We assume a pond to be a plane-parallel layer of melt water on an under-pond ice layer. Additionally we make the following assumptions: 1.

the melt water is pure, with neither absorbing contaminants nor scatterers;

Makshtas and Podgorny (1996) give the following formula for the albedo of a pond that satisfies the abovementioned assumptions: Aμ0=RFμ0+TFμ0exp⁡-εwz/μ0wfoutεwzAbn21-Abfinεwz, where RFμ0 and TFμ0 are the Fresnel reflectance and transmittance of the water surface for incidence angle θ0=arccos⁡μ0, n is the refractive index of water, μ0w is the cosine of the refractive angle, z is the pond depth, Ab is the pond bottom albedo, and εw is the extinction coefficient of water, equal to the sum of the water absorption (αw) and scattering (σw) coefficients: εw=αw+σw. We use the data of Segelstein (1981) for the water absorption and the power law for the spectral scattering coefficient (Kopelevich, 1983): σw(λ)=σ0λ0λ4.3,σ0=1.7×10-3m-1,λ0=550nm. Functions fin(x) and fout(x) are defined as fin(x)=2∫01Rinμwexp⁡-2xμwμwdμw,fout(x)=2∫01TF(μ)exp⁡-xμwμdμ, (we changed notation used by Makshtas and Podgorny (1996), where Rin is the internal reflectance of the water surface, and satisfy the relationship fout(2x)=n22E3(2x)-fin(x), where E3(x) is the integral exponential function of the third power: E3(x)=∫1∞e-xtt3dt. The first term in Eq. () describes the sun specular reflection from the water surface; the second one describes the light, multiply reflected between the pond bottom and its surface.

Albedo A(μ0) given by Eq. () is the albedo at direct incidence (in climatology often called the black-sky albedo) and Makshtas and Podgorny (1996) restrict their consideration by this reflective property. However, two other characteristics, closely related to A(μ0), are widely used both in climatology and in remote sensing. These are the albedo at diffuse incidence (white-sky albedo) and the bidirectional reflectance factor (BRF). Let us derive the structural formulas for these characteristics.

The albedo at diffuse incidence AD (white-sky albedo) is defined as AD=1π∫02π∫01Aμ0μ0dμ0dφ=2∫01Aμ0μ0dμ0 and can be found simply by integrating Eq. (): AD=RFD+fout2εwzAbn21-Abfinεwz, where RFD is the Fresnel reflectance for the diffuse incidence.

The albedo at direct incidence is expressed through the bidirectional reflectance factor Rμ,μ0,φ by the relation, analogous to Eq. (): Aμ0=1π∫02π∫01Rμ,μ0,φμdμdφ. Comparing Eqs. () and () for albedos of direct and diffuse incidence and keeping in mind the relationships () and (), we can immediately write the expression for the pond BRF: R=πμ0RFμ0δμ-μ0δφ+TFμTFμ0Abn21-Abfinεwzexp⁡-εwzμ0w-εwzμw. Equations (), (), and () give the structural formulas for the complete set of the pond reflective properties: albedos at direct and diffuse incidence and BRF. All the formulas are analytical and can be used straightforwardly, except Eqs. ()–() for functions fin(x) and fout(x), which should be calculated numerically. To speed up simulations one can calculate these functions once for a given set of wavelengths and then use a look-up table.

Note that Eqs. (), (), and () express the pond albedo and BRF in terms of the albedo of its bottom Ab, which therefore is the main (and most indefinite) factor that determines the pond reflective properties. Therefore, the main questions arising here are as follows: 1.

How is the pond bottom albedo expressed in terms of the inherent optical properties of sea ice and the ice layer thickness?

Let us consider the inherent optical properties (IOPs) of under-pond ice that forms the pond bottom.

The IOPs of a medium used in the radiative transfer theory are the spectral scattering σλ and absorption αλ coefficients and the scattering phase function p(θ). In the following consideration (see Sect. 2.3), as in other radiative transfer theory applications (see, e.g., Davison, 1958; Chandrasekhar, 1960), the transport scattering coefficient σt is used: σt=σ(1-g), where g is the average cosine of the scattering angle θ: g=〈cos⁡θ〉=12∫0πp(θ)cos⁡θsin⁡θdθ. The transport coefficient is useful in calculating the reflection and transmission by a scattering layer with a very forward-peaked phase function, particularly if one is interested in the layer albedo, rather than the angular structure (BRF) of the reflected light (Zege et al., 1991).

Main factors that determine optical properties of sea ice are its microphysical structure and values of complex refractive indices of its constituents; the dispersion of complex refractive indices determines the spectral properties of sea ice.

As the volume concentration of air bubbles in sea ice is small – only up to ∼ 5 % even in the extremely bubbly ice (Gavrilo and Gaitskhoki, 1970) – and the complex refractive index of brine is very close to that of ice (see Buiteveld et al., 1994; Warren and Brandt, 2008; and Sect. 2.2.2), we take the absorption coefficient of sea ice equal to that of solid ice. Impurities – sediment and organic pigments from sea water – could change absorption coefficients, particularly at shorter wavelengths. At this stage we neglect their effect, keeping in mind that their absorption spectra can be added, if necessary.

The scattering takes place at inhomogeneities in sea ice and is mainly caused by air bubbles and brine inclusions (Mobley et al., 1998; Light, 2010). Another source of scattering could be salt crystals, but they precipitate at low temperatures and are not observed in summer ice, where melt ponds are formed: precipitation temperatures for mirabilite (Na2SO4⚫10H2O) and hydrohalite (NaCl⚫2H2O) crystals are -8 and -23 ∘C, respectively (Light et al., 2003).

If both the absorption and transport scattering coefficients are known, the albedo of a layer can be calculated within the two-stream approximation, which is widely used for practical calculations: Ab=A01-exp⁡-2γτ1-A02exp⁡-2γτ, where A0 is the albedo of the semi-infinite layer with the same optical characteristics, γ is the asymptotic attenuation coefficient, and τ is the layer optical thickness. The version of the two-stream approximation developed by Zege et al. (1991) expresses these characteristics as follows: A0=1+t-tt+2,γ=34σtσt+αitt+2,τ=σt+αiH, with t=8αi3σt, where αi is the ice absorption coefficient; H is the ice layer thickness.

The two-stream approximation in the version given in Zege et al. (1991) has a wide range of applicability and can be used both for strongly and weakly absorbing media, for optically thin and thick layers. Hence, this approximation can be applied to all the variety of melt ponds: from young ponds, which are light blue and have comparatively optically thick under-pond ice, to mature dark ones, where under-pond ice is optically thin.

Thus, in the assumption of a Lambertian bottom and plane parallel geometry, which applies in the absence of strong wind, i.e., calm pond surface, the spectral reflection of ponds is determined by two values: water layer depth z and the albedo of the pond bottom Ab. The latter, in turn, depends on the transport scattering coefficient of under-pond ice σt and its geometric thickness H (or the transport optical thickness σtH). Note that only value αi in Eqs. ()–() has a spectral behavior, while the others – σt and H – are scalars.

Thus, in the absence of pollutants just three parameters determine the pond spectral reflectance: namely, the transport scattering coefficient σt and geometric thickness H of the under-pond ice and water layer depth z. This statement is confirmed by the coincidence of measured and modeled spectra demonstrated below. The outlined model of a melt pond is shown in short in Table 1.

Correct processing of the reflection measurement results requires the correct modeling of the illumination conditions. This is especially important for measurements in the Arctic, because of the low sun and the bright surface. When the sky is overcast, the incident light is close to diffuse, even if the solar disk is visually observed (Malinka et al., 2016b). In this case the measured albedo is the white-sky one. However, when the sky is clear and the sun is near the horizon, the direct solar flux is comparable to the diffuse flux from the sky, so the measured (blue-sky) albedo value is a mixture of those at direct (black-sky) and diffuse (white-sky) incidence. The black-sky albedo increases when the sun is approaching the horizon, so the difference between the white- and black-sky albedos is most essential at oblique incidence (see Fig. 3). The problem of the correct interpretation of the measured blue-sky albedo is considered in detail in Malinka et al. (2016b) for a homogeneous surface. However, the albedo of a melt pond can differ significantly from that of the surrounding background, e.g., white ice or snow. Some estimation for this case is given below.

Let R, A(μ0) and AD be, as before, the BRF, black-sky, and white-sky albedo of a melt pond, respectively. Let the surrounding background be Lambertian with albedo rb. Then the brightness of the incident radiance can be estimated as (Malinka et al., 2016b) B↓=t0(μ0)δ^+td(μ0)+rarb1-rarbT(μ0)E0μ0π, where t0(μ0) and td(μ0) are the direct and diffuse atmosphere transmittances, δ^=πδμ-μ0δφ-φ0/μ0 is the identity operator (δ(x) is the Dirac delta function), T(μ0)=t0(μ0)+td(μ0) is the atmosphere transmittance at direct incidence, and ra is the atmosphere bihemispherical reflectance at incidence from below. E0 is the extraterrestrial solar irradiance.

Thus, the light flux incident to a melt pond is F↓=T(μ0)1-rarbE0μ0. The radiance of light reflected by pond follows from Eq. (): B↑=Rμ,μ0,φt0(μ0)+A(μ)td(μ0)+rarb1-rarbT(μ0)E0μ0π=Rμ,μ0,φ-A(μ)t0(μ0)+A(μ)T(μ0)1-rarbE0μ0π. Therefore the reflected flux is F↑=A(μ0)-ADt0(μ0)+ADT(μ0)1-rarbE0μ0. For the measured value of the blue-sky albedo Ablue it follows that Ablue=F↑F↓=A(μ0)-ADt0(μ0)1-rarb+ADT(μ0). The equation for the blue-sky albedo can be written as a linear combination of the black and white-sky albedos: Ablue=wA(μ0)+(1-w)AD, with the proportion of direct radiance w: w=t0(μ0)T(μ0)1-rarb. Factor 1-rarb is responsible for multiple reflections between the atmosphere and surrounding background.

Modeled albedo spectra of a light melt pond (a pond with high reflectance) at different illumination conditions are shown in Fig. 4. The angle of incidence is 80∘ (the sun elevation is 10∘). The interval of albedo changes is limited by the values of white and black-sky ones. Also shown are the blue-sky albedos for clear sky and for sky with thin cirrus layer (with optical thickness of 0.1). Both are considered with different surrounding backgrounds: perfectly black (rb=0) and white (rb=1). As seen from Fig. 4, the effect of background is negligible (only small difference between the dots and the solid blue curve and between the crosses and the blue dashed curve), so the results of melt pond albedo measurements can be processed without a priori knowledge of the albedo of the surrounding background.

In contrast to the visible range, ice and water absorb a significant amount of light in the IR: a layer of ice a few centimeters thick or water completely absorbs radiation in the infrared range. Thus the melt pond optical response in the IR is restricted to the Fresnel reflection by the pond surface. In contrast, ice grains in the surface scattering layer are of the order of millimeters in size (and even smaller in snow). Due to this fact, specific features of the spectral behavior of the imaginary part κ of the refractive index of ice can appear. In particular, κ has a local minimum at 1.1 µm, which provides a slight peak of reflection in the interval 1.05–1.11 µm (Wiscombe and Warren, 1980). Figure 5 shows an example of the modeled albedo's spectral dependence for white ice, snow, and a melt pond. It clearly demonstrates that for wavelengths longer than 0.9 µm the melt pond reflection is restricted by the Fresnel reflection to a constant value, while snow and white ice demonstrate a local maximum at 1.1 µm. Thus, this slight peak can serve as a criterion for determining if a spectrum is taken entirely from an open pond or partially from snow/ice surface. If this peak is observed in a measured spectrum, it clearly indicates the presence of ice grains (of white ice or snow) in the receiver field of view.

In the description of the field data used in this study, most sky conditions were reported as overcast. Only a few measurements were taken under clear-sky conditions. Scattered clouds were not reported at all in the measurement series considered. In the cases of overcast sky, the measured albedo was interpreted as the white-sky one. In the clear-sky cases, the Rayleigh atmosphere with the Arctic background aerosol (Tomasi et al., 2007) was assumed. In this case the solar incidence angle was determined from the pond reflection in the IR: at the interval 1.25–1.3 µm (preferably) or 0.85–0.9 µm, if data at the former interval are not available. As the IR signal (both incident and reflected) is quite weak and hence some noise is always noticeable, we average the signal over one of the abovementioned intervals. The pond reflectance in these IR intervals is completely determined by the Fresnel reflection of its upper boundary. Atmosphere scattering in the IR is negligible (especially at 1.3 µm), so the incident light is unidirectional. In this situation the solar incident angle can be calculated through the Fresnel equations.

Measurements of the spectral albedo of different sea-ice surfaces were carried out during the R/V Polarstern cruise ARK-XXVII/3 (2 August–8 October 2012). Only in the second half of the cruise did the vessel leave the marginal ice zone and enter the ice pack. The ice thickness varied from 0.5 to 3 m with an average of 1–1.5 m. Melt ponds were observed in August. They were both open (with no skim ice) and frozen over (with a skim of ice), sometimes snow covered. The data were collected during stations, when the vessel was parked at an ice floe for several days. This gave the possibility to obtain several-day data sequences of melting sea ice and evolving melt ponds at the same location. The stations, where ponds were observed, were located from 84∘3′ N, 31∘7′ E to 82∘54′ N, 129∘47′ E. For more information about the cruise, including the coordinates and dates of the stations, see Boetius and ARK-XXVII/3 Shipboard Scientific Party (2012) and Istomina et al. (2016, 2017).

The ASD FieldspecPro III spectroradiometer used for these measurements has three different sensors that provide measurements from 350 to 2500 nm with the spectral resolution of 1.0 nm. A sensor measures the light signal supplied by a fiber optical probe, which collects light reflected by a 10 cm ×10 cm Spectralon white plate. The plate was held at about 1 m above the surface and was directed first towards the measured surface and then towards the sky. The ratio of these two measurements gives the hemispherical reflectance (albedo) of the surface. For some cases the water depth and ice thickness within the pond were measured.

For the model verification we considered the melt pond albedo in the spectral interval 0.35–1.3 µm. The retrieval procedure implies searching for the pond parameter values shown in Table 1. These three parameters comprise a 3-D vector, which is varied according to the Newton–Raphson method to provide the best fit (in the sense of the least squares) of the measured and modeled spectra (for details of the method see Zege et al., 2015). For the cases where the pond depth and underlying ice thickness were measured, the pond parameters retrieved were compared to the measured ones.

Some ponds were frozen over, i.e., they had a layer of newly formed ice on top of their surface. It is evident that a layer of flat, transparent ice at the pond surface practically does not change pond reflection, so we consider the ponds with ice crust in the same manner as open ones. However, if the upper ice layer is bubbly or snow covered, the pond reflectance can change drastically: the pond gets brighter and may become indistinguishable from the surrounding ice in the visible range. These snow-covered ponds would require other means for their characterization. We exclude such cases from consideration.

Figures 6–9 present photos of different ponds and their reflectance spectra, measured and simulated with the retrieved parameters (denoted as “retrieved” in the legend).

Figure 6 shows the photos as well as modeled and measured spectra of light-blue melt ponds, which have a uniform bottom on thick first-year ice under clear and cloudy skies, measured in the central Arctic on 10 and 22 August 2012, respectively. The albedo values are extraordinarily high. This could be related to the fact that the ponds are frozen over with a 2–3 cm layer of ice, which is likely not perfectly transparent. Figure 7 shows three cases of frozen over blue ponds with heterogeneous bottom under overcast skies measured on 11 and 22 August 2012, respectively. One can see darker parts in the ponds, which result from sea-ice melting from the lower boundary or lower bubble content. Figure 8 presents dark open melt ponds on thinner first-year ice under overcast skies, all measured on 26 August 2012. The albedo of these ponds is much lower than that of the previous ones: from about 0.07 to 0.14 in the visible and about 0.05 in the IR. Figure 9 presents two cases of light-blue ponds, both measured on 26 August 2012, and a dark pond contaminated with algae aggregates measured on 21 August 2012, all under overcast skies. Surprisingly, the spectrum of the pond with algae is reproduced quite well. This is because the contribution of the yellow algae spots to a total reflection is proportional to their area, which is not very large. However, their effect can be clearly seen in the spectrum: the measured values are less than the modeled ones in the blue range (0.3–0.5 µm) and greater in the yellow-green (0.5–0.6 µm).

The above ponds are quite different from one another. They range from dark to very light blue in color, open and frozen over, clear, and contaminated with organic matter. In spite of this, the model is able to reproduce the measured spectra in the visible region with high accuracy in all studied cases. The root-mean-square difference (RMSD) between the measured and simulated spectra has the average value of 0.01 for the whole considered spectrum (0.35–1.3 µm) and 0.007 for the visible range (< 0.73 µm).

The retrieved and measured geometrical parameters of the ponds, as well as the RMSD between the measured and simulated spectra, are presented in Table 2 and shown in Fig. 13.

Melt pond spectra were observed near Utgiaġvik, Alaska, USA (formerly Barrow) in 2008 as part of the SIZONET program, observing pond formation (Polashenski et al., 2012). Observations were collected at sites approximately 1 km offshore from Niksiuraq in the Chukchi Sea, near 71.366∘ N, 156.542∘ W on level, landfast first-year ice. For this work, a total of 27 measured melt pond spectra were used (no photographs were taken). All melt ponds were quite dark and polluted with sediments and their spectra look quite similar. Three of them are presented in Fig. 10. The albedo does not exceed the value of 0.3 in its maximum and shows a discrepancy in the blue range, presumably due to the presence of mineral sediments. Because of this, the RMSD between the measured and simulated spectra for the visible range (0.01) is greater than that for the whole spectrum (0.009). The ice thickness was not measured. The pond depths, measured and retrieved, as well as the RMSD, are shown in Table 2 and Fig. 13.

SHEBA was a year-long drift experiment conducted in the Beaufort Sea from October 1997 to October 1998 (Perovich et al., 1999; Uttal et al., 2002). Extensive measurements of the characteristics of sea ice were made. This included observations of the spatial variability and temporal evolution of the spectral albedo of the ice cover (Perovich et al., 2002).

One pond in this expedition was especially interesting, because its bottom had a region that was much brighter than the surrounding bottom. This region had sharp borders with rectangular corners (see the photo in Fig. 11). This likely was a broken piece of bubbly multi-year ice that was incorporated into the ice cover. This piece of ice had more air bubbles than the darker adjacent ice. This dual pond was observed during the entire period of its formation and development. The most intensive pond formation process was observed from 17 July through 14 August. The spectra were taken every 4 days during this period. The spectra processing results are shown in Figs. 11 and 12.

Figure 11 shows the spectra and the photos of the SHEBA dual pond. For the first five dates (17, 21, 25, 29 July, and 2 August) the retrieval is excellent (for the visible range RMSD = 0.0038 for 17 July and has a maximal value of 0.0061 for 29 July; see Table 2) and for the last three (6, 10, 14 August) the retrieval is a little bit worse, but still quite good (for the visible range RMSD = 0.0085 for 6 and 10 August). The reason for this difference is not obvious and we may assume that some contaminant got into the pond those days. Therefore, the regression analysis relies on the first five measurement dates.

Figure 12 presents the retrieved pond depth and ice thickness (for both parts independently) for these dates. The retrieved pond depth is 7 cm greater than the average reported pond depth (37 cm) at the light part of the pond and 13 cm greater at the dark part. Albedo of the light part (in the visible part of spectrum) is approximately twice greater than that of the dark part. In general, this agrees with the different nature of the pond's physical properties. The retrieved ice thickness in the light part is lower by 34 cm in average than that of the dark part. The slope of the linear regression for the retrieved ice thickness gives the melt rate of 1.9 and 2.6 cm day-1 for the light and dark parts, respectively. Taking the average surface and bottom melt for SHEBA ponded ice from 17 July to 14 August gives an estimated surface ice melt of 35 cm and bottom melt of 28 cm for a total of 63 cm, which gives a melt rate of 2.25 cm day-1 (Perovich et al., 2003).

Suppose that the difference between the transport scattering coefficient σt for the light and dark portion is due to air bubbles only, then the scattering coefficient by air bubbles can be estimated as σa=σtlight-σtdark1-ga. The retrieved values, averaged for the first five dates, are the following: the transport scattering coefficient for the light part σtlight is 5.6 m-1, for the dark part σtdark=1.0 m-1 (see Table 2). The slope of the regression line for these five dates is much less than the values scatter. Using the value of 0.86 for ga, we found that the average retrieved scattering coefficient by air bubbles σa is 33 m-1. In the bubble-saturated ice observed by Gavrilo and Gaitskhoki (1970) the air volume concentration was up to 5 % and the effective bubble radius was Ra=1.3 mm. If we suppose the same effective radius, the average air volume concentration in the light ice will be CaV=2/3Raσa=2.8 %, which is quite reasonable for bubbly ice.

The retrieved and measured pond parameters (melt water depth, and underlying ice thickness, and transport scattering coefficient), as well as root-mean-square difference (RMSD) between the measured and simulated albedo spectra, are given in Table 2. The RMSD is shown both for the whole spectrum and for the visible range (λ<0.73 µm). A scatter plot of the retrieved pond parameters is shown in Fig. 13. The maximal error is 55 %, the relative RMSD is 37 %, and R2=0.56. There could be different sources of error. One source could come from the fact that the most important parameter that determines the pond albedo is the transport optical thickness of under-pond ice τt that is a product of the transport scattering coefficient σt and ice thickness H:τt=σtH. Partially this explains the retrieval error: the sensitivity of the albedo to τt is twice greater than that to σt and H separately. Second, the under-pond ice might not be flat, especially its lower boundary. In this case the optical retrieval gives some average value, while the in situ measurement gives a random value taken in some particular point. The third source can be the presence of some impurities that affect the absorption spectrum. Additional absorption can affect the retrieval of the scattering coefficient and, consequently, of H. In addition, there could be other sources of uncertainties, like finite pond size, presence of snow in the receiver field of view, and clouds in the sky. Considering this, the retrieval of the underlying ice thickness seems reasonable. Let us note the fact that microwave sounding methods completely fail in ice thickness retrieval when ice is covered with water.

The retrieval of the pond depth is more uncertain: its value can differ up to 2 times from the measured one and RMSD = 65 %. This is to be expected, because the pond water depth has much less effect on the pond albedo than the underlying ice thickness. Nevertheless, the correlation for the entire dataset of the measured and retrieved pond depth values is quite high (R2=0.62) and 70 % of the retrieved values are inside the 50 % error range. The observed scatter in the retrieval results might partly be explained by the specifics of the field measurements of the water depth and ice thickness in the melt pond: ice drillings or water depth measurements are performed at one single point of the melt pond and do not necessarily represent the average ice thickness or water depth values, which can be highly variable. The transport scattering coefficient was not measured in any of the campaigns. We can only say that the retrieved values match the interval 0.1–10 m-1, indicated in Sect. 2.2.3.

Summarizing the verification, we can say that the spectra retrieval in the visible range is good for all cases considered. Some difference is observed in the blue, when colored organic matter or mineral sediments are present in the ice or melt water, and in the IR, where the reflectance is too low and the signal is noisy.