Based on radiative transfer simulations the impact of uncertainties of the snow albedo on the retrieved cloud properties is quantified. The spectral solar reflectivity above a liquid water cloud layer was simulated using the DISORT 2 radiative transfer solver embedded in the library for radiative transfer libRadtran, . A solar zenith angle of 63∘ representative of Arctic conditions around spring was chosen in the simulations. Typical Arctic boundary layer liquid water clouds located between 200 and 500 m altitude were assumed. Cloud optical thickness τ was varied from 1 to 20 and cloud droplet effective radius reff,C between 2 and 25 µm. For all clouds, simulations with different surface albedo covering effective snow grain sizes reff,S between 10 and 800 µm were performed. The spectral snow albedo ρλ was calculated with the parameterization by using the refractive index of ice by and a form factor A=5.8. This form factor is adopted from and accounts for the non-sphericity of snow grains; it represents a mixture of randomly oriented hexagonal plates and columns with rough surfaces. This mixture and therefore the form factor A may vary depending on the local snow properties. However, an uncertainty of A can be attributed to an uncertainty in the effective snow grain size as both properties have the same spectral impact on the snow reflection characteristics, such as spectral albedo. Snow impurities by black carbon were neglected as the absorption by black carbon is typically limited to wavelengths less than 1000 nm e.g. that are not used in cloud retrieval over snow surfaces. A set of calculated spectral snow albedo ρλ is presented in Fig. a, which illustrates the decrease of ρλ with increasing values of reff,S for wavelengths λ>1000 nm where the imaginary part of the refractive index of ice is high (Fig. c).

The simulated upward nadir radiance Iλ and downward irradiance Fλ were converted into spectral cloud reflectivity γλ defined by γλ=π⋅IλFλ. In Fig. b a set of γλ for typical values of τ between 2 and 16 is shown for a fixed cloud effective droplet size of reff,C=10 µm and a typical effective snow grain size of reff,S=100 µm. The simulations illustrate that τ impacts γλ primarily at wavelengths larger than 1000 nm where the snow albedo is lower than 0.8, while lower wavelengths are less sensitive to τ.

Based on the simulated cloud reflectivities, which are used as synthetic measurements, the commonly used bispectral cloud retrieval algorithm was applied to obtain cloud optical thickness and droplet effective radius. This retrieval method is similar to the cloud product of MODIS . The retrieval uses the different dependencies of γ1600nm (less-absorbing wavelength) and γ2100nm (high-absorbing wavelength) on cloud optical thickness and cloud droplet effective radius and basically follows the method by .

The bispectral retrieval grid obtained from the simulated cloud reflectivities is presented in Fig.  for three effective snow grain sizes: 50, 100, and 500 µm. The grids significantly differ and show a considerable snow grain size effect, especially for low values of γ1600nm, while at higher reflectivities the grids tend to converge. The reflectivity γ2100nm is less affected by changes of reff,S as the snow albedo is close to zero for reff,S>100 µm (see Fig. ). As the retrieval of τ is strongly linked to γ1600nm, the effect of the snow albedo on the retrieved τ is obvious. However, the non-rectangular shape of the grids indicates that both reflectivities are coupled to both cloud parameters and, thus, also the retrieved reff,C will be affected by changes in the effective snow grain size.

For liquid water cloud retrievals obtained over snow surfaces with unknown grain size, the snow grain size effect on uncertainties of retrieval results was quantified. Therefore, synthetic measurements obtained from the retrieval forward simulations as introduced in Sect.  are applied. For each synthetic measurement defined by τ, reff,C, and reff,S, a set of retrievals assuming different values of effective snow grain sizes were performed. The purpose of this exercise is to use the synthetic measurement (for which the original reff,S is known), and start the retrieval assuming reff,S is not known. The impact of these wrong assumptions about the retrieved cloud properties is then quantified.

In Fig. a and b the retrieval results are compared to the original cloud properties for synthetic measurements calculated with an original effective snow grain size of 50 µm, but retrieved from forward simulations assuming a effective snow grain size of 200 µm (crosses). The asterisks symbols in Fig. a and b indicate the opposite case: originally reff,S=200 µm is used to produce the synthetic measurement and then reff,S=50 µm is assumed in the retrieval of the cloud properties. While Fig. a shows retrieved τ for different reff,C indicated by the colour code, Fig. b presents retrieved reff,C for clouds of different τ also indicated by a colour code.

Assuming reff,S to be larger than originally present, the retrieved τ is systematically overestimated because the surface albedo is underestimated in this case (larger reff,S assumes lower snow albedo at 1600 nm; see also Fig. ). If reff,S is underestimated (surface albedo overestimated) the snow grain size effect is inverted, leading to an underestimation of τ. This is in agreement with the general surface albedo sensitivity of cloud retrieval as discussed by and . For the case presented in Fig. , the snow grain size effect, expressed by the percentage deviation of the retrieved from the original true value, ranges up to 83 % for low optical thickness τ=3.

The results for τ do not significantly depend on reff,C. By contrast, the uncertainties introduced in the retrieval of reff,C strongly depend on τ as illustrated in Fig. b. Especially for clouds of low optical thickness, the retrieved reff,C is significantly overestimated/underestimated when the effective snow grain size is assumed to be higher/lower than originally present. The snow grain size effect ranges up to 62 % for optically thin clouds (τ=3) with small reff,C=5 µm. If larger effective snow grain sizes are assumed, the absorption observed in γ1600nm is overestimated while γ2100nm is almost unchanged. This combination leads to an estimate of too low reff,C in the retrieval.

While the retrieval of τ is always biased due to uncertainties of reff,S, independently of reff,C, no snow grain size effect is observed for the retrieval of reff,C in the case of τ larger than about 10. For optically thick clouds, the high extinction of incoming radiation inside the cloud layer leads to a low amount of radiation that reaches the surface and interacts with the snow and is transmitted back to the cloud top. In this case the interaction of radiation with the surface can be neglected and therefore the surface albedo, or the assumption of reff,S, is not relevant.

The cases discussed in Fig. a and b represent the typical range of reff,S from 50 to 200 µm as expected in Arctic areas for snow surfaces. However, white sea ice and snow cover in midlatitudes may exhibit a higher variability leading to larger uncertainties. Therefore, Fig. c and d summarize the snow grain size effect on the retrieval of τ (Fig. c) and reff,C (Fig. d) for a set of combinations of assumed and original reff,S. A typical cloud with low optical thickness of τ=4 and reff,C=10 µm (green circles in Fig. a and b) was analysed. The red dots indicate the cases that are included in Fig. a and b.

The over- and underestimation of reff,S leads to almost symmetric effects for the clouds investigated here. The maximum snow grain size effect on the retrieval of τ covered by the simulations leads to a retrieval of τ=2 or τ=6, significantly deviating from the original value of τ=4. Compared to the original reff,C=10 µm the results for reff,C range between 6 and 13 µm. The effects are most pronounced when either smaller effective snow grain sizes are assumed or originally present; e.g. 50 µm is assumed but 300 µm is present, or 300 µm is assumed and 50 µm is present. Similar mismatches between assumed and original reff,S at larger grain sizes, e.g. 300 µm and 500 µm, cause lower errors in the retrieved τ and reff,C. This indicates that, especially in polar areas where snow grain sizes are typically smaller, the retrieval biases due to a wrong assumption of reff,S cannot be neglected.

The numbers presented here were obtained for a solar zenith angle of θ0=63∘. For simulations with different solar zenith angles in the range between 45 and 80∘ similar grain size effects are observed. In general, the magnitude of the grain size effect of τ does not significantly change with θ0. However, for low Sun (large θ0), the grain size effect on the retrieved reff,C was slightly reduced, while for higher Sun, small θ0, the effects increase. This is caused by the increased probability that radiation interacts with the surface in the case of a decreasing solar zenith angle.

Based on the spectral footprint of τ, reff,C and reff,S in the cloud reflectivity, a trispectral retrieval algorithm is proposed to derive the three cloud and snow parameter simultaneously. In this way, the conventional bispectral cloud retrieval was extended by a third measurement at a wavelength sensitive to reff,S, which adds the information on the snow grain size. Compared to retrieval algorithms that rely on a fixed assumption of reff,S, this new trispectral approach reduces the uncertainty of the retrieved cloud parameters. In particular, measurements at λ1=1040 nm which are most sensitive to reff,S, λ2=1650 nm most sensitive to τ, and λ3=2100 nm most sensitive to reff,C were chosen in the retrieval algorithm.

In the new trispectral retrieval algorithm, the radiance-ratio method introduced by , , and was applied. A normalization with the cloud reflectivity at λ0=860 nm was chosen. The corresponding ratios R1, R2, and R3 are calculated as R1=γλ1γλ0,R2=γλ2γλ1,R3=γλ3γλ2,withλ0=860nm,λ1=1040nm,λ2=1650nm,andλ3=2100nm. The normalizations additionally reduce the uncertainties of the retrieval by cancelling potential biases in the radiometric calibration of the measurements. Except for λ1, all wavelengths that were chosen for the algorithm are covered by the satellite imagers MODIS and VIIRS. To apply the algorithm to global observations by these instruments, λ1 can be exchanged by the 1240 nm wavelength band where cloud reflectivity is still most sensitive to reff,S.

Similarly to Sect. , the mean standard deviation σ with respect to τ, reff,C, and reff,S was calculated for the three reflectivity ratios. Table  shows σ for all possible parameter combinations. The higher σ, the more sensitive R1, R2, and R3 are with respect to changes of an individual cloud or snow parameter. One ratio with significantly higher σ is found for each parameter, τ, reff,C, and reff,S. For the cloud optical thickness the highest σ=0.145 is found for R2, which is almost twice as high as the sensitivity of R1 and R3. Similarly, R1 shows the highest sensitivity to reff,S and R3 the highest σ for reff,S and reff,C with σ=0.074 and σ=0.121, respectively. This indicates that R1, R2, and R3 are well suited to separate the information of τ, reff,C, and reff,S from spectral reflectivity measurements.

The set of cloud reflectivity simulations introduced in Sect.  served for the forward simulations (solar zenith angle of 63∘, liquid water cloud layer between 200 and 500 m). The simulated grids of R1, R2, and R3 used for the trispectral retrieval are presented in Fig. . While Fig. d shows the full 3-dimensional (3-D) grid covering τ=1–20, reff,C=5–25 µm and reff,S=25–800 µm, Fig. a–c presents two-dimensional projections of reflectivity ratios with one parameter being fixed. In Fig. a, reff,S is set to 50 µm (black) and 500 µm (red), while in Fig. b reff,C is fixed at 5 µm (black) and 16 µm (red) and in Fig. c τ is fixed to values of 4 (black) and 16 (red). The simulated grids show that the individual lines almost align orthogonally for wide ranges of the simulated parameter space. The different surfaces do not overlap. This indicates that the three chosen ratios R1, R2, and R3 separate the influence of the three parameters τ, reff,C, and reff,S on the cloud reflectivity and allow a retrieval of the cloud and snow parameters leaving few ambiguities. Only for reff,S lower than 50 µm is the grid more narrow, as seen in Fig. d (uppermost black grid) and Fig. a (black grid). Such a narrow grid increases the uncertainty of the retrieved τ and reff,C, especially for clouds of low optical thickness. This grid characteristic is caused by the higher surface albedo of snow for small values of reff,S, which reduces the contrast between cloud and snow surface for the wavelengths used to calculate the three ratios. However, a retrieval of cloud and snow properties is still possible in these ranges if the measurement uncertainties are sufficiently small. A more general quantification of the retrieval sensitivities and uncertainties can be derived by optimal estimation techniques, which is, however, beyond of the scope this paper.

Ambiguities appear in the retrieval for small cloud droplet effective radii of reff,C<5 µm (not shown in Fig. ). In that case the absorption of cloud droplets is weak and the cloud reflectivity is similar to a cloud with larger reff,C but smaller τ. Therefore, all solutions with reff,C<5 µm were excluded from the retrieval.

The retrieval algorithm was additionally adjusted to Arctic conditions where open water and ice floes can occur in close proximity. First, the algorithm determines whether the measurements are obtained over snow-covered sea ice or open water surfaces. Even in the case of cloud-covered scenes, the border between sea ice and open water is clearly discernable in the cloud top reflectivity as shown by . A surface with a high albedo always enhances the upward radiance above a cloud even in the case of optically thick clouds. Therefore, a fixed threshold of cloud reflectivity γ(λ0)>0.65 at a wavelength of 860 nm is applied to distinguish measurements obtained above snow or white sea ice from those over open water. This value may need to be adjusted for conditions of different solar zenith angles or clouds with higher optical thickness. The threshold assumes that the field of view of the radiance measurement (pixel size) is fully covered by a single surface type that is reasonable for airborne measurements. However, for satellite observations with typically larger pixel sizes, scenes dominated by broken sea ice might be misclassified.

If the cloud reflectivity at 860 nm is below the threshold, an open water surface is assumed. In that case the bispectral retrieval following the radiance ratio approach of is applied. Cloud reflectivities γ(λ2) and the reflectivity ratio R3=γ(λ3)/γ(λ2) are used to retrieve τ and reff,C. If γ(λ0) is above the threshold, a snow surface is assumed. Then the measurements are converted into the three reflectivity ratios R1, R2, and R3 and interpolated to the 3-D grid of simulated values introduced in Sect. .

However, 3-D radiative effects in the vicinity of sea ice edges significantly influence the reflected radiation and bias the retrieved cloud properties . Depending on cloud and surface geometry, this effect is important up to several kilometres from the ice edge. Therefore, measurements collected close to sea ice edges have been removed from the analysis.

The retrieval uncertainties are estimated considering the measurement errors of the reflectivity ratio and assuming a Gaussian distribution of the errors. In this case the uncertainties can be expressed by their double standard deviation 2σ. The retrieval is operated for varying each ratio, R1, R2, and R3, separately by adding and subtracting 2σ. This procedure results in six solutions for the trispectral retrieval over snow surfaces and four solutions for the bispectral retrieval over open water. This set of solutions is sufficient to represent the full solution space of the Gaussian distributed measurement uncertainties. The median of these solutions is used as the retrieval result of τ, reff,C, and reff,S, while the standard deviation of all solutions quantifies the retrieval uncertainty Δτ, Δreff,C, and Δreff,S.

Independent of these uncertainties caused by the measurements, systematic errors due to the assumptions in the forward simulations are considered. Currently, the retrieval algorithm is limited to liquid water clouds and, therefore, may suffer if the cloud contains ice. In the Arctic, mixed-phase clouds that are dominated by a liquid water layer at cloud top are frequent . In this case, the retrieval may provide unrealistic cloud properties as the ice crystals absorb solar radiation at similar wavelengths to the snow surface. The absorption by the ice crystals may add to the absorption by the snow surface and bias the results.

Furthermore, limitations of the snow albedo parameterization by applied in the forward simulations may introduce biases in the retrieved reff,S. The parameterization assumes a fixed snow grain shape quantified by the form factor A=5.8. Typical values of A range from 5.1 for fractals to 6.5 for spheres. For a retrieval of snow grain size in cloud-free conditions, the snow grain shape implies uncertainties in the retrieved effective snow grain size up to 25 %. However, as discussed by , the reflection characteristics of the snow, in particular the spectral albedo, are not affected at wavelengths used by the retrieval algorithm. Both properties A and reff,S change the snow albedo with similar spectral patterns, which allows the uncertainty of A to be attributed as an uncertainty of reff,S. Therefore, the retrieved cloud properties are independent of the assumption of A.

This emphasizes that the retrieved reff,S always has to be considered as an effective quantity. It represents the snow grain size that has to be used in the specific albedo parameterization (fixed form factor A) to provide the snow albedo which is most representative for the measurements. Spatial or temporal differences of the retrieved reff,S may result either from a change of the geometric size of the snow grains or from changes in the snow grain shape quantified by the form factor.

The observations in this case were collected exclusively over snow-covered sea ice and obtained between 16:54 and 17:21 UTC with a solar zenith angle of 63∘. The retrieved cloud and snow properties are presented in Fig. . Cloud optical thickness ranged between τ=6 at the beginning and τ=15 at the end of the flight leg. reff,C also shows a tendency of increasing values between 6 and 9 µm, while reff,S remained almost constant at values around 100 µm. In situ cloud microphysical measurements of reff,C had been obtained along the same flight track, about 1 h after the remote sensing measurements. At the cloud top, two derived vertical profiles show reff,C between 6 and 7.5 µm, which is in the range of the retrieval results.

The retrieval uncertainties are indicated by the shaded areas in Fig. . In the first part of the time series, up to 17:07 UTC, the retrieval uncertainties are lowest for τ with about Δτ±1, and range at Δreff,C±1.5 µm for reff,C, and Δreff,S±60 µm for reff,S. In the second part of the time series, the retrieval uncertainty Δτ significantly increases. This correlates with the increase of the cloud optical thickness. Similarly, Δreff,S shows a slight negative correlation with uncertainties being higher for low τ. This is also reflected by the retrieval grids presented in Fig. . The grid spacing is narrower for larger τ.

For all measurements, the behaviour of Δτ and Δreff,S as functions of the retrieved τ and reff,S is shown in Fig. . For Δτ, values of reff,S are colour-coded in each data point, while for Δreff,S colours indicate τ. Positive correlations are found for both parameters: the larger the retrieved τ or reff,S, the larger their uncertainties. For Δτ uncertainties are larger for small reff,S (colour code in Fig. a). In this case, small reff,S increase the snow albedo and lower the contrast between clouds and snow surface at λ=1650 nm, which then becomes less sensitive to τ. Similarly, Δreff,S depends on the retrieved τ (colour code in Fig. b), with higher uncertainties observed for large τ. For high optical thickness the clouds begin to mask the surface, and information of the surface is lost in the reflected radiation measured above cloud top leading to higher uncertainties of reff,S. For Δreff,C, similar dependencies are obtained but with less spread; Δreff,C ranges between ±1.2 and ±1.7 µm (not shown here). Δreff,C was found to increase slightly with decreasing reff,S and increasing reff,C.

The snow-covered sea ice below the clouds did have some open or only recently frozen leads, which were identified when Polar 5 flew below clouds after the remote sensing flight leg. From automatic photographs taken on board Polar 5, the proportion of leads was estimated to be lower than 5 %, which might explain some of the higher values observed in the retrieved time series of reff,S.

The retrieved cloud and snow properties were compared to satellite observations by MODIS . Figure  shows maps of cloud optical thickness τ (a), droplet effective radius reff,C (b) and effective snow grain size reff,S (c) retrieved by MODIS. The flight track of Polar 5 is indicated by a black line and overlayed by the retrieval results of SMART. τ and reff,C retrieved by MODIS along the flight track are additionally included in Fig. . Cloud properties are obtained by the MODIS cloud product Collection 6 for observations over snow or sea ice using band 6 at 1640 nm and band 7 at 2130 nm . The effective snow grain size is provided by the Snow Grain Size and Pollution amount (SGSP) retrieval algorithm by . The SGSP is limited to cloud-free pixels and therefore does not show values below the clouds observed in the same image. For Case I, the Aqua overpass of 20:00 UTC was analysed. Although the MODIS scene was taken about 3 h after the airborne measurements, the stable cloud conditions allow a comparison. Snow grain sizes typically change over longer timescales, except when precipitation occurs. The weather station in Tuktoyaktuk close to the coastline did report light precipitation of snow grains but is not necessarily representative for the clouds over the Beaufort Sea. However, a direct comparison of reff,S is not possible due to the missing data in cloudy pixels.

The MODIS cloud product in Fig.  shows τ and reff,C in the same range as that retrieved from the airborne measurements. Note that here a longer time series is shown than presented in Fig. . This includes areas with snow-covered land surfaces, for which the retrieval can be applied assuming that the snow layer is sufficiently thick and the snow albedo is not affected by the underlying surface . At the southern edges of the cloud field, lower values of τ are observed by both MODIS and SMART. For reff,C the values retrieved by SMART show the same tendency of lower reff,C at the southern cloud edge and increasing reff,C towards the western end of the flight track. For large areas of this cloud field the MODIS cloud product did not provide valid solutions, which illustrates the limits of the current cloud retrieval in Arctic regions.

For reff,S no direct comparison is possible. However,the retrieval using measurements by SMART fills the gap of the cloudy areas not considered in the SGSP retrieval for MODIS. The retrieved reff,S are in the same range as those observed by MODIS south and north of the cloud field and therefore are considered to be consistent with the SGSP product. Lower reff,S were detected by MODIS at higher latitudes with some bias due to large leads in the sea ice that are imprinted in the retrieval results. At lower latitudes, the snow is strongly influenced by accelerated metamorphism processes due to higher temperatures and therefore exhibits larger reff,S.

For the second case, observations collected on a flight leg crossing a sea ice edge between 16:45 and 17:12 UTC were analysed. This transition allows for testing the consistency of the proposed retrieval algorithm for observations over snow and open water. Compared to Case I, the cloud altitude was lower with 200 m cloud top altitude indicating a thinner cloud layer. The time of day and solar zenith angle were almost identical to Case I.

The retrieved cloud and snow properties for Case II are presented in Fig. . Additionally, the light grey shaded time sections indicate measurements above the open ocean while during non-shaded times snow-covered sea ice was present below the clouds. As suggested by for the given cloud base and top altitude (0–200 m), measurements within a distance of 400 m to the sea ice edge were removed from the analysis in order to minimize the impact of 3-D-radiative effects at the sea ice edge.

Over sea ice, the retrieved τ is almost constant with values around 5. Across the sea ice edge, τ decreases to about 2 and later slightly increases up to τ=10 with increasing distance to the ice edge. The systematic decrease of τ over the sea ice edge as retrieved by the airborne measurements extends up to 8 km. As indicated by , the radiative field across a straight sea ice edge is affected by the surface albedo transition only up to distances of about 400 m. Therefore, the coincidence of the decrease of τ with the sea ice edge observed here is attributed to the retrieval algorithm but rather has natural causes. It cannot be concluded from this single cross section whether the change of the surface and therefore the change of surface latent and sensible heat fluxes affected the cloud properties across the sea ice edge. As the high-resolution MODIS observations indicate, the cloud field showed an oscillating pattern, which might have coincided with the sea ice edge at the location of the airborne observations.

The retrieved reff,C vary slightly more strongly over open water, while the retrieved values over snow-covered sea ice are almost constant at about reff,C=8 µm with only short sections of higher cloud droplet sizes. Over open water, larger cloud droplets are found with an average of about 10 µm. Close to the sea ice edge, until 17:00 UTC, reff,C is found to slightly increase with increasing distance to the sea ice edge simultaneously with the increase of τ. The in situ microphysical measurements cover two cloud profiles along the same flight track, one observed above open ocean and one above sea ice. Both profiles showed no difference with reff,C of about 9 µm at cloud top, which is higher than Case I and in agreement with the retrieval results.

The retrieved reff,S shows a slightly higher variability and partly higher values ranging between 50 and 200 µm compared to Case I. The larger snow grains might result from the different location, the advanced time, and snow metamorphism, or the closer location to the open water. For example a systematic decrease of reff,S with distance to the sea ice edge is visible (16:52–16:54 UTC, about 6 km distance). In Case II the observations take place above compact fast ice without any leads. Photographs from a flight section in the same area below the clouds showed that the fast ice was partly free of snow, which may have caused the higher variability and the single peak of reff,S=300 µm.

The comparison of cloud and snow properties retrieved by SMART and MODIS is shown in Fig. , similarly to Case I. MODIS results along the flight track are additionally included in Fig. . For Case II, the MODIS image was observed at 21:25 UTC, more than 4 h after the airborne observations. Similarly to Case I, the temporal variation of the cloud properties is expected to be low as stable dynamic conditions in a high-pressure system prevailed during the time of observations and before. Figure  shows that the low values of τ observed by MODIS were also covered by SMART. The slight increase of τ in the western end of the flight leg is represented by the retrieval using SMART data. A similar pattern and agreement was found for reff,C.

The direct comparison of the time series in Fig.  confirms the general agrement, although differences in the location of cloud fluctuations are obvious. Above sea ice, MODIS observed a steady increase of τ and reff,C and a similar drop at the ice edge as retrieved by SMART. However, a more quantitative comparison of SMART and MODIS cloud products is not possible due to the time difference between the observations. Differences in the retrieved cloud properties may either result from the assumption of snow albedo or from temporal changes of the cloud. Over open water, the MODIS cloud product provided lower τ and reff,C. This is likely caused by the 4 h difference between both observations. Due to the subsidence in the anticyclonic conditions the cloud top continued to decline and reduced the amount of condensed water, τ, and reff,C. The effective snow grain sizes retrieved by SMART are in the range of reff,S retrieved by MODIS although the SGSP algorithm could provide results only in small cloud-free areas. The single measurements at the western end of the flight leg indicate that individual ice floes encountered on the ocean and show slightly lower reff,S. This is an indication of fresh snow precipitation in this area where the cloud optical thickness increased.