A snow layer is an aggregate of ice crystals with varying shapes and dimensions, embedded in air and laying on a surface. Two different approaches to define the IOP of a snow layer have been implemented in SCIATRAN: a database proposed by and analytical expressions for IOP developed by .

The former approach uses pre-calculated optical parameters of ice crystals with different shapes and dimensions. The SCIATRAN RTM adopts optical parameters for the following nine crystal shapes from the database by : aggregate of 8 columns, droxtal, hollow bullet rosette, hollow column, plate, aggregate of 5 plates, aggregate of 10 plates, solid bullet rosette, and column. The implemented database contains optical properties of the ice crystals in the spectral range of 0.2–15.25 µm for particle sizes (described by the maximum dimension) ranging from 2 to 10 000 µm. Details of implementation and calculation of bulk optical parameters for polydisperse and habit mixture models can be found in .

The technique suggested by is based on a stereological approach and geometrical optics applied to a random mixture of irregular ice grains and air gaps. Below, this approach will be referred to as the stochastic model of the snow layer. In the framework of this model, the snow layer is considered as a two-phase random mixture of ice particles and air gaps. Irregularly shaped ice particles and air gaps within the layer are characterized by the distribution of lengths of random chords, which are defined as straight lines laying within an ice particle or air gap and connecting any two points on its boundary.

The grain concentration in a snow layer is characterized by the volume fraction of ice particles, Nv, which is given by 1Nv=ρbulkρice=aa+h, where ρbulk and ρice are bulk density and ice density, respectively, a and h are mean chord length of ice particles and air gaps, respectively.

In accordance with , for any ensemble of independent convex particles the mean chord, a, is related to the mean particle volume 〈V〉, surface area 〈S〉, and projection area 〈Sp〉 as 2a=4〈V〉〈S〉=〈V〉〈Sp〉. Considering the standard definition of the effective radius of grains in snow (see e.g., ) it can be easily related to the mean chord in the ensemble of convex particles as given by Eq. (): 3re=3〈V〉〈S〉=3〈V〉4〈Sp〉=34a. Typically, snow grain size in a snow layer is much larger than the wavelength of light in the visible (VIS) and near-infrared (NIR) spectral ranges. Furthermore, in these spectral ranges, the imaginary part of the ice refractive index is relatively small compared to the real part. This ensures the applicability of the laws of geometrical optics to the light scattering in snow layers. Employing the geometrical optics and stereological approach, analytical expressions for the IOP of snow layers can be obtained.

In particular, the extinction coefficient σe is expressed through the extinction efficiency Qe, a mean particle projection area 〈Sp〉, a mean particle volume 〈V〉, and particles volume concentration Nv: 4σe=Qe〈Sp〉〈V〉Nv. Combing this relationship with Eqs. () and () for the volume fraction and mean chord, respectively, we have 5σe=QeNva=Qea+h. In the approximation of the geometrical optics (without the contribution of the diffraction part), the extinction efficiency, Qe, is equal to the unity. We adopted this assumption based on the work of , with a detailed analysis provided therein. Therefore, the final expressions for the extinction and scattering coefficients are given by 6σe=Nva=1a+h,σs=ωσe=ωNva, where ω is the single scattering albedo. Expressions for the single scattering albedo and phase function are given in previous publications, see e.g. , and , and not provided here for a sake of brevity.

The SCIATRAN model offers a possibility to introduce a vertical inhomogeneity within a snow layer. In this case, the vertical coordinate within a snow layer, denoted as x, is defined using the concept of the dimensionless “altitude” introduced by : x=(ht-z)/(ht-hb), where ht and hb are the top and bottom heights of the snow layer, respectively. To introduce a vertical inhomogeneity, users need to specify Nv(z) and a(z) as functions of the dimensionless vertical coordinate x ranging from 0 to 1.

While pure snow absorbs only weakly in the UV and visible (UV-VIS) spectral ranges, a snow layer might contain contaminants that exhibit significant absorption. Examples of such contaminants are particles of sediments from the atmosphere (such as clay, silt, and sand particles), as well as Dissolved Organic Matter (DOM) – also known as yellow substance – in sea ice. These contaminants tend to absorb in the blue spectral region .

In the current version of SCIATRAN, the analytical expression for the DOM absorption coefficient is implemented in the following form : 7σy(λ)=σy(λ0)e-0.015(λ-λ0),λ≤500nm,σy(λ0)e-0.015(500-λ0)-0.011(λ-500),λ>500nm, where λ0=390 nm and σy(λ0) is the DOM absorption coefficient at the wavelength λ0. The absorption caused by sediments from the atmosphere is taken into account assuming the volume fraction of impurities to be small. In this case the scattering by impurities can be ignored, and absorption coefficient is approximated according to as 8αλ(z)=-6πλImmλ2-1mλ2+2Ca, where Im denotes the imaginary part of a complex variable, mλ=nλ-iκλ is the refractive index of absorber material, and Ca is its volume concentration. The refractive index mλ and volume concentration Ca are input parameters of SCIATRAN.

Similar to the snow parameters described in Sect. , the absorption coefficient σy(λ0) and the volume concetration Ca can be defined as functions of the dimensionsless vertical coordinate x to introduce a vertical inhomogeneity of impurities within a snow layer.

The volume concentration of air bubbles can reach about 5 % in the upper layers of sea ice and typically decreases with the depth. According to , the shape of air bubbles is mostly spherical, although it can be more complex in the case of multi-year ice. Near the surface, bubbles are highly interconnected and form a complex network in hummock ice. The size of bubbles varies depending on whether they are within brine inclusions or within the ice itself. Previous studies have shown that the observed size range for air bubbles in ice is in the range of 0.1 mm ≤ r ≤ 2 mm . The size distribution function and its parameters, however, are known to depend on the type of ice .

As reported by , even the smallest air bubbles within brine inclusions have a dimensionless size parameter (x=2πr/λ) of ∼ 45 in the visible spectral range. As a result, the scattering by these bubbles can be described in the framework of geometrical optics. Thus, the scattering efficiency, Qsca, of air bubbles is equal to 2, and their scattering coefficient can be calculated using the following expression: 9σa=2SaNa, where Sa is the average cross-section area and Na is the numeric concentration. Accounting for the definition of effective radius Ra=3Va/4Sa, the scattering coefficient of air bubbles can be expressed in another equivalent form: 10σa=3Cv,a2Ra, where Cv,a=NaVa is the volume concentration. The transport scattering coefficient is given by 11σat=σa(1-ga), where ga is the asymmetry parameter of the scattering phase function. The later was calculated by using Mie theory in the spectral range 0.35–0.95 µm. The obtained values ranged from 0.851 to 0.865 with the mean value of 0.860. reported that the asymmetry parameter, ga, at 670 nm was found to be 0.86, which agrees well with the value obtained by . Using Eq. () and the measurement results presented by (see the inset in their Fig. 9), the scattering coefficient of air bubbles can be estimated as 32 m⁻¹ at the top of the ice layer and 9 m⁻¹ at 0.5 m depth. The SCIATRAN software does not include any database of optical parameters of air bubbles. User need to provide input files containing the extinction coefficients and single scattering albedo at a desired wavelength grid and dimensionless depth grid. In general case, the phase function of air bubbles needs to be provided at discrete numbers of scattering angles or in the form of expansion coefficients at a desired wavelength grid. For the Henyey-Greenstein phase function, the asymmetry parameter at a desired wavelength grid needs to be provided.

Brine inclusions in sea ice are usually vertically oriented, irregularly shaped and have varying lengths . In accordance with , the size of individual inclusions ranges from 0.01 to 8.0 mm in length and from 0.01 to 0.23 mm in diameter. A distribution function of these inclusions was studied by , and .

In previous publications, scattering properties of the brine inclusions were modelled by representing the inclusions either as an array of roughly cylindrical shapes or as vertically oriented prolate spheroids with a 5 : 1 ratio of major to minor axes , or as irregularly shaped particles for which the Wentzel-Kramers-Brillouin approximation can be used . However, since the size of brine inclusions is typically significanlty larger than visible wavelengths, their optical properties can be modelled in the framework of geometrical optics. In this case, the scattering efficiency, Qsca, is independent of the wavelength and equals 2. The scattering coefficient of brine inclusions is spectrally neutral and given by the following equation simular to Eq. (): 12σb=2SbNb, where Sb is the average cross-section area and Nb is the numeric concentration. The asymmetry parameter of optically soft particles was derived analytically by as 13gb=1-log⁡2x-1x2, where x=1/(nb-1) and nb=1.024 is the refractive index of brine relative to ice for the temperature of -2°. As a result, gb=0.998 for brine inclusions. We adopted this assumption based on the work of , with a detailed analysis provided therein. In accordance with , the Mie-predicted mean cosine of the scattering angle for the brine pockets is 0.99. Using Eq. () and measurement results presented by the scattering coefficient of brine pockets can be estimated. For the top 4 cm layer of first-year ice this results in 224 m⁻¹ and for layers dipper than ∼ 30 cm in 80 m⁻¹. reports the value of 220 m⁻¹ for σb for a sample of typical first-year ice at -15 °C.

As for air bubbles, the SCIATRAN software does not include any database containing optical parameters of brine inclusions. User need to provide input files containing the extinction coefficients and single scattering albedo at desired wavelength grid and dimensionless depth grid. In general case, the phase function of brine inclusions needs to be provided at discrete numbers of scattering angles or in the form of expansion coefficients at a desired wavelength grid. For the Henyey-Greenstein phase function, which is also included in SCIATRAN, the asymmetry parameter at a desired wavelength grid needs to be provided.

Phytoplankton and algae have been observed not only in oceanic water but also within ice layers. For example, reported on a dark pond contaminated with algae aggregates. Similarly, various microalgae, including diatoms, flagellates, dinoflagellates, and chrysophytes, have been identified in sea ice samples collected in the Canadian Arctic . Since the maximum absorption of phytoplankton occurs at wavelengths around 400 nm, where the absorption of pure ice is very weak, it is crucial to consider the absorption of phytoplankton and algae when calculating radiative transfer, particularly in the visible part of the solar spectrum.

In this regard, an interface has been implemented in the SCIATRAN software to account for absorption by any phytoplankton groups and algae. User need to provide input files containing the absorption coefficient as a function of the wavelength and the concentration of pigments as a function of the dimensionsless vertical coordinate (see Sect. ) in ice. To facilitate the usage of this option for inexperienced users, the SCIATRAN database includes the input files to account for the absorption of chlorophyll-a, diatoms, dinoflagellat, emiliania, and yellow substance.

In this section, we compare the spectral albedo modelled by SCIATRAN with the results from measurements over the bright white ice, which is an ice layer covered by a layer of aged show, and bright white ice covered by fresh fine-grained snow . Below, the latter scenario will be referred to as the snow-covered ice. The measurements were performed on 11 August and 5 September 2012 (Polarstern stations PS80/224 and PS80/323).

In the case of the decoupled model, the lower boundary condition of RTE, see Eq. () in Appendix , was set in accordance with the analytical reflectance model suggested by . The modelling parameters to be determined from Eq. () are the optical thickness of the snow layer, τsnow, effective radius of ice crystals, reff, and absorption coefficient of the yellow substance, σy(λ0), see Eq. (). Although effective IOP are meant here, they refer mostly to the upper layer of the fresh fine-grained snow for the snow-covered ice and to the aged snow otherwise. This is because the IOP of a snow layer are mostly determined by its upper few centimetres. In the case of the coupled model, the geometrical thickness of the ice layer, Δzice, needs to be determined in addition while the sensitivity of the spectral albedo to the extinction coefficients of the air bubbles and brine inclusions within the interior ice was found to be negligible for τsnow about 10 and above.

Two runs were done with the coupled RT model employing different approaches to calculate the IOP of the snow layer, one using the stochastic model and the other assuming monodisperse droxtal ice crystals. Following , only measurements in the spectral range of 350–1350 nm were used in the fitting procedure to avoid strong noise contamination especially under low sun conditions. The retrieved modelling parameters for the coupled and decoupled models are presented in Table .

Figures  and show comparisons of the modelled and measured spectral albedo for the snow-covered and bright white ice, respectively. The left panels in both plots depict scenarios with high contamination by the yellow substance, while the retrieved concentration of the yellow substance for the scenarios shown in the right panels of both plots is negligible. It is seen from the plots that general spectral behaviour of the measured spectral albedo is well reproduced by the model for both scenarios. In particular, one clearly observes a high reflection in the visible spectral range and a low reflection in NIR and SWIR (Short Wave InfraRed). The latter is caused by the ice absorption . The local maximum of the spectral albedo at 1.05–1.11 µm is attributed to the local minimum of the imaginary part of the ice refractive index at 1.1 µm .

For the snow-covered ice (Fig. ), the relative differences between the modelled and measured data are mostly within 2 % in the visible spectral range increasing to up to 5 % in NIR and SWIR. These differences are well within the general requirement for the absolute accuracy of the surface albedo (0.02–0.05) in climate models . For the bright white ice (Fig. ), the model overestimates the absorption of the snow layer in the SWIR spectral range. At wavelengths above 1.2 µm, the difference between the modelled and measured data increases with wavelength to up to -22 %. A possible reason for this discrepancy might be a vertical inhomogeneity of the grain size. A detailed investigation of this topic is, however, beyond the scope of this paper.

In general, the SCIATRAN model represents well the spectral albedo of the snow-covered and bright white ice for both uncontaminated and contaminated by yellow substance snow layers. However, in the SWIR range, larger differences are identified for the bright white ice scenario. Analysing retrieved modelling parameters presented in Table , the following findings can be formulated:

Differences between the retrieved parameters for the coupled and decoupled models are rather small (∼ 20 %). This demonstrates that, for the selected scenarios, the coupling effects play rather a minor role.

In this section, we compare the spectral albedo modelled by SCIATRAN with the results from measurements over melt ponds on sea ice for clear sky and cloudy scenes . The measurements were performed in the central Arctic on 10 and 26 August 2012. As stated by , the selected measurements represent typical types of melt ponds.

In the case of the decoupled model, the lower boundary condition of RTE, see Eq. (), was set in accordance with the analytical reflectance model of melt ponds on sea ice suggested by . The modelling parameters to be determined from Eq. () are the ice thickness, water depth, and transport scattering coefficient of the ice layer.

For the coupled model, the spectral albedo was calculated for both Open melt Pond (OP) and Frozen melt Pond (FP) scenarios. In the OP case, the lower medium includes a water layer on the top of sea ice. In the FP scenario, a very thin ice layer (ice crust) is added on the top of the OP lower medium. For both scenarios, the modelling parameters to be determined from Eq. () are the top and bottom depths of the sea ice layer and scattering coefficient of the brine inclusions. For the FP scenario, the geometrical thickness of the ice crust is set to a fixed value of 1.5 cm. The scattering within the ice crust layer is assumed to be only due to air bubbles with the scattering coefficient of 80 m⁻¹, which corresponds to an optical thickness of 1.2. The retrieved modelling parameters obtained for the coupled and decoupled RT models are presented in Table .

Figures  and show comparisons of the modelled and measured spectral albedo of frozen melt ponds for the clear sky and cloudy conditions, respectively. Although melt ponds generally exhibit a lower albedo in the visible spectral range as compared to bright white ice, it still can reach values of about 0.7 for wavelengths between 400–500 nm for clear sky conditions, see Fig. . This might be due to the reflection of the solar light back to the atmosphere by the thick ice layer located below the meltwater . For cloudy scenes, the melt ponds albedo shows similar spectral behaviour as for clear sky conditions but has significantly lower values in the visible spectral range.

A distinguishing feature of the melt pond reflection is a strong decrease of the spectral albedo in the NIR range. Unlike the reflection by the bright white ice, there is no local maximum around 1.1 µm and the dependence of the spectral albedo on the wavelength in the NIR spectral range is quite flat. This is because the wavelength dependence of the melt pond reflection in this spectral region is determined by the Fresnel reflection rather than by the absorption of ice crystals.

As it is seen from Figs.  and , the spectral albedo of the melt ponds modelled by SCIATRAN agrees well with the measured data for both clear sky and cloudy conditions. For both the coupled model using OP scenario and the decoupled model, the modelling accuracy is similar resulting in similar RMSE, as shown in the lower panels of the plots. Both approaches, however, result in an underestimation of the spectral albedo in the NIR range. For these two model runs, the maximum relative difference between the modelled and measured spectral albedo is 15 %–20 % for the clear sky and 10 %–30 % for conditions.

The use of the coupled model with FP scenario significantly improves the modelling accuracy in the NIR spectral range reducing the disagreement between the modelled and measured data to below 5 % for clear sky and below 10 % for cloudy conditions. In the latter case somewhat larger differences are still seen in the SWIR spectral range. The observed difference between the modelling results when using FP and OP scenarios can be explained by the fact that the absorption of the light occurs within the ice crust layer in the former case and in the water in the latter case. A distinct difference in the imaginary parts of the refractive indices of ice and water in this spectral range (not shown here) confirms this explanation.

In this section, the HDRF modelled by SCIATRAN is compared to multi-directional measurements of the upwelling radiation conducted on 25 May 2015 at the GreenEdge ice camp at the beginning of the snow melting . The measurements were performed at a solar zenith angle of 63.56° and the sky conditions were reported as variable.

In accordance with , the geometrical thicknesses of the snow layer and of the underlying ice layer were about 30 and 130 cm, respectively. Employing the stochastic model of snow (see Sect. ) to calculate the IOP, the optical thickness of the snow layer was estimated to be about 300. This was done by calculating the snow layer extinction coefficient using Eq. () with the volume fraction of ice, Nc, set to 0.3 and the mean chord length of ice crystals, a, set to 0.3 mm. The latter value was selected in accordance with a typical snow grain radius of beginning melting snow of about 0.4 mm reported by and references therein. The volume fraction of ice was calculated from Eq. () using the ice density of 0.917 g cm⁻³ and the snow density of 0.28 g cm⁻³. The latter was selected in accordance with observations during the SnowEx17 campaign .

A sensitivity study performed using the coupled SCIATRAN model showed that, for an optical thickness of the snow layer of about 300, the HDRF depends only weakly on the optical parameters of the underlying ice layer, on the variation of snow optical thickness, and on the effective radius of ice crystals. Therefore, no fit of the modelled HDRF with respect to measured data was performed.

As clouds in the atmosphere can change the angular distribution of the downwelling diffuse radiation and cloudiness situation during the measurements was not precisely described by , the comparison between the modelled and the measured HDRF was done for several model runs: one for clear sky conditions and three for cloudy scenes with different cloud optical thicknesses (COT).

The impact of clouds on the angular distribution of the reflected radiance modelled with SCIATRAN is illustrated in panels (b)–(e) of Fig. . It is evident that in the backward directions (azimuth angle φ∈[90,270°]), the HDRF for cloudy scenes is higher compared to that for clear sky conditions and shows better agreement with the measurements. However, in the forward directions (azimuth angles φ∈[0,90°] and φ∈(270,360°)), the situation drastically changes as glint conditions are approached. Here, the HDRF for the clear sky is significantly higher than that for cloudy scenes and exceeds the measured HDRF. This behaviour is explained by the fact that the radiation reflected around the glint direction originates mostly from the reflection of the direct solar light while for off-glint directions contribution from the downwelling diffuse radiation is essential. Thus, the observed differences in the agreement between the modelled and measured data for different directions and cloud scenarions might be caused by the horizonthal inhomogeneities in the cloud coverage.

In the framework of a 1D RT model, the presence of a cloud results in a strong attenuation of the direct solar radiation. This attenuation might, however, be inappropriate to desribe observations under inhomogeneous cloud coverage conditions. To demonstrate the impact of the attenuation of the direct solar radiation by a cloud, we calculated the HDRF combining the diffuse downwelling radiation modelled for cloudy conditions and direct solar radiation calculated for clear sky conditions. The results obtained for cloudy scenes with COT of 2, 4, and 8 are presented in Fig. f–h, respectively. It is apparent that the usage of the direct solar radiation not affected by clouds significantly improves the agreement with the measurements in the forward direction region. However, the model still underestimates the HDRF for the backward directions.

For the results presented in panels (f)–(h) of Fig. , the minimum RMSE occurs for COT = 8, where the RMSE is about 36 % lower compared to that for clear sky conditions and 22 % smaller compared to that for the cloudy scenario with COT = 2. For this reason we chose COT = 8 to illustrate the agreement of the measured and modelled HDRF at different wavelengths, see Fig. . Contrary to the measured data, the modeled HDRF does not exhibit any irregular patterns at any wavelength. The irregular patterns seen in the relative differences between the modeled and measured HDRF coincide with those in the measured data. This is a clear indication of an atmospheric inhomogeneity (such as inhomogeneous cloud coverage) or variations in the surface reflection properties, which currently cannot be considered in the fremework of the SCIATRAN RTM. In the glint range, the difference between the modelled and measured data reaches about 5 % for all wavelengths.

A similar comparison was performed for the decoupled SCIATRAN model. In this case, the BRDF model developed by was used to represent the snow reflectance. All other settings remained unchanged. The comparison with the measured HDRF for the decoupled model (not shown here) reveals similar RMSE as for the coupled model. Similar comparison results for the coupled and decoupled models in the considered case are expected because the main assumptions of the latter, such as weak absorption and large but finite optical thickness, are valid.

In this section, the HDRF modelled by SCIATRAN is compared to multi-directional measurements of the upwelling radiation conducted on 1 June 2015 over bare ice (Fig. 6d–f in ). The sea ice surface was manually cleaned from the snow cover and described by as a flat grey-bluish ice surface with a thin surface scattering layer (< 2 cm) consisting of coarse ice grains. The measurements were performed at a solar zenith angle of 59.52° at clear sky conditions.

In the SCIATRAN model, a bare ice surface with a scattering layer on its top and a flat interface between the scattering layer and the interior ice was assumed. This parameterization follows , who suggested to represent the structure of sea ice by up to three distinct layers. Specifically, they state that “when the melt season began, most of the brine drained out of the upper layers of the ice, producing a `surface scattering layer' that persisted throughout the melt season”. A photo of a representative ice core with a drained surface layer is shown in Fig. 4 of .

In accordance with , the geometrical thickness of the scattering layer was set to 2 cm and that of the ice layer to 130 cm. The IOP of the scattering layer were calculated assuming the stochastic model (see Sect. ). The IOP of the interior ice were modeled assuming scattering by brine inclusions, air bubbles, and absorption of pure ice.

The selected HDRF model of the bare ice requires the input of two parameters of the scattering layer (effective radius of ice grains and optical thickness) and of two parameters of the interior ice (scattering coefficient of air bubbles, σa, and of brine inclusions, σb). To reduce the number of parameters to be retrieved, we followed and set the effective radius of ice grains in the scattering layer to a fixed value of 1.5 mm. This approach is reasonable beacase of a weak absorption of ice in the considered spectral range resulting in a weak sensitivity of the directional reflectance to the assumed grain size.

Although the HDRF was measured at 10 260 directions, simultaneous estimation of both σa, and σb was impossible because of a strong correlation between their weighting functions. Therefore, the fitting was performed assuming that the scattering in the ice layer is caused by either air bubbles or by brine inclusions. This means that the retrieval process was performed twice for each wavelength. First, setting σb=0, we retrieved σa and the optical thickness of the scattering layer, τ. Thereafter, setting σa=0, we retrieved σb and τ. In the following discussion, two values separated by a slash denote fitting results obtained assuming the scattering by either air bubbles or by brine inclusions. The calculation of the HDRF was performed assuming wavelength-independent asymmetry parameters of the Henyey–Greenstein phase function of 0.86 for air bubbles and 0.99 for brine inclusions.

Initial retrievals showed that σa/σb at 438 nm were almost twice as small as those at 628 nm.

This contradicts the fact that the scattering coefficient of a particle which is much larger than the wavelength must be independent of the wavelength. This discrepancy can be explained by a presence of an additional absorber in the interior ice whose absorption coefficient is maximal at 438 nm and minimal at 628 nm. As there was no information about absorbing species in the ice layer from in situ measurements performed during the measurement campaign, the yellow substance (see Sect. ) was assumed to be a possible contaminant. To estimate the absorption coefficient of the yellow substance, the fitting process was repeated independently for wavelengths 438, 560, and 628 nm setting σy(λ0) to 0,0.1,0.2,0.3,0.4, and 0.5 m⁻¹. We remind that λ0=390 nm. It was found that σy(λ0)=0.2 m⁻¹ provides minimum difference between the retrieved scattering coefficients of ice at 438 and 628 nm. Therefore, this value was considered a reasonable estimation of the yellow substance absorption coefficient.

An overview of the retrieved parameters is presented in Table . The optical thickness of the scattering layer is estimated to be 0.19–0.23 depending on the wavelength. Note that this refers to the optical thickness of the scattering layer, not the geometric thickness of the ice. The transport scattering coefficient of the bare ice is estimated to be 7.2–10.5 m⁻¹. The value at 560 nm is significantly smaller (∼30 %) than those at 438 and 628 nm. This suggests that either the absorption by yellow substance, as described by Eq. (), is too weak at the wavelength of 560 nm or other absorbers are present.

Figure  displays a comparison of the measured and simulated HDRF for the bare ice. The top row of the plot shows the measured HDRF at three different wavelengths: 438, 560 and 628 nm. The observed irregularity of the HDRF angular distribution indicates a presence of horizontal surface inhomogeneities. As suggested by , these inhomogeneities might be attributed to remaining snowpacks on the ice surface.

In this section, HDRF for melt ponds on sea ice modelled by SCIATRAN is compared with measurements conducted at the GreenEdge ice camp on 7 June 2015 as reported by . In the SCIATRAN model, the optical properties of the under-pond ice layer were represented by absorption by pure ice and scattering by brine inclusions and air bubbles. In accordance with , the pond depth and under-pond ice thickness were set to 11 and 120 cm, respectively.

The model initialisation parameters to be obtained by the fit are the root mean square value of the isotropic Gaussian shape, σ2, and ice scattering coefficient (represented by the scattering coefficient of air bubbles, σa, or brine inclusions, σb). The fitting in the principal plane, where the impact of specular reflection is maximal, results in σ2=1.7×10-4. Similar to the bare ice, simultaneous estimation of both σa and σb was impossible because of a strong correlation between their weighting functions. Therefore, the fitting was performed twice assuming that the scattering in the ice layer is caused solely by either air bubbles or by brine inclusions. In the text below, the corresponding fitting results are separated by a slash. An overview of the retrieved parameters is presented in Table . The transport scattering coefficient of the interior ice is estimated to be in the range of 1.42–1.62 m⁻¹ which is significantly smaller than for the bare ice.

Figure  shows the measured and modelled HDRF at 438, 560, and 628 nm wavelengths. A strong forward peak caused by the Fresnel reflection on the atmosphere-ocean interface is observed in the measurements and well reproduced by the SCIATRAN model. However, the peak in the measured data is wider, which might be related to a slight roughness of the water surface or a finite angular dimension of the solar disk, which is not accounted for in the SCIATRAN model. Typically, the relative difference between the modelled and measured data is within ± 15 %. Larger relative differences at 628 nm result from a significantly lower reflection at this wavelength as compared to that at 438 and 560 nm. The model calculations were performed assuming a cloud free atmosphere, although rapid variations in sky conditions (alternation of clear sky and diffuse cloud coverage) were reported by . Contrary to the melting snow scenario discussed in Sect. , no improvement in the agreement between the measured and modelled data was observed when performing modelling runs for a cloudy atmosphere.