Airborne measurements of directional reflectivity over the marginal sea ice zone

The directional reflection of solar radiation by the Arctic Ocean is dominated by two main surface types: sea ice (often snow-covered) and ice-free (open) ocean. However, in the transitional marginal sea ice zone (MIZ), the reflection properties of both surface types are mixed, which might cause uncertainties in the results of retrieval methods of atmospheric parameters over the MIZ using airborne and satellite measurements. To quantify these uncertainties, respective measurements of reflection properties of the MIZ are needed. Therefore, in this study, an averaged hemispherical-directional reflectance factor (HDRF) of 5 the inhomogeneous surface (mixture of sea ice and open ocean surfaces) in the MIZ is derived using airborne measurements collected with a digital fish-eye camera. For this purpose, a sea ice mask was constructed to separate the reflectivity measurements from sea ice and open ocean pixels. The separated data sets were accumulated and averaged to provide separate HDRFs for sea ice and open ocean surfaces. The respective results were compared with simulations and independent measurements available from the literature. Using the sea ice fraction derived in parallel from the digital camera images, the mixed HDRF 10 describing the directional reflectivity of the inhomogeneous surface of the MIZ was reconstructed by a linear weighting procedure. The result was compared with the original measurements of directional reflectivity over the MIZ. It is concluded that the HDRF of the MIZ can be well reconstructed by linear combination of the HDRFs of homogeneous sea ice and open ocean surfaces, accounting for the special conditions present in the MIZ compared to homogeneous surfaces.

2 Methodology and measurements

Definition of reflectance quantities
The spectral BRDF f BRDF of a surface describes the directional distribution of the reflected radiation and is defined as: 90 (Nicodemus et al., 1977) . F i represents the spectral irradiance (in W m −2 nm −1 ) illuminating (subscript "i") a surface at wavelength λ from the direction characterized by the incident zenith and azimuth angles, θ i and ϕ i , respectively. I r quantifies the radiance (in W m −2 nm −1 sr −1 ) reflected (subscript "r") into the direction characterized by the reflection zenith and azimuth angles, θ r and ϕ r , respectively, and depends additionally on the incident angles. The BRDF has the unit of inverse steradiant (sr −1 ). Often the bidirectional reflectance factor (BRF) R BRF is used instead of the BRDF. The reflectance factor (without 95 unit) is defined as the ratio of the BRDF of the actual surface to the constant BRDF of a Lambertian surface, which is equal to 1/π sr −1 . Thus: Since the illumination under atmospheric conditions is a combination of a direct and a hemispherical diffuse irradiance component, F dir and F diff , respectively, both BRDF and BRF cannot be measured practically. Therefore, the hemispherical-directional 100 reflectance factor (HDRF, without unit) R HDRF is introduced (e. g., Schaepman-Strub et al., 2006). However, if the diffuse fraction of the incident radiation is sufficiently small, the HDRF represents a good approximation of the BRF. The HDRF is determined by: R HDRF (θ i , ϕ i , 2π; θ r , ϕ r ) = π sr · I r (θ i , ϕ i , 2π; θ r , ϕ r ) Here, the additional argument 2π refers to the diffuse radiation incidence over the whole hemisphere while θ i and ϕ i refer to 105 the direction of the direct component. The spectral dependence is omitted here.

Airborne campaign
In this paper, data collected during the Arctic CLoud Observations Using airborne measurements during polar Day (ACLOUD;  campaign, which took place in May and June 2017, are analyzed. During ACLOUD, the MIZ was 110 located at about 80 • N in the region north-west of Svalbard (Norway). 19 measurement flights were conducted with each of the two research aircraft Polar 5 and Polar 6 from Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research (AWI; Wesche et al., 2016). Both aircraft were equipped with downward-looking 180 • fish-eye cameras measuring the upward directional radiances . Additionally, on Polar 5 the Spectral Modular Airborne Radiation measurement SysTem (SMART; Wendisch et al., 2001) was operated to measure the spectral downward solar irradiance.

180 • Fish-eye camera
The upward radiance was measured by a CANON EOS-1D Mark III digital camera equipped with a 180 • fish-eye lens. Images with a resolution of 3906 × 2600 pixels were taken every 6 s. As common for commercial cameras, each pixel covers three 125 spectral channels (RGB) centered at wavelengths of 591 nm (red), 530 nm (green), and 446 nm (blue) with a full width at half maximum (FWHM) of about 80 nm Carlsen et al., 2020). Figures 1b and 1d show two examples of raw true-color images taken by the fish-eye camera. The camera was calibrated in terms of geometrical, spectral and radiometric characteristics, which allows a conversion of the measured raw data into radiances as described in detail by Carlsen et al. (2020).
In contrast to Carlsen et al. (2020), who applied a stellar method for the geometrical calibration, images of checkerboards taken 130 from different perspectives served as reference. The images were analyzed by the open source routine cv2.fisheye from the free programming library OpenCV (http://opencv.org; Jiang, 2017). The backward model described by Urquhart et al. (2016) was applied to calculate the camera-fixed viewing zenith and azimuth angles, θ v and ϕ v , respectively, of each image pixel using the OpenCV output parameters.
The fish-eye camera was fixed to the aircraft frame. To obtain radiance measurements with respect to an Earth-fixed coordi-135 nate system, the viewing angles (θ v , ϕ v ) of each camera pixel were corrected to consider the aircraft attitude angles (roll, pitch, yaw). Euler rotation matrices were applied to transform the viewing angles into the reflection angles (θ r and ϕ r ) (Ehrlich et al., 2012). The azimuth plane of the images was rotated with respect to the relative position of the Sun, such that the Sun (and the forward direction) is located at the right in all polar plots shown in this paper. The footprint of one single image varied between 380 m and 915 m for the altitude range of the 20-minute leg assuming that the effective FOV of the fish-eye lens is 160 • .

Calculation and uncertainty of the HDRF
Combining the downward irradiance F i measured by SMART  and the angularly resolved radiances I r from the fish-eye camera  allows the calculation of the HDRF in flight altitude (Eq. 3), whereby the spectrally resolved irradiances were converted into the spectral range of each camera channel using the individual relative spectral response function from the spectral calibration.

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The uncertainty of the radiometric calibration of the fish-eye camera was estimated by 4 % (Carlsen et al., 2020). Further uncertainties stem from the sensor characteristics, the geometric calibration, and the aircraft attitude correction, leading to a total uncertainty of the fish-eye camera radiance measurements of about 4.2 %. However, during ACLOUD complementing radiance measurements were performed with SMART and a spectral imager, which demonstrated deviations of up to 35 % for the blue channel, while reflected radiances measured in the green and red channels ranged within the measurement uncertainties 150 . As a consequence, the fish-eye camera was inter-calibrated. However, for the further analysis in this study only radiances measured in the red channel were selected, since they have shown the best agreement within the instrumental intercomparison. The root mean square deviation between the radiances of the digital camera and SMART in the red channel amounts to 0.01 W m −2 nm −1 sr −1 and the correlation coefficient is 0.98.
According to Bierwirth et al. (2009), the total uncertainty of the SMART irradiance measurements is 3.2 % in the visible 155 spectral range. However, an updated transfer calibration (3 % error) and a larger cosine correction error (2 %) due to the larger solar zenith angle lead to an increased total uncertainty of 4.3 %. Thus, the total uncertainty of the calculated HDRF amounts to about 6 %.
Radiative transfer simulations performed with the library for radiative transfer (libRadtran; Emde et al., 2016) were used to estimate the impact of the atmosphere between the ground and the maximum flight altitude (165 m) on the measured HDRF. ocean, pixels with HDRF values above a second threshold of h 2 = 0.6 are mostly related to sea ice but can also be assigned to open ocean in case of specular reflection observations. HDRF values between both thresholds, which are mostly linked to the ice floe edges, amount to roughly 3 % of the data and were excluded from further analysis.

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In the sunglint region, also the HDRF of open ocean can feature values similar to or even exceeding typical values of sea ice.
This sunglint area was identified by two additional criteria. Firstly, HDRF values larger than a third threshold of h 3 = 1.3 were considered as sunglint and assigned to open ocean. Secondly, a color ratio defined by the ratio of the radiances measured in the red and the blue camera channel was used to identify the edges of the sunglint zone. Since the sunglint area appears more yellowish than the sea ice, its color ratio is higher. A color ratio threshold of c = 0.95 was chosen for the separation of sea ice and sunglint when the HDRF was between h 2 and h 3 . In Fig. 3a the color ratio of the exemplary scene (same as in Fig. 1d) is shown with values higher than 0.95 highlighted in red. Together with Fig. 3b, which shows the surface types identified by the sea ice mask, it demonstrates that the mask is capable to separate sea ice, open ocean, and sunglint. The complete decision process of the sea ice mask is summarized in Fig. 4. An uncertainty analysis of the sea ice mask with respect to the applied thresholds is given in the following section based on the calculated sea ice fraction.

Sea ice fraction
Using the sea ice mask, the sea ice fraction was calculated for each image. The sea ice fraction usually refers to a horizontal surface. The fish-eye lens of the camera, however, weighs the individual pixels equally although each pixel refers to a different surface area. Nadir pixels cover a smaller area than pixels close to the horizon. These different pixel projections were taken into account when the sea ice fraction averaged over a whole image was calculated.   of pixels is obvious for extreme cases, the uncertainty of the sea ice fraction due to the sea ice mask is estimated to be less than 4 %. types.  Fig. 6b. Although there is also some directional variability in the pixel-based sea ice fraction, it is higher than 0.8 for most reflection directions.

HDRF of open ocean
In the observations presented here, the sunglint does not have the shape of a typical Gaussian distribution as reported in 220 literature (e. g., Cox and Munk, 1954;Gatebe and King, 2016;Ehrlich et al., 2012). Instead, several smaller local maxima are obvious beside a global one and the sunglint is slurred towards the horizon. The irregular shape of the sunglint is likely a result of the low amount of observations and is imprinted in the pixel-based sunglint fraction (grey line in Fig. 6b). The pixel-based sunglint fraction is defined as the ratio of the number of observations identified as sunglint to the number of all observations assigned to open ocean by the sea ice mask for the respective direction. It implies that the shape of the sunglint is 225 highly variable among the images used for averaging, ranging from pure specular reflection of the Sun to sunglints cut by the edge of an ice floe (i. e., Fig. 1b) and widely blurred sunglints (i. e., Fig. 1d). The width of the sunglint primarily depends on the surface roughness, which is related to the surface wind speed (Cox and Munk, 1954). However, the distribution of sea ice and open ocean in the MIZ affects the surface roughness and its dependence on the wind speed. The development of waves is weaker in the gaps between ice floes than in the homogeneous open ocean (Kohout et al., 2011). This leads to the hypothesis  (second y-axis).
of different surface types (Mayer and Kylling, 2005) including open ocean (Cox and Munk, 1954   Interestingly, the simulated open ocean HDRF outside the sunglint is significantly lower than the separated one. Nearly independent of the wind speed, the mean simulated HDRF of the shadow side in the solar principle plane (−90 • to 0 • ) is around 0.02, which is about one order of magnitude lower than for the separated HDRF. It is likely that these differences are 265 due to horizontal photon transport. In the MIZ some photons reflected from the sea ice surface are scattered in the atmosphere such that they are detected in directions that actually point to open ocean as discussed by Schäfer et al. (2015). This effect may increase the HDRF of open ocean in the MIZ compared to homogeneous ice-free ocean.

HDRF of snow-covered sea ice
Similar to open ocean (Fig. 6), the average HDRF of the separated snow-covered sea ice areas (separated sea ice HDRF) is 270 shown in Fig. 8. Note, that the ranges of the y-axes in Figs. 6b and 8b are different. In comparison to open ocean, the HDRF values are significantly higher for a large part of the angular domain with values around 0.9. The HDRF is slightly enhanced in the forward direction, which is obvious in both the polar plot (Fig. 8a) and the HDRF along the solar principal plane (Fig. 8b), and is in accordance with the literature (e. g., Bourgeois et al., 2006;Gatebe et al., 2003;Goyens et al., 2018). Furthermore, the variability of the separated sea ice HDRF (standard deviation below 0.20) is much lower than that of the separated open 275 ocean HDRF due to the high sea ice fraction providing a higher number of images used for averaging.
In order to assess potential differences between the HDRF of homogeneous snow-covered sea ice surfaces and sea ice areas in the inhomogeneous MIZ, the separated sea ice HDRF is compared to homogeneous sea ice and snow HDRFs obtained from two studies Carlsen et al., 2020). The ground-based fish-eye camera measurements described by  these measurements was similar to the SZA observed in our case study (about 58 • ). The comparison of the HDRF along the solar principle plane is shown in Fig. 9.
The snow HDRF from Carlsen et al. (2020) (green line) is larger than the separated sea ice HDRF (black line) for all 285 reflection directions of the solar principle plane. Although the shape of both HDRFs is similar, the difference between them (0.19 at nadir) decreases with increasing reflection zenith angle in the forward direction (min. 0.12 at 59 • ). The HDRF of snowcovered sea ice (yellow line) observed by Goyens et al. (2018) agrees with the separated sea ice HDRF only for reflection zenith angles less than −60 • . With increasing reflection zenith angle, the difference between the HDRFs increases to 0.95. While the minimum of the separated sea ice HDRF seems to be around nadir, the snow-covered sea ice HDRF from Goyens et al. (2018) 290 is lowest in the backward direction (at about −60 • ). However, the well documented increase of the HDRF towards the horizon in forward direction is also visible in their observations. The larger anisotropy of the snow-covered sea ice HDRF observed by Goyens et al. (2018) can be explained by the smaller snow grain size (e. g., Warren et al., 1998). Although the exact opticalequivalent snow grain size is not known, the snow sampled by Goyens et al. (2018) is referred to as cold snow which typically has smaller grains than snow in the MIZ. Analyzing sections with high sea ice fraction, the snow grain size retrieval following 295 the approach by Zege et al. (2011) showed grain sizes of about 200 µm during ACLOUD. However, the optical-equivalent snow grain size on the Antarctic Plateau retrieved by Carlsen et al. (2020) was also significantly smaller (78 µm) than in the MIZ, but the anisotropy of their HDRF is not enhanced compared to the anisotropy of the separated HDRF. The variability of the snow-covered sea ice HDRF observed by Goyens et al. (2018) is slightly larger than that of the other aforementioned HDRFs. One reason might be the smaller footprint of ground-based measurements compared to airborne observations and, Although the HDRF increases towards the horizon in forward direction, the shape of the bare ice HDRF is very irregular. Its variability is even larger than that of the snow-covered sea ice HDRF. According to Goyens et al. (2018) this is due to the presence of thawed ice nearby highly reflective ice grains, which often occurs at the beginning of the melt season.

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In summary, the separated sea ice HDRF is lower compared to the HDRFs of homogeneous snow-covered surfaces. Although the snow grain size is larger in the MIZ, the weak absorption of snow for visible wavelengths cannot explain these differences. While the measurements from Carlsen et al. (2020) were performed in the green camera channel (central wavelength of 538 nm), the red channel (628 nm) measurements by Goyens et al. (2018) were used for comparison. However, the spectral difference of white snow in the visible range should be small. Rather, the HDRF in the MIZ could be slightly smaller 310 due to the effect of horizontal photon transport mentioned above. However, since several other properties (such as snow grain shape, snow pack density, or impurity concentration) can affect the snow HDRF, the comparison illustrates the variability of the snow and sea ice HDRF in Arctic environments.

HDRF of the MIZ as function of sea ice fraction
In the next step, the average HDRF of the inhomogeneous sea ice-open ocean surface of the MIZ (mean MIZ HDRF, obtained 315 by averaging all images without separation) is analyzed (Fig. 10a). Despite the high sea ice fraction, the mean MIZ HDRF shows features of the HDRFs of both open ocean and sea ice surfaces. The strongly enhanced reflectance in the sunglint region is clearly visible. However, because of the high sea ice fraction, its maximum HDRF (about 2.6) is significantly lower compared to the separated open ocean HDRF (see Fig. 6a). Outside the sunglint but still in forward direction, the slightly enhanced HDRF characteristic for the sea ice surface is imprinted in the mean MIZ HDRF (compare Fig. 8a). For all other directions, the HDRF 320 is more or less isotropic with values slightly lower (mean of 0.74 on the shadow side) than observed in the sea ice HDRF (0.85), due to the contribution of open ocean surfaces.
In the following, the HDRF of the MIZ is reconstructed (HDRF recon ) assuming a linear combination of individual HDRFs of open ocean HDRF ocean and sea ice HDRF ice weighted by the sea ice fraction f ice : v eff is considered as an effective wind speed, that would produce the same surface roughness and, thus, the same open ocean HDRF if the ocean was ice-free. v meas is the wind speed measured at flight altitude and scaled to 10 m. It has to be noted that this very basic relation between surface wind speed and sea ice fraction aims only to illustrate the effects in a qualitative view.

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Numbers may change if observations for different sea ice conditions are available.
The reconstructed MIZ HDRF calculated for f ice = 0.83 (Eq. 4) is shown in Fig. 10b and is compared to the mean MIZ HDRF (Fig. 10a). The difference between both HDRFs is less than 0.1 for more than 96 % of the pixels. For more than 86 % of the pixels, the difference lies within the uncertainty range of the HDRF measurements (6 %). The reconstructed MIZ HDRF appears more smoothed than the mean MIZ HDRF for statistical reasons. The smoothness was quantified by the 340 standard deviation of the HDRF calculated with respect to all reflection directions of the shadow side (to exclude the sunglint contribution). For the reconstructed MIZ HDRF the standard deviation is slightly lower (0.035, 4.7 % of the mean value) than for the mean MIZ HDRF (0.044, 5.9 % of the mean value). The difference between both HDRFs originates from the calculation Figure 11. Sea ice fraction observed at each pixel throughout the entire image sequence (pixel-based sea ice fraction).
of the reconstructed MIZ HDRF with the mean sea ice fraction, where homogeneous conditions are assumed. This means that the sea ice fraction is assumed to be uniform for each pixel (direction). In the observations this homogeneity is not given due 345 to the limited number of images used for averaging. Figure 11 shows a polar plot of the pixel-based sea ice fraction, that was introduced in Sect. 4. It is obvious, that for each pixel the pixel-based sea ice fraction is different, covering a wide range HDRF on the sea ice fraction (Eq. 5) is neglected in Fig. 10c, the sea ice fraction-dependent simulations are used in Fig. 10d.
Based on the linear combination (Eq. 4), both panels show a decrease of the magnitude of the sunglint peak with increasing sea ice fraction. However, in Fig. 10d, also the shift of the sunglint towards the horizon is visible as the effective wind speed increases with decreasing sea ice fraction. Furthermore, the maximum of the sunglint contribution is significantly decreased in Fig. 10d. For sea ice fractions lower than observed, this is partly due to the higher effective wind speed. Additionally, the 360 sharp peak that is visible in in Fig. 10c as a result of the pure specular reflection observed in some images, is not present in the simulations, which reduces the HDRF maximum. This difference in the sunglint region is the major discrepancy between the reconstructed MIZ HDRF for the observed sea ice fraction (0.83, purple line in Fig. 10d) and the mean MIZ HDRF. Outside the sunglint, they show good agreement. For 65 % of the directions of the solar principle plane the relative difference between both HDRFs is within the uncertainty of the HDRF measurements.