the spectral resolution is coarser – 12 nm with 6 nm sampling. From the SSFR measurements, spectral albedo, net flux, and absorption can be derived.

SSFR is typically flown in conjunction with an active leveling platform (ALP, also built at LASP), which was developed for counteracting the changing aircraft attitude to keep the zenith light collector horizontally aligned (the nadir light collector was fix-mounted). This is particularly important in the Arctic, where low sun elevations lead to large systematic errors for fix-mounted or poorly stabilized sensors (Wendisch et al., 2001). One reason is that radiation from the lower hemisphere (for example, from clouds below or at the aircraft altitude) is registered by the zenith detector when it is tilted, which leads to systematic biases that cannot be corrected. Another reason lies in the specific design of the SSFR light collectors, which are realized as integrating spheres with a circular aperture on top. They diffuse the incoming light collected by the aperture and bundle it into a fiber optics cable that transmits it to the radiometer system inside the aircraft (Schmidt and Pilewskie, 2012). The integrating sphere has an imperfect response to the incidence (polar) angle θ (Kindel, 2010), in contrast to the response of broadband radiometers such as BBR, which are closer to cos⁡(θ) as required for irradiance. At high sun elevations, a so-called hot-spot arises from a baffle that prevents light from being directly transmitted into the fiber optics. Since the response deviates significantly from cos⁡(θ), the direct and the diffuse light need to be corrected. This is done by separating the diffuse and direct components, using radiative transfer calculations in conjunction with SPN1 measurements (details are provided in Appendix A) and further assuming that the downwelling diffuse radiation is close to isotropic. This assumption is an approximation, which becomes invalid if parts of the lower hemisphere are in the light collector's field of view.

The light collector's angular response to the azimuthal angle also needs to be considered. Throughout the course of the mission, the zenith data revealed a dependence on the relative azimuth of the sun to the aircraft. This dependence was characterized at the end of the mission, by two calibration circles flown on 2 October. The non-homogeneous azimuthal response of the zenith light collector occurred for solar zenith angles greater than 66∘. Generally, an azimuthally variable response could be attributed either to aircraft interference (e.g., by the tail and/or propellers of the host aircraft) or to the light collector itself. For the former, BBR and SPN1 (both fix-mounted on the C-130) would also be affected. To assess their azimuthal response, the attitude-corrected BBR data (Bannehr and Schwiesow, 1993; Bucholtz et al., 2008; Long et al., 2010) were compared with the SPN1 global irradiance data, as well as with radiative transfer calculations. This comparison revealed that in this case, aircraft interferences were minor compared to atmospheric effects (e.g., cirrus) and that only SSFR measurements, but not BBR and SPN1, had a significant azimuthal dependence, suggesting the SSFR light collector as the source, rather than aircraft interferences. In order to determine the azimuthal dependence, the SSFR measurements were referenced to the BBR measurement

Since BBR has a near-ideal angular response, the attitude correction with respect to the polar angle can be performed by software as long as data are limited to small deviations from level. By contrast, SSFR with its non-ideal angular response requires an active leveling platform.

The angular dependence of SSFR was verified in the laboratory. In addition, wavelength and radiometric calibrations were performed before and after the mission. The wavelength calibrations ensured spectral accuracy by referencing the SSFR measurements to several line sources. The primary radiometric calibration, performed with a NIST-traceable (National Institute of Standards and Technology) calibrated lamp, links SSFR measured digital counts to spectral irradiance. The radiometric calibration was also transferred to a so-called secondary radiometric field standard, which monitored the stability of the radiometers throughout the mission.

A downward-looking video camera (referred to as “nadir camera”) is often included as a standard payload on NASA aircraft. It is a standard, commercially available video camera and typically records scenes for context only and is not radiometrically or geometrically calibrated. Despite this shortcoming, the videos recorded by the nadir camera are used for quantitative image analysis. From the video, we first extract image frames with an average rate of 2 Hz (two frames per second). The extracted image has a pixel resolution of 2592 (width) × 1944 (height). To co-register the aircraft nadir imagery with the measurements from other instruments, the times for the individual image frames are needed, but the image frames themselves did not contain a digitally stored time. They include a timestamp located at the lower left side that contains time information, and we used optical character recognition (OCR) to retrieve the time from this information.

In the second step, the nadir camera imagery was used to quantify the fractional snow coverage. The snow fraction, which is the fraction of bright pixels of the image, was estimated. To this end, the image was converted from RGB (red, green, and blue) into grayscale by 1gray=0.299R+0.587G+0.114B for each pixel. The weights come from standardized encoding recommendations for television (referred to as BT.601

https://www.itu.int/dms_pubrec/itu-r/rec/bt/R-REC-BT.601-7-201103-I!!PDF-E.pdf, last access: 17 October 2020

One issue of the nadir camera imagery was the darkening effect from the center to the edge of its field of view, which is known as the vignette effect. To compensate, the brightness of the image was linearly increased from edge to center through an image blending and interpolation technique by Haeberli and Voorhies (1994): 2out=1-β×black+β×gray, where black is a black image with the same dimensions as gray, and β is the image blending factor, a 2-D matrix with increasing values of 1.1–1.5 from the image center to the edge. The operator “×” denotes element-by-element multiplication. To avoid the vignetting extremes in the corners, only the imagery within a concentric sampling area was used to derive snow fraction (left panel of Fig. 2a). The key step of the snow fraction detection algorithm is the separation of dark versus bright pixels. To do this, an adaptive thresholding technique was applied. It is an approach for handling an image with unevenly distributed intensities by dividing the image into subimages and assigning different thresholds for each of the subimages (Gonzalez and Woods, 2002). The details of the adaptive thresholding are described in Appendix C. The snow fraction is then estimated by 3Frac=NbrightNtotal, where Nbright is the number of pixels above the variable threshold, and Ntotal is the total number of pixels within the sampling area. The imagery and detection results are illustrated in Fig. 2a, whereas Fig. 2b shows the simultaneously measured upwelling and downwelling spectral flux. The uncertainties associated with the estimated snow fraction are discussed in Appendix D.

The NASA C-130 aircraft was equipped with a thermometer and a hygrometer to measure air temperature and relative humidity, but it did not carry a dropsonde system. Figure 3b shows the profiles derived from the C-130 during a descending leg from 19:31:14 (altitude: 6.447 km) to 19:50:05 (altitude: 0.258 km) on 13 September 2014. Due to a malfunction of the hygrometer on 11 September 2014, no water vapor profile from the C-130 is available on this day. Instead (Fig. 3a), we used the temperature and water vapor content profiles from MERRA-2, which is an atmospheric reanalysis dataset from NASA (Bosilovich et al., 2015). MERRA-2 (M2I3NVASM) provides 3-hourly assimilated 3-D meteorological fields (dimensions: 576 in longitude; 361 in latitude; 72 pressure levels from 985 to 0.01 hPa). The comparison of the in situ profiles and MERRA-2 (Fig. 3b) shows good agreement, although the reanalysis does not reproduce the details of the vertical profile. A more systematic comparison of reanalysis and in situ data from ARISE is done by Rozenhaimer et al. (2018) and is not the focus of this paper. The observations reveal much drier and slightly colder conditions than captured in the subarctic climatology from Anderson et al. (1986), referred to here as AFGL. Nevertheless, we used the climatology above 6.5 km to provide complete temperature and water vapor profiles from 0 to 120 km, after rescaling them to the observed temperature and water vapor values at 6.5 km. The constructed atmospheric profiles were then used in the RTM (described in the next subsection) to obtain irradiance calculations.

The publicly available pixel-level MODIS cloud products (MOD/MYD06, collection 6.1), which are provided in 5 min granules (Platnick et al., 2017b), are used in this study. The MODIS cloud product includes COPs such as COT, CER, and cloud thermodynamic phase, which are essential parameters for calculating cloud radiative effects. As described before, the MODIS COT and CER are retrieved simultaneously using a bispectral reflectance method (Nakajima and King, 1990). To minimize the influence of the surface on cloud retrievals, the 1630 and 2130 nm bands are used since the snow and ice surfaces are relatively dark at those two bands (Platnick et al., 2001; King et al., 2004). These retrievals are included in the MOD/MYD06 files and will be referred to as the “1621” cloud product. Limited in situ observations suggested that the clouds consisted primarily of liquid water, and the MODIS cloud phase product showed less than 2 % of ice clouds along the flight track. Therefore, the clouds were assumed to be liquid.

The MODIS 1621 product includes COPs for cloudy and partially cloudy conditions. The latter are denoted as “PCL” in the MODIS data variable name. The product was extracted along the flight track and then input into a radiative transfer model (RTM) to calculate spectral and broadband irradiance at flight level. A 1-D RTM (libRadtran version 2.0.1, Emde et al., 2016) was used for the calculations. It requires the following inputs: 1.

day of the year (for accurate sun–Earth distance);

From the simultaneous measurements of spectral downwelling and upwelling irradiances (F(λ)↓ and F(λ)↑), the surface albedo 4α(λ)=F(λ)↑F(λ)↓ can be derived through atmospheric correction (Appendix E) from low near-surface legs under clear-sky conditions. Clear-sky measurements were a rare occurrence because low-level clouds were ubiquitous. In this study, we used clear-sky measurements of SSFR–BBR from 20:00:26 to 20:10:51 UTC on 13 September (referred to as “0913-clear-sky”). A time-synchronized video of the flight is provided in the Supplement (S1 “s1_flight-video_0913-clear-sky.mp4”). This video shows that the Arctic surface varied significantly – from snow scenes to scenes with a large amount of dark ice. Clear-sky scenes (no clouds above or below) were identified from the forward and nadir cameras. During the 0913-clear-sky case, the aircraft flew at an altitude at around 149 m.

To make full use of the direct measurements of the spectral surface albedo from SSFR–BBR, we parameterized the surface albedo by snow fraction, which can be estimated from the nadir camera imagery (described in Sect. 2.3). The parameterization was done through a data aggregation technique that combines collective measurements in a partially snow-covered environment. Figure 4 shows the surface albedo at 640, 1240, and 1630 nm plotted versus the snow fraction. The uncertainties of the surface albedo and snow fraction are indicated as vertical and horizontal error bars, respectively (details are provided in Appendix D). The data showed that linear regression can be used to establish a simple relationship between snow fraction and albedo, assuming that each observed spectrum is a mixture of only two so-called end-members: the spectral albedos of a dark and a bright surface. These end-members can vary depending on the local conditions. For example, the dark component can be either open ocean or young ice. The bright component can be either thick ice or a snow-covered surface. The resulting spectral surface albedo for a mixed sampling region is established through the slopes sλ and intercepts iλ of the linear fit, with the snow fraction SF ranging from 0 to 1 as the independent variable: 5αλ=iλ+sλSF. The linear regression coefficients (iλ and sλ) and associated uncertainties were obtained through orthogonal distance regression (Boggs and Rogers, 1990) for all the SSFR wavelengths except for the water absorption bands and those less than 350 nm or greater than 1800 nm because of a low signal-to-noise ratio. We provided the coefficients in the Supplement (see S2 “s2_surface-albedo-coefficients.h5”). This simple surface albedo parameterization has obvious drawbacks; for example, the implicit linear-mixing assumption, the variability of the end-members, and data sparsity of the individual end-members (in the example in Fig. 4, snow fractions below 0.6 rarely occur).

The snow spectral end-member (snow fraction of 1) of the mixed-scene spectral surface albedo (referred to as “2014-09-13 surface albedo”) is shown in Fig. 5. The error bars of the surface albedo are larger in the shortwave than in the near-infrared range. As expected, the surface albedo is high in the shortwave range from 400 to 900 nm and decreases in the near-infrared range. The SSFR–BBR-derived albedo spectra resemble the ground-based measurements of thick snow over ice near Davis Station, Antarctica (Brandt et al., 2005), and they are also close to springtime aircraft measurements near Utqiaġvik – formerly known as Barrow (Alaska; Lyapustin et al., 2010). Figure 5 also shows the surface albedo with zero snow fraction. As pointed out above, snow fractions below 0.6 were extremely rare during 0913-clear-sky. Nevertheless, the mixed-surface data, extrapolated to 0 snow fraction, compares surprisingly well to ground-based measurements of young gray ice, taken during the Australian National Antarctic Research Expeditions (ANARE) in 1996 (Warren et al., 1997). The spectra shape of the surface albedo at 0 snow fraction (along with the nadir camera imagery from S1) suggests that during the sampled time period, the dark pixels were ice at various freezing states instead of open ocean. As mentioned above, the binary representation of surface types oversimplifies the actual mixture of ice and snow but is adequate to serve as surface albedo input for the RTM to constrain the irradiance calculations over mixed surfaces, which is our primary goal here.

In this section, we show broadband irradiance comparisons between SSFR and BBR measurements and MODIS COPs based RTM calculations at aircraft flight level for an above-cloud case (referred to as “0911-above-cloud”) and a below-cloud case (referred to as “0913-below-cloud”), collected by the research flights on 11 and 13 September, respectively.

The RTM irradiances were calculated for wavelengths from 200 to 3600 nm. Since the SSFR–BBR-derived surface albedo described in the previous subsection was not available at wavelengths shorter than 350 nm, in gas absorption bands, and for wavelengths greater than 1800 nm due to a low signal-to-noise ratio, several techniques were applied to fill in the surface albedo spectra (details in Appendix F). For both the 0911-above-cloud and 0913-below-cloud cases, the surface albedo along the flight track was calculated from SF as a driving parameter to Eq. (). The SF was determined differently for the two cases. For 0913-below-cloud, SF was obtained from the camera imagery; for 0911-above-cloud, that was not possible because the surface was not visible through the clouds, and SF was instead set to a constant value of 76.4 %, which was obtained by varying SF in Eq. () until α1640nm reproduced the observed clear-sky baseline for the upwelling irradiance at 1640 nm. Since SF is inferred from the albedo at a single wavelength in this case, it may reflect an “effective”, rather than the actual snow fraction (as obtained from the camera imagery where available), unless the spectral shape of the albedo as acquired for 0913-clear-sky matches the one of 0911-above-cloud. This is not necessarily the case. For example, changes in snow grain size between the two cases would disproportionately affect 1640 nm. It should also be noted that the observed albedo is the blue-sky albedo (direct beam and diffuse light conditions), whereas the albedo required for the cloud cases is the white-sky albedo (diffuse light only). However, MODIS-derived surface albedo in the Arctic (not shown here) shows that usually the blue-sky albedo of snow does not deviate significantly from the white-sky albedo. The difference between the two is discussed by Gardner and Sharp (2010). In addition, it is assumed that the simple parameterization as expressed in Eq. () holds for the whole study region. This is justified because the measurements occurred in the same general area. Figure 6 shows the surface albedo calculated for SF = 76.4 % for 0911-above-cloud. Comparing with dry- and wet-season surface albedo climatology from Kay and L'Ecuyer (2013), the wet-season climatology agrees well with SSFR–BBR-derived surface albedo in the shortwave range (wavelength less than 900 nm) except for wavelength 660 nm, where climatology has a higher surface albedo. In the shortwave near-infrared range (wavelength greater than 900 nm) however, the dry-season climatology agrees better with SSFR–BBR-derived albedo than the wet-season climatology. It is worth noting that the surface albedo assumed in MODIS 1621 cloud retrievals (Platnick et al., 2018) agrees with the surface albedo we obtained from SSFR–BBR.

Figure 7 shows the broadband irradiances from SSFR–BBR, BBR, and the calculations (Fig. 7a: downwelling; Fig. 7b: upwelling) for 0911-above-cloud, where the aircraft was flying at an altitude around 7 km. The observed variability in the downwelling signal is due to the occurrence of cirrus above the aircraft, which is confirmed by the forward camera (Fig. 7a 1–3). In Fig. 7a, cirrus-free regions are highlighted in green. It shows that the cirrus decreases the measured downwelling irradiance by up to 10 % (40 Wm-2). However, there is no appreciable cirrus cover in the regions where low-level clouds are present. Since those are the focus of the paper, cirrus were not considered in the RTM. For the upwelling irradiance, the MODIS-derived baseline value of 230 Wm-2 corresponds to locations where MODIS did not detect any clouds. It is important to note that the value of the baseline indicates the RTM calculations under clear-sky conditions, which would change if a different surface albedo parameterization or a different snow fraction were used. For a SF = 76.4 %, the calculations agree with the measurements within 10 %. The cloud optical thickness along the flight track (included in Fig 7b) ranges from 0.5 to 15.3, with a median of 5.7, suggesting that MODIS does not retrieve clouds with an optical thickness below 0.5. In contrast to the calculations, the measurements show a continuous variation from leg to leg, suggesting that the clouds actually extended beyond the locations where MODIS detected them. Since the SSFR–BBR sensors integrate the cosine-weighted radiances hemispherically, they do not have the same field of view (FOV) as MODIS pixels. The clouds detected by SSFR–BBR but not by MODIS could therefore be caused by clouds located outside the FOV of MODIS. To take this into account, we assume a 90∘ (± 45∘) FOV for the SSFR–BBR that encompasses roughly half of the irradiance signal for an isotropic radiance distribution. When the aircraft was flying at 7 km, the FOV diameter of SSFR–BBR was 14 km (indicated as horizontal bars in Fig. 7b, translated into a time range using the aircraft speed). This is larger than the 1 km MODIS pixel-level product FOV. However, the results indicate that cloud portion missed by MODIS exceeds the FOV of the aircraft radiometer and therefore cannot be explained by the mismatch in the observational geometry. To further corroborate that the MODIS algorithm is indeed missing clouds, a sequence of nadir camera imagery (Fig. 7bi–iv) is considered. At close inspection, the images reveal wave patterns, suggesting the existence of thin clouds in regions where MODIS does not detect any. In this case, undetected, optically thin clouds made up more than one-fifth of the points along the flight track. Figure 7b indicates that these undetected clouds lead to an underestimation of the upwelling irradiance by 30 Wm-2 averaged over these pixels (> 10 % discrepancy). Figure 8 shows the histograms of the (a) downwelling and (b) upwelling irradiance of the cirrus-free data (marked in green in Fig. 7a). Without including the data affected by the cirrus, the downwelling irradiances from RTM agree with observations within measurement uncertainty. The upwelling irradiances from the RTM show two distinct modes. The mode on the left corresponds to clear sky and the mode on the right to a range of somewhat higher reflected irradiance due to those clouds that are detected by MODIS. From the clear-sky mode (black dashed line on the left at 233 Wm-2) to the thinnest detectable cloud (green line at 246 Wm-2), there is a gap, which, according to the measurements, is actually filled with a continuum of values from thinner clouds and/or from a variable surface albedo. Because of this gap, the low bias due to undetected clouds is smaller than or equal to 13 Wm-2. Whereas this bias caused by undetected clouds in the upwelling irradiance is almost negligible, it becomes significant for the transmitted irradiance (see below). The calculated irradiances for the locations where MODIS does detect clouds are only 10 Wm-2 lower than the measurements (4 %), which is only slightly larger than the BBR–SSFR measurement uncertainty (3 %, see Appendix D) and can be explained by either (a) incorrect COPs (optical thickness, effective radius, or thermodynamic phase) and/or (b) inaccurate or variable surface albedo. To quantify the contributions of these effects to the total discrepancy, the spectral information from SSFR is used in Sect. 3.3.

After the investigation of the above-cloud case for MODIS-derived irradiance, we turn our attention to the below-cloud case – 0913-below-cloud, which relates to near-surface irradiance. The primary cloud layer consisted of stratocumulus cloud and was located between 0.8 and 1.2 km. The cloud optical thickness (indicated in Fig. 9a) ranges from 4.1 to 8.1, with a median of 5.8. A secondary cloud layer close to the surface, located below the aircraft's minimum flight altitude of 500 ft (approximately 150 m), frequently occurs due to a temperature inversion close to the surface, where leads and cracks in the ice provide the necessary moisture for their formation. These clouds also need to be considered to quantify the radiative surface budget, but they are excluded from the analysis here because the aircraft could not underfly them. As a result, only the data from 22:21:00 to 22:25:48 UTC (minimal occurrence of the secondary cloud layer as indicated by the forward and nadir camera imagery) were selected for comparison. A time-synced video for this flight leg is provided in the Supplement (see S3 “s3_flight-video_0913-below-cloud.mp4”). As mentioned before, in contrast to the above-cloud case where the surface albedo was held constant in the RTM, the surface albedo variability on the below-cloud leg was considered here. Figure 9 shows the upwelling and downwelling broadband irradiance comparison between calculations and observations from SSFR–BBR and BBR. When incorporating the “13 September surface albedo” into the RTM, the upwelling irradiance calculations resemble the SSFR–BBR and BBR measurements (Fig. 9b). The calculations agreed well with SSFR–BBR and BBR when clouds were detected except for the time period before 22:22:48 UTC when the aircraft was entering the cloud field. The MODIS granule from Aqua was a snapshot of the cloud scene at 22:10, 10 min prior to the beginning of the flight leg. Measurement–model discrepancies for specific pixels can therefore be explained by changes of the cloud field over time. The bimodal behavior that is apparent in the time series (Fig. 9a and b) as well as in the histograms (Fig. 10) stems from time periods with and without clouds in the model input. The observations show no evidence of any cloud gap – hence only one mode appears. The “cloud gaps” apparent in the satellite but not aircraft measurements could be caused by different viewing and sun-sensor geometries between the satellite and aircraft instruments. For example, tall clouds could block the direct sun beam measured by the aircraft radiometer when flying below clouds under low-sun conditions. By evaluating the fields of cloud optical thickness and radiance at 860 nm from MODIS (Fig. 9c and d) and the flight video S3, we found that any cloud gaps are not large or frequent enough permit the direct beam to be transmitted. This leads to a smooth irradiance time series in the aircraft measurements. The gaps (circled in Fig. 9c, most likely at sub-grid scale for the 1 km product), however infrequent, do occur in the satellite retrievals. From the histograms of Fig. 10, one can estimate the pixel-level bias caused by undetected clouds. In this case, the thinnest detectable clouds are associated with 234 Wm-2 in the calculations, as opposed to the clear-sky value of 330 Wm-2 (bias of 86 Wm-2). For the upwelling irradiance, the bias is 57 (245–188) Wm-2, and the net irradiance high bias due to undetected clouds is therefore 29 Wm-2.

Although the model–measurement biases in the broadband irradiances are negligible when clouds were detected, the time series as shown in Fig. 7b do not quite match, especially for the thin parts of the clouds near the edge of a field. To diagnose the cause, we use the spectrally resolved measurements by SSFR–BBR in this section.

For 0911-above-cloud, Fig. 11 presents the spectral upwelling irradiance comparison at 860 and 1640 nm. To put these results into context, the RTM calculations were not only performed with a surface albedo from Eq. () (with SF = 76.4 %), but also with climatological surface albedos of the Arctic dry and wet seasons (0.85 and 0.75) for 860 nm from Kay and L'Ecuyer (2013). As shown in Fig. 11a, the baseline of the clear-sky RTM calculations varied significantly with surface albedo. The clear-sky measurements from 21:12:25–21:15:35 UTC are slightly below the SF = 76.4 % baseline calculation for 860 nm and above for other times. It is impossible to tell whether the variability at this wavelength stems from surface albedo variability or from undetected clouds. For 1640 nm (Fig. 11b), however, the clear-sky baseline is much more defined and less variable than other wavelengths, which is why we determined the SF value based on that wavelength.

Since any inaccuracies in the spectral surface albedo will propagate into model biases for both cloudy and clear-sky conditions, an operational surface albedo retrieval in the Arctic would be highly desirable. In this context, it is important to note that the small broadband model–measurement discrepancy of 8 Wm-2 from Fig. 7 is only achieved when the SSFR–BBR-derived surface albedo is used in the RTM calculations; when using a climatology instead, it would be larger. In other words, in absence of an operational product, the surface albedo variability dominates the uncertainty in clear- and cloudy-sky irradiance calculations.

At 1640 nm (Fig. 11b), there is good model–measurement agreement for the clear-sky baseline and for cloudy pixels that MODIS detects. That is because snow is dark in the shortwave infrared and because MODIS COPs in the Arctic are primarily based on these wavelengths. Because of the obvious distinction between cloudy and clear pixels in the measurements and calculations, it is possible to estimate the fraction of partially or fully cloudy pixels that are not detected by MODIS. Among all the cloudy pixels along the flight leg (i.e., pixels with clouds above or below the MODIS detection threshold), 27 % (highlighted in green) are actually cloudy even though MODIS identifies them as clear sky. One interesting finding from the broadband irradiance comparison (Fig. 7b) is that the calculations are low-biased relative to the observations. However, from the spectral comparison (Fig. 11), the calculations have larger (similar) values than the SSFR measurements at 860 (1640 nm). To reconcile the apparently contradictory results, we use the full spectrum from the calculations and observations at 21:24 UTC on 11 September, when the broadband calculation indicates a 6 Wm-2 low bias.

Figure 12a and b show the spectral upwelling irradiance from the RTM calculations and from the SSFR–BBR measurements, as well as the difference between RTM and SSFR–BBR. In addition to the RTM calculations with atmospheric profiles from MERRA-2 (referred to as RTMMERRA), we provided the calculations with the atmospheric profile climatology (AFGL subarctic summer, Anderson et al., 1986, referred to as RTMAFGL). The agreement between RTMMERRA and SSFR-BBR in the water vapor absorption bands indicates that MERRA-2 is sufficient to prescribe the water vapor content in the calculations. The broadband irradiance difference between RTMMERRA and RTMAFGL due to water vapor is 13.5 Wm-2. Outside of the gas absorption bands, the calculations agree with the measurements at wavelengths smaller than around 850 nm but are slightly low-biased at near-infrared wavelengths. Spectral discrepancies are caused by the use of inaccurate (1) surface albedo (2) cloud optical parameters, some of which compensate for each other in the broadband integral. Such error compensation may lead to an improved model–measurement agreement for the “wrong reasons”; therefore, validation efforts should include spectrally resolved measurements.

So far, the analysis did not reveal whether the observed model–measurement discrepancies are due to biases in the COPs or in the surface albedo. Figures 13 and 14 are an attempt to disentangle both sources of uncertainty despite the limited number of observations during ARISE. Figure 13 shows the ratio between modeled (labeled “RTM”) and measured (“SSFR–BBR”) upwelling broadband irradiance at flight level as a function of the retrieved COT for the collection of cloudy pixels from 11 September. At large COT, clouds dominate the upwelling irradiance, whereas the surface dominates in the limit of zero COT (as stated above, the retrieved minimum is 0.5). The ratio of RTM / SSFR–BBR can be used to indicate how biased the surface albedo is in the RTM when COT is approaching 0 and how biased the cloud optical properties are when the COT approaches large values. The data reveal a functional relationship between COT and the RTM / SSFR–BBR ratio. An exponential fitting of 6r=a-eb⋅COT+c is used to parameterize the upwelling irradiance ratio as a function of COT. The black curve in Fig. 13 suggests that the surface albedo in the calculations is biased low by about 8 %, whereas almost no bias is detectable in the cloud properties (a of ∼ 1.01). Figure 14 shows the spectral fits for the wavelengths between 350 and 1800 nm. Two spectra are calculated: the spectrum of the ratio when COT=0 (denoted as r0(λ)), corresponding to cloud-free conditions, and the spectrum of the ratio at infinite COT (denoted as r∞(λ)), corresponding to cloudy conditions. The r0(λ) spectrum (red) is consistently lower than 1.0 at short wavelengths (< 1300 nm) and slightly greater than 1.0 for wavelengths longer than 1500 nm. This suggests that the surface albedo is underestimated for the shorter wavelengths and overestimated for the longer wavelengths. Since changing the snow fraction will only increase or decrease surface albedo for all wavelengths, simply changing the snow fraction does not improve the agreement for both long and short wavelengths. As mentioned before, the albedo we used in the RTM is the so-called blue-sky albedo, which differs from the white-sky albedo that is captured by the measured upwelling irradiance under cloudy conditions. Instead, the discrepancies could be caused by the physical changes of the surface, different sun angles, and/or instrument performance changes. The r∞(λ) spectrum (blue) oscillates around 1.0 for the shorter wavelengths and is consistently larger than 1.0 for longer wavelengths, which suggest that the retrieved effective radius is slightly biased. Unfortunately, owing to limited sampling time, the below-cloud flight (13 September) leg does not lend itself to any conclusions from a cloud transmittance perspective since it is not the same cloud field as on 11 September. In future flight campaigns, coordinated above- and below-cloud legs will furnish more information on bias analyses than possible from ARISE.