Melt ponds occupy a large part of the Arctic sea ice in summer
and strongly affect the radiative budget of the atmosphere–ice–ocean system.
In this study, the melt pond reflectance is considered in the framework of
radiative transfer theory. The melt pond is modeled as a plane-parallel layer
of pure water upon a layer of sea ice (the pond bottom). We consider pond
reflection as comprising Fresnel reflection by the water surface and
multiple reflections between the pond surface and its bottom, which is
assumed to be Lambertian. In order to give a description of how to find the
pond bottom albedo, we investigate the inherent optical properties of sea
ice. Using the Wentzel–Kramers–Brillouin approximation approach to light
scattering by non-spherical particles (brine inclusions) and Mie solution for
spherical particles (air bubbles), we conclude that the transport scattering
coefficient in sea ice is a spectrally independent value. Then, within the
two-stream approximation of the radiative transfer theory, we show that the
under-pond ice spectral albedo is determined by two independent scalar
values: the transport scattering coefficient and ice layer thickness. Given
the pond depth and bottom albedo values, the bidirectional reflectance factor
(BRF) and albedo of a pond can be calculated with analytical formulas. Thus,
the main reflective properties of the melt pond, including their spectral
dependence, are determined by only three independent parameters: pond depth
The effects of the incident conditions and the atmosphere state are examined.
It is clearly shown that atmospheric correction is necessary even for in situ
measurements. The atmospheric correction procedure has been used in the model
verification. The optical model developed is verified with data from in situ
measurements made during three field campaigns performed on landfast and pack
ice in the Arctic. The measured pond albedo spectra were fitted with the
modeled spectra by varying the pond parameters (
Melt ponds occupy a large fraction of the Arctic sea-ice surface in summer: up to 60 % on multi-year ice according to Maykut et al. (1992) and up to 80 % on landfast ice according to Langleben (1971) with more typical values between 20 and 40 % (Polashenski et al., 2012; Rösel et al., 2012; Istomina 2015b). They reduce the ice albedo significantly and, therefore, increase the flux of absorbed sunlight energy and speed up the process of melting, thus amplifying the positive ice–albedo feedback effect (Curry et al., 1995; Eicken et al., 2004; Pirazzini, 2008; Schröder et al., 2014). Recent observations show that the melt onset is shifting earlier and the whole melt season is getting longer (Serreze et al., 2000; Dethloff et al., 2006; Perovich et al., 2008; Markus et al., 2009; Pistone et al., 2014). Moreover, as the prevailing sea-ice type has changed from multi-year ice to first-year ice in the recent decades (Comiso, 2012; Maslanik et al., 2007, 2011), the topography of the sea ice evolves from rough to uniform, flatter surface. As the melt pond fraction is closely connected to the relief of the sea ice (Polashenski et al., 2012), the maximum pond fraction is expected to increase as well. Therefore, including light reflection by melt ponds into climate models is an important task, particularly in light of the environmental changes observed recently (Flocco et al., 2010, 2012; Hunke et al., 2013; Lüpkes et al., 2013). A physical model of the reflective properties of melt ponds is needed for understanding the physics of sea ice, as well as for the correct interpretation of the results of remote sensing and field measurements (Herzfeld et al., 2006; Tschudi et al., 2008; Rösel et al., 2012; Zege et al., 2015).
The observed albedo of melt ponds varies over a wide range. They can change from light-blue ponds, when just formed, to dark mature ones, late in melt, meaning that the character of the ponds is important in addition to their coverage (Perovich, 1996; Barry, 1996; Nicolaus et al., 2010; Sankelo et al., 2010; Polashenski et al., 2012). Although there are quite a lot of measurements of melt pond spectral albedo (e.g., Perovich, 1994; Morassutti and Ledrew, 1996; Perovich et al., 2002, 2009), an adequate physical and optical model of melt pond reflection is still absent. Makshtas and Podgorny (1996) gave the analytical formula expressing the pond albedo in terms of the albedo of its bottom. However, despite asserting that bottom albedo is the main factor that determines the albedo of a pond as a whole, they did not address how to calculate it. This essential gap exists up to now. In this work we propose a simple solution for determining the pond bottom spectral albedo. This solution has required the detailed consideration of the inherent optical properties of sea ice, which forms the pond bottom. In addition, the question of the angular distribution of light reflected by a melt pond is still open. The angular distribution is highly important for understanding Arctic energy balance, because only the bidirectional reflectance is measured by satellite optical sensors and it is necessary to model the bidirectional reflectance distribution function (BRDF) to determine surface albedo from satellite data. Additionally, the processing of the reflectance measurement data, both satellite and ground-based, requires atmospheric correction, especially for polar regions. All these points are discussed in this work.
The paper is arranged as follows. First, our model of melt pond reflectance is described in Sect. 2. Section 2.1 presents the formulas for pond reflectance at various incident conditions. Inherent optical properties (IOPs) of sea ice are considered in Sect. 2.2. A simple analytical solution for bottom albedo in terms of the ice IOPs and its thickness is given in Sect. 2.3. Section 2.4 gives a final summary of the model developed. Section 3 discusses how illumination conditions are accounted for in processing and how the experimental results are interpreted. The atmospheric correction of experimental data is considered in Sect. 3.1. A possibility to use the near-IR reflectance as evidence of the ice grains' presence is discussed in Sect. 3.2. Notes about processing experimental data when the incident angle is unknown are given in Sect. 3.3. Then, Sect. 4 presents the verification of the developed model with the three datasets of in situ measurements (Polarstern-2012, Barrow-2008, and SHEBA-1998). The conclusion sums up the paper.
In this work we propose a simple optical model that enables the parameterization of the pond bottom albedo with a few physical characteristics and thus determines the spectral reflective properties of the melt pond as a whole, including its bidirectional reflectance.
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:
the melt water is pure, with neither absorbing contaminants nor
scatterers; the Rayleigh scattering in water is negligible compared to the water
absorption, so a ray inside the pond is attenuated according to the
exponential law; the pond bottom reflects light by the Lambert law (the reflected
radiance is independent of the direction).
Makshtas and Podgorny (1996) give the following formula for the albedo of a
pond that satisfies the abovementioned assumptions:
Albedo
The albedo at diffuse incidence
The albedo at direct incidence is expressed through the bidirectional
reflectance factor
Note that Eqs. ( How is the pond bottom albedo expressed in terms of the inherent
optical properties of sea ice and the ice layer thickness? What are the main optical characteristics of under-pond ice that
really determine the pond bottom albedo and, hence, the pond reflectance?
We address these questions in the following subsections.
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
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
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
(
The upper layer of sea ice (20–30 cm) usually contains a significant amount
of air bubbles (Gavrilo and Gaitskhoki, 1970; Mobley et al., 1998), with
volume concentration which can reach values of 5 % and which decreases
with depth. (We do not consider here the surface scattering layer that is
formed on top of sea ice during the water drainage process.) Air bubbles in
sea ice are mostly spherical (Gavrilo and Gaitskhoki, 1970; Mobley et
al., 1998; Light, 2010). Light (2010) gives the following size distribution
for bubbles in first-year sea ice:
However, since air bubbles in ice are optically hard (the refractive index of
air differs strongly from that of ice) and do not absorb light, scattering by
bubbles of this size range is described by the laws of geometrical optics.
Thus, the scattering characteristics do not depend on the bubble size (unless
considering the strictly forward and backward directions), and the shape of
the size distribution is also insignificant. Particularly, the scattering
efficiency
The refractive index of air (relative to ice) in the interval
0.35–0.95
The main features of brine inclusions are the following: they are optically
soft, i.e., their refractive index
The size of brine inclusions, which can be on the order of several
millimeters, is so much larger than the wavelength of visible light that
their optical properties can be considered in the limit of infinitely large
particles, despite their refractive index
Spectra of the relative refractive index “water to ice”: distilled
water (symbols), fresh water at 0
Light-scattering properties of sea ice are a combination of those of brine
inclusions and air bubbles. The total and transport scattering coefficients
are the sum of the respective values:
Generally, the IOPs of sea ice depend on its microstructure. In view of the
fact that both bubble and brine inclusion size is much larger than the
wavelength, the scattering coefficient equals
The phase function (and consequently its average cosine
We conclude that the phase function (and consequently
Phase functions of the mixture of air bubbles and brine inclusions
at
For example, Light (2010) gives the value of 110 m
The bubbly ice reported by Gavrilo and Gaitskhoki (1970) has
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:
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.
Melt pond characteristics.
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
Thus, in the absence of pollutants just three parameters determine the pond
spectral reflectance: namely, the transport scattering coefficient
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
Black-sky albedo of a light melt pond (
Thus, the light flux incident to a melt pond is
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
Modeled spectra of melt pond albedos at various sky conditions and
background albedo at sun elevation 10
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
Typical spectral albedo of melt ponds, snow, and white ice,
calculated for the following parameters: light pond – depth
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
Light frozen (2–3 cm layer of ice) blue ponds. Polarstern-2012,
Stations 1
Three different datasets with in situ field measurements were used for the evaluation of the pond model. They are described in the next subsections.
Measurements of the spectral albedo of different sea-ice surfaces were
carried out during the R/V
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
For the model verification we considered the melt pond albedo in the spectral
interval 0.35–1.3
Frozen blue ponds. Polarstern-2012, Stations 1
Dark open ponds. Polarstern-2012, Stations 4.
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
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
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
Measure and retrieved pond parameters derived from the spectral
range 0.35 to 1.3
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.
SHEBA-1998 dual pond: photos and spectra (Grenfell et al., 2016), measured (dashed) at the light (blue) and dark (red) parts and simulated (solid). The photographs are taken at the early and late melt season (on 3 July and 8 August, respectively).
Retrieved pond depth
Ice thickness and pond depth, measured at different stations and
retrieved. For ice thickness
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
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
Suppose that the difference between the transport scattering coefficient
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 (
The retrieval of the pond depth is more uncertain: its value can differ up to
2 times from the measured one and RMSD
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.
This work presents an optical model of melt ponds on sea ice. The melt pond model described in this work relates the optical properties of a melt pond (spectral albedo and angular reflectance) to its physical characteristics (microphysical ice properties, water depth, sea-ice thickness, sediment amount) at various sky conditions.
We assume a pond to be a plane-parallel layer of melt water on an under-pond ice layer. We paid particular attention to the pond bottom albedo as the main factor that determines the pond reflectance. The albedo of the under-pond ice is calculated within the modified two-stream approximation (Zege et al., 1991), which relates the layer albedo to its thickness and to the transport scattering coefficient of a medium. The analysis of the spectral behavior of the inherent optical properties of sea ice, using the WKB approximation approach to light scattering by non-spherical particles (brine inclusions) and Mie solution for spherical particles (air bubbles), has shown that the average cosine of the scattering phase function, and therefore the transport scattering coefficient of sea ice, is spectrally neutral. Hence, the pond can be characterized by only three independent parameters that determine its reflectance through the visible and near-IR spectral range: the pond depth, the under-pond ice thickness, and the ice transport scattering coefficient.
The model developed proposes the simple analytical formulas to calculate the main reflective characteristics of a melt pond: the bidirectional reflectance factor and the black- and white-sky albedo. The model is simple in its implementation, because it is entirely based on analytical formulas. The derivation of the analytical formulas becomes possible due to the assumption of the Lambert reflection by the pond bottom. Although this commonly used assumption has no reliable experimental basis, the model verification with a wide set of field measurements (SHEBA-1998, Barrow-2008, and Polarstern-2012) confirms that this assumption is reasonable, at least concerning the spectral albedo. Its validity for the pond bidirectional reflectance requires further investigations.
Additional attention is paid to correctly accounting for the illumination conditions during the field measurements. It is shown that multiple reflections of light between the atmosphere and surrounding background can be neglected, so the a priori knowledge of the background (surrounding ice) albedo is not necessary. However, the sky conditions (overcast or clear, presence of cirrus or aerosol load) should be specified to interpret the pond albedo as the white-, black-, or blue-sky ones. In the last case it is highly desirable to know the spectrally resolved atmospheric optical thickness during the measurements. Unfortunately, such information is rarely available for field measurements of the sea-ice reflective properties.
The model can be used to study the distribution of melt pond physical properties and temporal evolution of the small-scale sea-ice morphology during summer melt. The melt pond model is necessary to retrieve the melt pond fraction from optical satellite data of moderate resolution (with pixel size starting from hundreds of meters), where melt ponds become subpixel. In turn, the amount of melt ponds on Arctic sea ice determines the sea-ice reflectance and transmittance and thus allows estimating the energy balance above, within, and under sea ice and its response to climate change. The temporal evolution of melt ponds consists of melt stages, which are specific to sea-ice type (landfast ice, first-, second-, or multi-year ice). The spring melt pond fraction predicts the autumn Arctic sea-ice extent. Therefore, the melt pond fraction dataset obtained from satellite data is required to derive the sea-ice extent and type during summer melt.
The model presented has been already used in an algorithm for sea-ice albedo and melt pond fraction retrieval from MERIS satellite data (Zege et al., 2015; Istomina et al., 2015a, b). The model provides accurate description of the melt pond reflective properties: not only pond albedo but also pond bidirectional reflectance, which is of great importance for processing satellite data. Moreover, the approach presented can be easily extended to describe the light transmittance through sea ice, which is also important for the radiative budget of the Arctic Ocean. The model presented is able to reproduce a variety of melt pond types observed in the field. It can be applied to the problems of physics of sea ice and to monitoring the melt of the Arctic and Antarctic sea-ice cover. Also, it makes it possible to improve the parameterization of the underlying surface in various atmospheric remote sensing retrievals over the Arctic summer sea ice (clouds, aerosols, trace gases) and potentially re-evaluate the climatic feedbacks and radiative budget of the Arctic region at a new accuracy level.
The field data from the R/V
The field data from the Barrow-2008 expedition are available at the Arctic Data Center: spectral albedos – Polashenski et al. (2016a); line photos – Polashenski et al. (2016b).
The field data from the SHEBA-1998 expedition are available at
The authors declare that they have no conflict of interest.
The work was supported by the Institutional Strategy of the University of Bremen, funded by the German Excellence Initiative, and by the TR 172 “ArctiC Amplification: Climate Relevant Atmospheric and SurfaCe Processes, and Feedback Mechanisms (AC)3”, funded by the German Research Foundation (DFG).
The authors are grateful to the scientific party of the ARK XVII/3 cruise for making the spectral albedo measurements possible. Special thanks are expressed to Marcel Nicolaus for organizing the logistics and to the Sea Ice Physics group on board for assisting with the measurements. The article processing charges for this open-access publication were covered by the University of Bremen. Edited by: Jean-Louis Tison Reviewed by: two anonymous referees