Data were collected on first-year ice north of Svalbard during the ICE12 expedition on R/V Lance in July–August 2012, between 82.5 and 82∘ N . The ice cover was generally less than 1 m thick and was at an advanced stage of melt, with a melt pond coverage of 25–30 % . The observations for this study were made on one floe that was studied during an 8-day drift . Based on aerial surveys the ice was representative of the area .

Downward irradiance at the ice–ocean interface was measured at intervals of about 1 m along three different transects by a diver holding a RAMSES Acc-Vis sensor (TriOS Mess- und Datentechnik GmbH, Rastede, Germany). An aerial photo of the study area, with the three transects indicated, is shown in Fig. . To guide the diver, a rope was stretched under the ice between two poles mounted at the beginning and end of a transect, at a depth of roughly 1 m below the ice. The rope was marked every 1 m, but due to the elasticity of the rope, the actual distance between markers was somewhat longer, depending on how tight the rope was. An irradiance spectrum was collected directly at the ice bottom above each marker. In order to estimate the true position of the measurements along the transects, variations in the measured ice topography (Fig. ) was compared with variations in pressure measured by the RAMSES sensor. Rope stretch factors of 1.125, 1.05, and 1.1 were estimated for the transects 1, 2, and 3, respectively (see Table ).

Incident irradiance was measured coincidentally, with the same type of instrument as below the ice, but mounted on a tripod by the dive hole, near the start of the transect. We assume no horizontal variability of incident irradiance between the measuring site and the transect. Spectra were collected simultaneously from the two sensors, and as RAMSES sensors are generally not calibrated to exactly the same wavelengths, the spectra were interpolated to a common wavelength grid, with 1 nm spacing.

Ice cores for determining the vertical structure of the ice, in terms of temperature, salinity, and microstructure, were not collected from the transects themselves, but temperature measurements from cores taken from bare ice elsewhere on the same ice floe showed that the ice was warm, generally around 0 ∘C in the surface, decreasing to about -1.2 ∘ at the bottom. Assuming a bulk salinity of 3, and temperatures up to -0.5 ∘C for the interior ice, the brine volume fraction may be up to 12–30 % . No samples were melted and filtered for measurement of absorption by particulate matter and coloured dissolved organic matter (CDOM).

Ice thickness and freeboard were measured directly by drilling through the ice at 1 m intervals, after the irradiance measurements were done. The measured ice thickness was used to identify the measurements that were far enough away from a different surface type to not be influenced by it. For each boundary between pond and bare ice along a transect, the ice thickness hi for bare ice that was closest to a pond edge was picked out, and all measurements that were done within a distance of 2hi of the pond edge were identified as being close to a pond edge. The value of 2hi was chosen based on , who, using a 3-D Monte Carlo radiative transfer model, found that 90 % of the light hitting a sensor at the ice bottom was incident within a radius of twice the ice thickness, for uniform ice.

The radiative transfer model used is AccuRT (formerly c-disort). AccuRT is a 1-D plane-parallel coupled atmosphere–ice–ocean radiative transfer model based on DISORT , and uses the discrete ordinates method to solve the radiative transfer equation. The model domain consists of two horizontal slabs with different refractive indices, and multiple layers in each slab to resolve vertical variations in inherent optical properties (IOPs). Snow can be represented as a layer of ice spheres in the bottom of the upper slab (atmosphere), and ice can be added to the top of lower slab. Brine pockets are represented by spheres of pure sea water with a given radius and volume fraction, and air pockets are likewise represented by spheres of air. Their inherent optical properties are calculated using a parameterization based on Mie calculations .

AccuRT outputs multiple properties, including downward and upward planar (E) and scalar (E0) irradiance at specified wavelengths and vertical levels. For the planar irradiance, both the direct beam and the diffuse part of the radiation field are available. The inherent optical properties of each layer are also available.

Analysis and plotting were done using Python, libraries including numpy , scipy and pillow, with plots made in matplotlib . Python code for reading model output from AccuRT and calculating heating rate, albedo, and transmittance is available at https://github.com/TorbjornT/pyAccuRT.

In addition to the upward (E0↑) and downward (E0↓) scalar irradiance at specified depths, AccuRT outputs the total absorption coefficients (a) of each layer. The spectral heating rate at a given depth, Hλ(z), is determined by the change in net irradiance (upwelling minus downwelling planar irradiance, Enet=E↑-E↓) with depth, Hλ(z)=dEnetdz. This can be related to the scalar irradiances and absorption coefficients output by the model through Gershun's law e.g., dEdz=-aE0, where the total scalar irradiance E0=E0↑+E0↓. From Eqs. () and (), we use the model output to calculate local spectral heating rates as Hλ=a|E0↑|+|E0↓|.

Bulk transmittance over the range of wavelengths from λ1 to λ2 is defined as Tbulk=∫λ1λ2T(λ)Ein(λ)dλ∫λ1λ2Ein(λ)dλ=∫λ1λ2Etra(λ)dλ∫λ1λ2Ein(λ)dλ, where T is spectral transmittance, Ein is incident total irradiance (downward diffuse plus direct beam), and Etra is transmitted total irradiance. Unless otherwise specified, the wavelength range over which the bulk transmittance is calculated is 400–900 nm. Bulk albedo is calculated similarly, but with reflected irradiance instead of transmitted irradiance in the numerator.

Irradiance was calculated every 2 nm, and the resulting spectra smoothed with a Gaussian filter, to approximate the spectral resolution of the RAMSES sensors.

Model parameters were chosen through an iterative process to obtain a good correspondence with the measured data.

In the radiative transfer model, the sea ice under ponds is represented by two layers (Fig. ). The upper layer has a thickness of 10 cm, with an air volume fraction that depends on the average pixel intensity of the aerial photo (see below), with higher air volume fractions for brighter areas. Brighter areas appeared to be caused by a layer containing a larger amount of air, but the thickness of this layer in the simulations (10 cm) was chosen arbitrarily. The lower layer extends from 10 cm below the pond to the measured ice thickness, and has a fixed air volume fraction of 0.1 %. Bare ice is also represented with two layers, but here the thickness of the upper layer, corresponding to the surface scattering layer (SSL), is determined by the measured freeboard. The upper layer, which consisted of granular ice, was simulated as large-grained snow, i.e. ice spheres in air, with a radius of 2.5 mm. All ice, with the exception of the SSL, was simulated with a 20 % brine volume fraction. The effective radius of air bubbles in all simulations was set to 0.25 mm, which is a bit higher than the upper bound for the air inclusions found by , though it should be noted that was for ice at a temperature of -15 ∘C, whereas the ice in our study was warmer, around -1 ∘C. For brine, the effective radius of inclusions (1.5 mm) was taken to be constant for all ice layers in the simulations.

Values of the effective scattering coefficient, beff=b(1-g), are shown in Fig. . used beff of 102, 15, and ≈ 1 m-1 in the SSL, drained layer and interior ice, respectively. Hence, our SSL has less scattering than that of , but we did not include a drained layer. On the other hand, our interior ice has higher scattering. For a review of beff used in various publications; see Table 2 in .

Ponds are represented by adding a layer of pure sea water on top of the sea ice, with the thickness of the water layer equaling the measured pond depth. Water depth in the region where the measurements were taken is around 3 km, and that depth is used as the total water depth in the radiative transfer simulations. The sea water itself was pure sea water with the addition of absorption and scattering measured during the Norwegian Young Sea Ice Cruise in 2015 “pre-bloom” case from. The solar zenith angle was set corresponding to the time and approximate location of measurements, with values between 64.5 and 66.3∘.

The number of streams used in the upper slab, i.e. the atmosphere, was set to NU=32. The number of streams in the lower slab (ice and ocean) is set automatically by AccuRT to NL=NU⋅mmax2, where mmax is the maximum refractive index of the lower slab in the simulated wavelength range, which gives NL=57 for ponded ice and 61 for bare ice. These high numbers of streams were needed to avoid a numerical artifact in the calculation of albedo over the highly scattering bare ice.

Varying cloud cover was present on all days, causing large variations in incident irradiance. Integrated from 400 to 900 nm, values ranged from 84 to 266 Wm-2. In all simulations clouds are represented by a 0.5 km thick layer of water droplets, with a base height of 2 km, and an effective droplet radius of 10 µm. The volume fraction of cloud particles was adjusted so that the simulated incident irradiance at 460 nm was within about 5 mWm-2nm-1 of the measured incident irradiance. The applied volume fractions ranged between 1.3×10-10 and 9.1×10-7, corresponding to optical depths between 0.007 and 45.7.

Some other transmittance data from the same year and region can be found in the literature. The bare ice transmittance in our study is comparable to the measurements by (transmittance of ∼ 0.2 in mid-July), though this was for a different wavelength region (350–800 nm). measured transmittance up to about 0.3 in mid-August, but used a wider wavelength region (320–950 nm).

Not limited by location and time, there are multiple other studies to be found. , also for the range 320–950 nm, showed transmittances that were generally below 0.1, but this was for thicker ice. Our PAR (photosynthetically active radiation, 400–700 nm) transmittances are similar to those measured by , but generally somewhat higher for bare ice, likely due to thinner ice in our study. Where we have PAR transmittance of 0.20–0.25 for bare ice and 0.50–0.59 for ponded ice, reported 0.07–0.20 and 0.34–0.65 for bare and ponded ice, respectively. Our results are also in the upper range of those reported by , where PAR transmittance was 0.03–0.22 (bare ice) and 0.13–0.58 (ponded ice). showed PAR transmittances no higher than 0.4, for a study comprising several ROV deployments at different locations in the Arctic Ocean. That the transmittance in this study is high compared to other studies is likely due to this being relatively thin first-year ice.

Concerning the ice thickness and pond depth used in simulations, there are several factors that give rise to uncertainties regarding these. First, the rope was located about 1 m below the ice, but the diver held the radiometer at the ice bottom. The diver therefore had to estimate the position at the ice bottom to be as close as possible to the mark on the rope. Second, rope stretching makes the distances of the mark on the rope from the start of the transect an estimate. Finally, the rope stretching also means that the under-ice irradiance measurements were not performed in the same locations as the thickness drillings, with differences possibly up to 0.5 m. To account for the rope stretching, ice thickness, pond depth, and freeboard were interpolated linearly to the estimated locations of the irradiance measurements (Fig. ), though there remains some uncertainty in the exact locations of both the irradiance measurements and the thickness measurements.

The aerial photo (Fig. ) from which the along-transect average intensity has been determined was captured on 28 July, i.e. 1 day after transect 1 was sampled, 2 days before transect 2 was sampled, and 3 days before transect 3 was sampled. As the surface may have changed somewhat in the interim, the average intensity obtained from the image may not exactly represent the conditions at the time of the radiometric measurements.

As a different way of looking at how the model performs, we take a closer look at the spectral shape of measured and simulated transmittance. Figure  shows the mean gradient of the transmittance spectra (dT/dλ), for bare and ponded ice separately, calculated using numpy's gradient method. While there is generally a good correspondence between gradients from observed and simulated transmittance spectra, some differences stand out. At around 730 nm the observed spectra have a notably steeper gradient for ponded ice, whereas the bare ice fits better; 730 nm is a wavelength where the absorption coefficient of pure water is higher than that of pure ice, and a higher brine volume gives a steeper gradient here. As such, this difference between simulated and observed data could indicate that the specified brine volume of 20 % is too low.

The troughs in the gradient seen near 690 and 760 nm are likely caused by absorption by atmospheric oxygen, which changes the angular distribution of the light field, and therefore may give a local increase in transmittance as described in .

Near 800 nm the gradient for observed spectra exceeds zero, whereas the gradient for simulated spectra is always negative. This discrepancy, which is particularly visible in the case of ponded ice, was discussed in , but no satisfying explanation was found.

The simulated ice has an air volume fraction (0.0015) that is about an order of magnitude smaller than that of the ice studied by . At the same time, the effective radius of 0.2 mm used here is much smaller than the 0.93 mm used by for fast ice in Kongsfjorden, Svalbard, but more consistent with the first-year fast ice from Point Barrow, Alaska, studied by . Compared to , the differing volume fraction and effective radius has opposite effects on transmittance. The smaller air volume fraction leads to less scattering, but a smaller radius gives a higher number of air bubbles, which will increase scattering, with the scattering efficiency peaking for sizes comparable to the wavelength of the incident light. That smaller spheres with constant volume fraction increases scattering is true to a certain point – for very small spheres scattering approaches the Rayleigh regime with a very small cross section per particle.

It should be noted that brine inclusions in sea ice are generally not spherical . related an equivalent spherical radius to the length of brine inclusions, but only for ice at a temperature of -15 ∘C. To represent both absorption and scattering accurately, both total volume and total area need to be the same for the spheres as for the original inclusions. show (their Fig. 9a) that for a brine inclusion length of 3 mm, the radius of equivalent spheres, conserving both area and volume, is about 0.1 mm, while for a brine inclusion length of 1 cm, the equivalent sphere radius increases to about 0.15 mm. The ice in our study was much warmer than -15∘, so brine inclusions are likely comparatively large. Hence, even when using the highest equivalent radius from , 0.15 mm, for brine inclusions in AccuRT, while keeping the brine volume at 20 %, the scattering increases so much that, even with zero air bubbles, transmittance decreases. However, this likely just indicates that the properties of our ice are very different from the colder FYI studied by , so that a larger equivalent radius is more appropriate.

The ice was in a late stage of melt, with temperatures near 0 ∘ at the top and around -1.2∘ at the bottom. On the bare ice regions there was a surface scattering layer above the freeboard, which consisted of deteriorated ice that was in a granular form, similar to coarse grained snow. Figure  shows that the interior ice was visibly porous, with some deteriorated internal layers. The ponds on the surface had drained to sea level, indicating that most brine pockets or channels would have been filled with melt and/or sea water. Detailed structural analysis and identification of air- versus liquid-filled inclusions was hampered by the immediate drainage of the liquid inclusions when a core was taken up, but it is safe to say that the ice did not retain the traditional structure of sea ice, determined by freezing equilibrium relationships. As a result of the heavily modified state of the ice, our estimates of brine and air volumes and sizes may be highly uncertain.

Further, the vertical orientation of brine inclusions commonly seen in sea ice appears to cause anisotropic scattering in the ice , which AccuRT cannot account for. This issue may be important for determining the exact radius over which edge effects are important.

As little information about the vertical structure of the ice is available, the number of layers in the model is kept at a minimum. While assumptions could be made based on typical profiles of salinity, for the sake of simplicity, and reducing the number of assumptions, we have chosen to not do this.

The results in Fig.  are what one might expect – simulations of bare ice underestimate the transmittance when compared to measurements near a pond, and simulations of ponded ice overestimate the transmittance when near bare ice. However, due to these opposing effects, the averaged bulk transmittances are similar, being 0.28 and 0.29 for simulated and measured values, respectively, including outliers. Hence, for area-wide averages the edge effects may largely cancel each other out.

Looking at a simpler case, we simulate a typical ponded and bare-ice case for each transect, using the mean pond depth, freeboard, and ice thickness. Ice properties are as described in Sect. , with the amount of air in the upper 10 cm of the under-pond ice as the median of the values found based on the luminosity (Sect. ). The total length of each surface type is found (see Fig. ), and a length-weighted average transmittance is calculated using T¯=LpTp+LbTbLp+Lb, where Lp and Lb are the length of the transect covered by ponds or bare ice, respectively, and Tp, Tb are the corresponding transmittances. This approach can easily be applied to area fractions of surface types by replacing the lengths by areas.

The mean measured bulk transmittances for the three transects are respectively, 0.27, 0.33, and 0.27 for transects 1, 2, and 3. Using Eq. () and transmittance based on simulations using mean values, we get 0.30, 0.30, and 0.29 for the three transects individually, and 0.29 for all points. While the average simulated values correspond well to measured values, one should note that the ponds in this study are generally wide compared to the ice thickness. A similar correspondence would not necessarily have been seen if the ponds were so small, or the ice so thick, that the light field beneath most or all of the ponded area were affected by the surrounding bare ice.

Figure  shows contour plots of the spectral heating rate Hλ, which is the energy absorbed per unit volume per wavelength per unit time, given by Eq. (). The absorbed energy is calculated from the simulated scalar irradiance field and total absorption coefficient of the layer. The scalar irradiance field was calculated every 1 cm in the pond and ice. For easier comparisons, layer thicknesses were equal for both ponded and bare ice. That is, the thickness of the SSL was set equal to the pond depth (14 cm), and the thickness of the interior ice was 60 cm in both cases, giving a total SSL/pond + interior ice thickness of 74 cm.

Due to the higher absorption coefficient of both ice and water in the near-infrared range compared to the visible range , the near-infrared has the highest surface heating rate. Beyond 1400 nm nearly all the energy is absorbed in a layer just a few centimetres thick. In the visible range, where the incident irradiance is higher, the absorption is significantly lower, meaning that more energy is deposited deeper in the pond-ice-system, and in the ocean.

High concentrations of algae in the bottom layer of the ice could influence the energy deposition . In Fig. c and d we show the result of a simulation where absorption and scattering corresponding to 500 mgm-3 chlorophyll a, as described in , is added to the bottom 5 cm of the ice. measured chlorophyll a concentrations of over 5000 mgm-3 in the Canadian Arctic, albeit for 2.5 cm ice; thus 500 mgm-3 is a realistic concentration in sea ice. Figure  shows the corresponding profiles of cumulative absorption. Note that the diffuse attenuation coefficient for PAR in the algae layer is around 5 m-1, which is significantly lower than the ∼ 22 m-1 estimated by for the same chlorophyll a concentration.

To compare the amount of heating within the ice for the two cases, Fig.  shows the cumulative heating of the ice (or ice with pond) relative to the incident irradiance, Qrel, for both the range 350–2200 nm and the PAR range. The heating rate Eq. () is first integrated over the wavelength range, and then cumulatively integrated from the surface and downwards. This result is divided by the energy entering the ice E↓, i.e. the downward planar irradiance less the specularly reflected irradiance just above the atmosphere–ice interface. Hence, Qrel(z)=∫0zH(z′)dz′∫λ1λ2E↓dλ, where H(z′)=∫λ1λ2Hλ(λ,z′)dλ.

We see in Fig.  that about 19 % of the incident energy is absorbed in the bare ice, and 49% in the pond and ice below it. Considering all 99 cases corresponding to measurements, the mean (standard deviation) is 22 % (6 %) and 51 % (3 %) for bare and ponded ice, respectively. Considering only the SSL and pond, the values are 15 % (5 %) and 35 % (4 %). With algae present the values are 21 and 54 %, respectively. That a smaller fraction of the incident energy is absorbed in the bare ice is caused by the higher amount of light scattered back to the atmosphere from the surface scattering layer. In general, a thicker SSL in the simulations gives a higher albedo, and a smaller fraction of energy absorbed in the ice. Simulations from ponded ice show no clear correlations between the various parameters, such as pond depth, ice thickness, albedo, and absorbed energy.

On another note, some of this backscattered light is again scattered back towards the surface when it hits the thin cloud layer present in both simulations. As a result of this interaction with the clouds, the simulated incident irradiance is 45 % higher over the bare ice compared to the darker ponded surface (213 Wm-2 vs. 147 Wm-2). Hence, where the modelled relative absorption is equal for bare and ponded ice, the bare ice has a higher absolute absorption. This effect of higher incident irradiance over bare ice will likely not be present at all in reality, as the horizontal scale of pond features is small, of the order of a few metres. A radiation enhancement effect due to scattering between surface and clouds may be present, but the area scattering radiation back to the clouds will be a mixture of bare ice and ponded ice, so there will likely be very little or no spatial variability of incident irradiance along the transects.

In both cases the strongest heating rate is seen right below the surface, due to the strong absorption by both water and ice in the near-infrared range (see Fig. ). The absorption and scattering coefficients in the interior ice are the same in both cases, as the properties of the ice are the same. Therefore, the stronger heating rate seen in the interior ice under the pond is caused by the higher amount of available light.