The energy balance of melting thin first-year Arctic sea ice was a focus of the 8-day ICE12 drifting ice floe experiment on R/V Lance, conducted from 26 July to 3 August 2012, north of Svalbard in the southwestern Nansen Basin (82.3∘ N, 21.5∘ E). Figure  shows the Lance drift track that was in an area of very close (≥ 90 %) drift ice. The corresponding operational ice chart produced by the Norwegian Ice Service of the Norwegian Meteorological Institute (NMI, www.met.no) from 1 August is shown superimposed onto the map. The ice floe (ICE12 floe hereafter) that Lance was moored to during the drift had a size of approximately ∅ 600 m and a modal ice thickness of 0.8 m, deducted from drillings and measurements using a Geonics EM-31 electromagnetic induction device . The floe was mainly represented by level ice, with ridging over less than 10 % of the area. Based on airborne surveys of ice thickness using another electromagnetic induction device, the EM bird , and analysis of aerial photography, the floe was found to be representative for the area. The sea ice was in the latter stage of melt, covered by melt ponds some 15–30 cm deep with steep margins. The majority of ponds were connected to complex networks, often with an outlet to the ocean. Some of the ponds had actually melted through the ice slab, corresponding to stage III of surface-melt and melt-pond evolution according .

The aggregate albedo of a spatial mosaic of surface types is generally defined as

α=g(αj,fj):=∑jαjfj;αj,fj∈0,1, where summation is over all surface types used, here j={ow, bi, bp, dp}, with the corresponding fractional coverage fj. Note that for convenience we use the fractional total melt-pond coverage, fmp, relative to the sea-ice area. Coefficients fbp and fdp are defined as fractions of bright and dark melt ponds with regard to the relative melt-pond coverage, i.e., fbp=(1/(1+r))fmp and fdp=(r/(1+r))fmp. This transforms Eq.  for α to α=αowfow+αbi(1-fmp)(1-fow)++αbpfbp(1-fow)+αdpfdp(1-fow).

For any arbitrary set fmpi,fowi,Si, i=1,…,N, the set-based aggregate albedo αs is therefore calculated in the same way as the local estimate using Eq. , with the values of fows and fmps derived as fows=∑iSifowi/∑iSifmps=∑iSi(1-fowi)fmpi/∑iSi and referring to the set-based estimates of open-water and melt-pond fractions.

Deriving particular values of interest from the analysis of individual sea-ice images is analogous to sampling from a random data field with an a priori unknown theoretical distribution and a covariance structure. Any empirical statistic calculated from a set of analyzed images is therefore a derivative of the available data sample and should be considered an estimate accurate to within some unknown probability density.

Since the probability distribution of the local, image-based albedo αi is non-Gaussian, the large number of available samples makes the bootstrapping (i.e., sampling with replacement) technique an optimal choice to assess the probability density and the accuracy of the estimated image-set albedo. In our setting, the sets are formed of the swaths of images prone to the presence of autocorrelation in the variables used. It suggests the use of the moving block bootstrap approach .

For each flight the application of this method to the sequence of fowi,fmpi,Si involves the following steps:

The series of fowi,fmpi,Si of length N is split into N-K+1 overlapping blocks of length K; the block length is determined empirically from the data using the procedure described in the next subsection.

Steps 2–3 are repeated L times to generate L×M estimates of the swath-based aggregate albedo αs. The assigned values of {L,M}=200 yield a total of 40 000 samples of αs combined to generate the bootstrap pdf of the swath-based αs. The 95 % confidence interval (CI0.95) on the estimate is then calculated as {2.5,97.5} % of the empirical bootstrap pdf of αs.

This section presents the results of the analysis of sea-ice imagery along the six selected flight tracks that took place during the ICE12 cruise (Table ). All but one flight (flight 1, on 31 July) were combined EM bird/ICE camera flights, which fixed the helicopter flight altitude to approximately 35 m above the sea-ice surface except for some shorter periods of climbing to 150–200 m for EM bird calibration.

Figures  and show the summary statistics of melt-pond and sea-ice/open-water fractions along the tracks of flights 2 and 6, derived using the technique presented in Sect. . The data for the other four flights are presented in the Supplement in Figs. S1, S3, S5 and S7. Note that for flights 1–5, carried out from 31 July to 2 August, the results are similar, with a typical fmp of about 26 % relative to the sea-ice area and a similarity in the shapes of the respective pdf. In 50 % of these images, the observed fmp was between 15 and 36 %. We found that in some occasions the melt ponds could cover as much as 66 % of the ice surface within the image frame, yet for some 10 % of images with sea ice in the field of view, the sea-ice surface exhibited no or very little melt-pond coverage (fmpi<4 %). The average open-water fraction of fowi=11 % was characteristic of very close drift ice and varied for the analyzed images between 0 and 8 % in 50 % of cases, with fewer than 1 % of images showing 100 % open water. This variability lies within the uncertainty of the estimates and corresponds well to the respective operational ice charts for the area (see Fig. ).

Flight 6, on 3 August, was conducted while moving southwards out of the close drift ice. The flight track traversed the marginal ice zone (MIZ) with extensive areas/strips of open water. Thus the estimates of fow (30 %) and fmp (20 %) for flight 6 are substantially different from those inferred from survey flights conducted the previous days in the close pack ice (see Fig. ).

The EM bird surveys conducted during flights 2–6 further corroborate the inference of regional-scale homogeneity in the properties of the sea-ice cover. The probability density functions on sea-ice thickness presented in Fig.  suggest fairly similar shapes of the distributions, with the modal ice thickness ranging within 0.7–0.9 m for flights 2–5. The pdf for flight 6 reveals a tendency towards generally thinner ice, with a modal ice thickness of about 0.6 m. We note, however, that there can be a negative bias associated with a much higher open-water coverage observed during this flight.

Figure  summarizes the latitudinal distribution in melt-pond fraction and open-water coverage in the study area. Due to the nearly east–west orientation of the MIZ within the study area, Fig.  reflects the variability in these parameters towards the sea-ice edge. We note that in the time between the first and fifth flights the ice drifted southwards some 20 km, somewhat smearing the actual distribution in this direction. Flight 6 in turn provided a snapshot across the marginal ice zone. The figure reveals a fairly stable melt-pond coverage across a range of latitudes associated with very close drift ice during the experiment. In the ≈ 30 km wide MIZ the melt-pond coverage shows a gradual decline to values below 10 % close to the edge of the ice pack, in parallel with an increase in the open-water fraction. The transition occurs when the mean open-water fraction exceeds a threshold of approximately 20 % and is most likely associated with a generally more intense melt and a decrease in the typical ice floe size in the area. As the ice floes tend to break up preferentially along the existing melt ponds and melt channels, subsequent transformation of ponds into open water leads to a decreased fmp in the MIZ.

The bootstrap technique described in Sect.  is applied to the flight-track data of surface type variability fmpi,fowi,Si, i=1,…,N and in situ albedo measurements from the ICE12 drift experiment to yield the upscaled estimates of swath-based αs and a regional albedo of the study area αr. In addition we use the same technique to calculate the albedo of the ponded sea ice alone (αsi). Figures a and a show local (i.e., based on individual images) aggregate albedo estimates, αi, made from the helicopter imagery along the two selected flights with contrasting surface conditions presented in Sect. . The results for other tracks are presented in the Supplement and further summarized in Table . Note that in this case the image-based albedo variability is estimated from the data treated “as is” without taking the uncertainties into account.

Figure b and corresponding figures in the Supplement demonstrate fairly similar pdfs of local aggregate surface albedo for the flight tracks 1–5, suggesting a homogeneous state of sea-ice cover in the area within approximately 80 km of the ICE12 floe. We note that the empirical probability density functions of local albedo are skewed substantially towards zero due to the contribution of open-water areas. This suggests that an estimate of the regional-scale albedo of melting sea ice pack made by simple averaging of the respective quantities from a sequence of local scenes can be negatively biased. This may have implications for areal estimates of the surface energy budget both in observational and modeling studies.

Panels c in Figs. , , S2, S4, S6 and S8 display the generated bootstrap probability density of the swath-based αs for the six flights. Table  shows the calculated values of the average swath-based albedos and their respective bootstrap CI0.95. The respective values of ϕ from the transition matrix varied in the range of 0.78–0.88, whereas the probability of retaining the “open-water” state was lower: 0.51–0.57. These results are summarized in Table . The block length K was calculated as a ratio of N/Neff, yielding a block size of 9–12 images for four of the six transects, which corresponded to approximately 500–700 m of the flight track. For the tracks with the lowest (flight 1) and highest (flight 6) open-water fractions the derived block lengths were 18 and 7 images, respectively.

For all tracks the αs probability density is approximately Gaussian, with 95 % confidence according to the Lilliefors goodness-of-fit test of composite normality . The respective fits are shown together with the bootstrap pdfs in Figs. c, c, S2, S4, S6 and S8. The calculated standard deviations of the fitted Gaussian distributions are σαs=0.01 for flights 1–5 and σα6s=0.02 for flight 6.

Flight tracks 1–5 demonstrate similar values of the swath-based aggregate albedo αs of about 0.39, all lying within the estimated confidence intervals (see Table ). This suggests that the data from these five flights can be combined to provide the regional-scale albedo estimate for the ice pack outside the MIZ. This is implemented using the same technique applied to the concatenated sequence of fmpi,fowi,Si for all flight tracks but flight 6. When flight 6, representing mainly the marginal ice zone, is included in calculations, it decreases αr to a value of 0.37. The latter is related to the presence of extensive open-water areas in the some 30 km wide MIZ. The results of calculations are presented in Fig. a, and Table  further summarizes the results of the analysis.

The effect of open-water areas on the spatial albedo is demonstrated in Fig. b showing the bootstrap pdfs of sea-ice albedo αsis for tracks 1–6. We note that the spread in the inferred αsis pdfs between the individual tracks is much less pronounced compared to the respective αs. The regional bootstrap αsir of about 0.44 thereby provides a good estimate of the albedo for melting sea ice about 0.7–0.9 m thick for the entire study area.

The data on fow and fmp, merged from all six flights, were further binned in 0.1∘ wide latitudinal bins in a way similar to what was presented in Sect. . We calculated the bootstrap areal and sea-ice albedo for each latitudinal subset to yield the latitudinal distribution of these quantities. Figure  presents the results, demonstrating fairly stable values of αs and αsis for the area outside the MIZ, in accordance with the corresponding results on fow and fmp from Fig. . Within the MIZ increasing (decreasing) values of fow (fmp) towards the ice edge drive opposite trends in the bootstrap albedos αs and αsis. This suggests that a decrease in melt-pond fraction towards the ice edge may have a weak compensating effect on the areal albedo, slowing down the sea-ice surface melt in the MIZ. For solar radiation conditions observed during the drift experiment , the net effect of increasing the sea-ice albedo to about 0.5 would be to remove roughly 5 W m-2 of solar energy input, enough to reduce melt by about 1.5 cm of pure ice over the period of the experiment. One should note, however, that the upscaling results in this area with a more intense bottom and lateral sea-ice melt should be interpreted with caution. Potential for bias in the EM sea-ice thickness measurements due to effects of open water in the footprint of the EM instrument and a large dependence of sea-ice albedo on thickness for the thinner ice makes the application of the in situ albedo measurements made outside the MIZ less certain.

In order to infer the relative contribution of the spatial variability in melt-pond/open-water coverage and the uncertainty of in situ albedo measurements to the overall variance of the swath-based and regional albedo estimates, we also repeated the numerical experiments with the albedo of surface types treated as constants. The result demonstrated a substantial reduction in the standard deviations σαs and σαr down to 0.003 and 0.002, respectively. This indicates that in the defined framework, about 90 % of the estimated variance of αs and 95 % in αr is due to variability and uncertainties in the in situ albedo measurements. Only a minor part of the variance is due to all other errors and variability accounted for in the model.

The notion of aggregate scale for an environmental variable refers to the minimal spatial scale at which the contribution of local sampling variability to its total variance is diminished . The concept is directly related to the weak law of large numbers, provided that the samples are drawn from a stationary distribution. Knowledge of this scale is crucial for an accurate upscaling of local measurements and subsequently linking them to larger-scale climate models. We note that in a hierarchy of spatial scales, the present study focuses specifically on the range of meters to hundreds of kilometers, which encompasses the scales typical for in situ measurements up to regional models and CGCM.

The aggregate scale for the regional albedo was estimated using sets of (pseudo-)independent samples of different size drawn from the whole collection of classified images. The sample size varied from 10 to 1000 images, and for each sample size 10 000 subsets were drawn at random, without replacement, to gain the necessary statistics on the aggregate albedo distribution as a function of sample size and total sample area. As the image areas within each sample were not identical due to variations in the flight altitude, the average total area for each sample size was used. Images with an area over 6000 m2, corresponding to a flight altitude above 55 m, were not included in the analysis.

Figure  (black lines) shows the fraction of sample-based aggregate albedo estimates, falling within the interval of ± 1 and ± 2 standard deviations of the regional aggregate albedo (Table ), as a function of sample area. The results demonstrate a rapid growth in the proportion of accurate estimates of the regional albedo with an increase in the number of images drawn for analysis. The curves level out when the total sample area exceeds the threshold of about 0.7 km2, when some 95 % of the subset-based estimates lie within the interval of 2 SD of the regional bootstrap albedo. One should emphasize that these estimates are specific to this study's setup, time period and region. For the range of flight altitudes typically sustained during the operation of the EM bird, the 0.7 km2 aggregate scale corresponds to a set of at least 300 independent images spatially representative of the study region.

In order to simulate higher flight altitudes and examine the effect of smaller sample sets and/or sub-kilometer scale spatial autocorrelation in the state of sea-ice cover on the estimate of the aggregate scale, the numerical experiment was repeated with successive images combined into blocks of different length. The validity of this experiment relies on the assumption of smaller-scale anisotropy in statistical properties of the sea-ice surface. The red and grey lines in Fig.  show the fraction of the accurate estimates of the regional albedo for image blocks of length 10 and 25 images, respectively. Results suggest an increase in the aggregate scale to values above 2 km2 which would correspond to sets of at least 80 (30) area-representative images captured from an altitude of about 100 (170) m. Notably the estimated thresholds (aggregate scales) have an order of magnitude similar to the respective estimate of >1 km2 obtained by during the SHEBA experiment in a different region of the Arctic.