This study integrates multiple data products from satellites and reanalyses listed in Table , which we carefully select to ensure high-quality coverage of the entire pan-Arctic region on a daily basis. By intersecting the temporal and spatial domains of the data products, the overlapping period is from 2013–2020, during the months of March to September when sunlight is present in the whole pan-Arctic region. The final dataset consists of five sea ice and five atmospheric input features. For better readability, we refer to input features as features.

Building on the principles outlined by and , this study employs a Pareto optimality-based workflow as illustrated in Fig. .

The numbers in brackets indicate the averaged ranking across the ten subsets. When no bracket is indicated, the ranking of a feature remains consistent across all subsets. Let Pd be a polynomial of degree d∈{1,2,3,4}. The SFS algorithm reveals the following feature rankings for P1-4 and NN: P1:hsnow→T0m→T2m→rain→snowfall→hice→RH→wind speed→ice speed→ageP2:hsnow→T0m→hice→T2m→snowfall→RH→ice speed(7.5)→rain(7.9)→wind speed(8.6)→ageP3:hsnow→T0m→T2m→hice→snowfall→RH→age(7.1)→wind speed(8.8)→rain(8.9)→ice speed(9.2)P4:hsnow→T0m→hice(3.2)→T2m(3.9)→RH(6.6)→age(6.8)→wind speed(7.4)→snowfall(7.6)→ice speed(8.2)→rain(8.4)NN:hsnow→T0m→hice(3.2)→T2m(3.8)→snowfall(5.6)→RH(7.0)→rain(7.8)→wind speed(8.0)→ice speed(8.2)→age(8.4)

In all model families, there is a consistent pattern in the ranking of the most informative features. All model types identify hsnow as the most informative predictor and T0m as the second most informative predictor. hsnow being the most informative predictor is plausible since snow is among the most reflective medium in natural surfaces, especially when fresh and dry. When present, snow represents the uppermost layer where solar radiation initially impacts. Snow has a low optical depth due to the scattering of incoming solar radiation in diffusive directions, implying that a snow layer of a few centimeters significantly increases surface albedo . Additionally, at the spatial scales of our dataset (25 km), snow depth is likely related to snow fractional coverage which is also impactful for albedo. T0m as the second most informative predictor is in agreement with the fact that T0m is a proxy of whether the surface is under melting or freezing conditions, as the presence of melting water reduces albedo. For instance, fresh snow exhibits a higher albedo compared to wet snow . Sea ice albedo parametrisations that do not explicitly resolve melt ponds include the radiative effect of melt ponds implicitly with T0m (e.g. PW79). Excluding the linear model, the top predictors after hsnow and T0m are hice and T2m. As sea ice has a higher optical depth than snow, hice ranked below hsnow seems plausible, implying that hsnow provides larger marginal predictive improvement than hice.

The inclusion of T2m among the most informative predictors is unexpected given the high correlation (0.92) between T0m and T2m (see Appendix ), which would suggest redundancy in T2m, ranked below T0m. Nevertheless, SFS quickly chooses T2m as an additional predictor after T0m is accounted for, indicating that α is not only dependent on surface conditions, but is also influenced by atmospheric conditions near the surface, affecting the optical properties of the sea ice surface. Additionally, this may be in part due to the fact that T2m can go above the melting point, whereas T0m cannot. Another consideration is that ERA5 does not assimilate sea ice or snow thickness, nor near-surface Arctic observations, except for surface pressure from stations and drifting buoys. Previous studies have shown that this leads to warm temperature biases in ERA5 over the Arctic, particularly during polar winter clear-sky events . Such biases could introduce inconsistencies between the satellite-derived T0m and the ERA5-biased T2m , which might partly contribute to the predictive skill attributed to T2m. The documented warm bias is particularly large during polar winter stable boundary conditions. Our exclusive use of polar-day samples thus helps mitigate the influence of this bias. However, this documented warm bias in ERA5 and data inconsistencies between surface and 2 m air temperatures may play some role in our results, although we believe it is unlikely to fully explain the relationship identified here. To the best of our knowledge, existing sea ice albedo parametrisations in ESMs with an implicit scheme of melt pond representation do not include T2m. Sea ice models that explicitly resolve melt ponds, e.g. and , use T2m to compute the surface melting rate to calculate the melt water accumulation in the ponds.

In implicit schemes, the transition of T0m around the freezing point of sea ice is used as information to implicitly determine melting and freezing conditions, which characterise the wetness of sea ice surface, altering sea ice optical properties. Examining in-situ measurements of T2m from the MOSAiC expedition and satellite swath data of melt pond fraction with a resolution of 1.2 km, have reported that T2m is one of the main driver of the formation and evolution of melt ponds, explaining short-lived changes in melt pond fractions and thus, decreasing albedo. Although they concluded that ERA5 reanalysis data are not well suited to study local melt pond characteristics due to the coarse spatial resolution, here we show that T2m of ERA5 is a valuable predictor to understand the large-scale mechanisms that modulate sea ice albedo in the pan-Arctic region.

The ranking of the remaining features shows some variability across model families, but some patterns can be identified. For instance, snowfall and RH tend to be ranked higher than wind speed, ice speed, and age in most model families. On a large scale, features related to thermodynamics are more relevant to describe sea ice albedo than features related to sea ice motion.

PySR selects the four best ranking features chosen by the SFS algorithm for the NN (see Sect. ), namely hsnow, T0m, hice, and T2m, and neglects snowfall. This results in the following physically consistent equation with the lowest MSE 4αhsnow,hice,T2m,T0m=tanh⁡2(p̃snowhsnow2+p̃icehice+ã)b̃-tanh⁡(p̃T2mT2m-p̃T0mT0m+c̃). Equation () contains seven coefficients for which the optimised values are as follows {p̃snow,p̃ice,p̃T2m,p̃T0m,ã,b̃,c̃}=63.131m2,0.111m,0.141°C,0.301°C,0.84,2.19,0.95, where p̃x denotes the weight of a feature x. The following Sections highlight the main physical findings discovered by Eq. () and which role the coefficients play.

Figure  presents the five best-performing equations in terms of MSE discovered by PySR (see Appendix  for the equations ranked second to fifth with the respective PySR configurations) and baseline models, including polynomials and NNs, on an error-complexity plane (see Sect. ). Optimising PW79 using the Nelder–Mead method reduces the MSE from 0.08–0.03. Despite this improvement, all models outperform the tuned PW79.

Notably, increasing the polynomial degree from one to two yields a significant reduction in MSE of approximately 0.005. However, further increasing the degree to three or four does not result in substantial performance gains, indicating that model complexity beyond this point does not lead to significant improvements. Moreover, increasing feature dimensionality leads to a convergence of model performance within each model family, typically after adding the fourth or fifth feature. This suggests that the first four or five features chosen by the SFS algorithm represent the key features that govern albedo, while the remaining features are redundant, contributing less marginal information or introducing noise.

The full-set NN exhibits slight overfitting (MSE=0.0125), since it is less skilful than the 7-feature NN (ΔMSE=0.0002). Thus, we find that sparsity can help the NN to generalise. Interestingly, polynomials and NNs show similar performance, with polynomials requiring additional features to match the accuracy of NNs. For instance, comparable performance is observed in models like 1-feature NN and 3-feature P1, and 2-feature NN and 4-feature P2. The comparable model performances suggest that simpler polynomial models are sufficient to capture the underlying patterns between the features and albedo, and are as effective as NNs, which may be overly complex for this problem. Furthermore, the need for additional features in polynomials may be beneficial, as it can help to compensate structural uncertainty in the parametrisation.

Figure a compares the sea ice albedo distributions during the validation period between the reference observation and model predictions, all illustrated within the physical range between 0 and 1. The model predictions are: Eq. (), 4-feature P3 (Eq.  in Appendix ), and 4-feature NN to compare models with the same number of features. The Hellinger distance measures the similarity between two discrete univariate probability distributions P and Q 10H(P,Q)=def12||P-Q||2.

The three model predictions exhibit a bimodal distribution similar to the reference observation (with peaks at 0.46 and 0.84). Among the models, 4-feature P3 shows the greatest similarity to the reference observation, with a Hellinger distance of 0.218 and MSE of 0.0145. This is followed by the 4-feature NN, with a Hellinger distance of 0.294 and MSE of 0.0133, and Eq. (), with Hellinger distance of 0.356 and MSE of 0.0156. However, some of the predicted sea ice albedo values from 4-feature P3 fall outside the physical range of 0 and 1, as illustrated in Fig. b, violating the first PC. Additionally, none of the models fully capture the long tail of the reference observation towards higher albedo values. Instead, both 4-feature NN and Eq. () demonstrate a notable peak at higher albedo values (0.82 and 0.83, respectively), with Eq. () having an upper limit for sea ice albedo at 0.83. The reference observation shows a peak of 0.84. Hence, the models exhibit a slight shift to the left. At the lower end of the albedo scale, the 4-feature NN best captures the long tail, although all model peaks at lower albedo values are more shifted to the left compared to the reference observation with a peak at 0.46: 0.42 for the 4-feature P3, 0.39 for the 4-feature NN, and 0.42 for Eq. ().

During training, we do not account for uncertainties associated with various satellite products. For the VIIRS product, the overall uncertainty for albedo retrieval is 0.1, and for surface temperature, it is 1.98 K, based on RMSE comparisons with in-situ measurements from the SHEBA campaign . assessed the albedo of eight individual sea ice surface types of sea ice based on field measurements from the MOSAiC expedition, finding that early autumn snow exhibits the highest albedo values between 0.8 and 0.9, while dark ponds have the lowest albedo values between 0.12 and 0.25. Thus, we conclude that the long tails of the albedo in the reference observation, values below 0.12 and above 0.9, are likely due to measurement, data processing, and retrieval errors. Moreover, our dataset has a spatial resolution of 25 km, which covers a variety of sea ice surface types, providing spatially averaged albedo values. In contrast, reports highly localised albedo values for each surface type. Furthermore, as examined in Sect. , the lower and upper limits of Eq. () are determined by the coefficients ã=0.84 and b̃=2.19, which have been optimised using the pan-Arctic dataset. With these coefficient values, Eq. () is unable to reproduce extreme albedo values. On the basis of these considerations, we infer that the upper albedo limits prescribed by Eq. () is physically plausible, given that the reference observation is noisy and reflects average albedo over a large area.

Figure  illustrates the sea ice albedo exemplarily for 23 May 2020, with the reference observation (Fig. a), computed with the tuned PW79 (Fig. b), and Eq. () (Fig. c). Figure d and e depict deviations from the observed albedo. The tuned PW79 demonstrates two areas distinguishing between high and low albedo zones due to its constant albedo values representing surface types, namely snow-covered ice (α=0.66), and melting snow (α=0.40). As PW79 is not a smooth function, PW79 causes a sharp border between the two surface types. Conversely, the spatial variability of the observed albedo is better captured with Eq. (), reducing the MSE by about a half (0.0156) compared to the tuned PW79 (0.0300). Biases remain in the Hudson Bay and along the sea ice edges, but are much reduced in the Central Arctic.

Figure  presents the seasonal albedo cycle of the reference observation, Eq. (), and tuned and untuned PW79 for the period March–September. The data are averaged acrossed the years 2013–2020 to highlight the typical seasonal pattern and reduce interannual variability. For completeness, the seasonal cycles averaged seperately over the training period (2013–2018) and validation period (2019–2020) are provided in Appendix , where they exhibit similar behaviour, indicating that the following analysis is robust across both training and validation periods.

The untuned PW79, unlike the other parametrisations, maintains a constant albedo at about 0.81, showing no seasonality. In contrast, the other three datasets display strong seasonality, with maximum sea ice albedo occurring during the winter period between March and April, followed by a steady decrease during the melting period between May and July, where it reaches its minimum. Equation () demonstrates a strong agreement with the reference observation, achieving an R2 score of 0.94, whereas both untuned and tuned PW79 reach 0.57. During the winter period, the tuned PW79 starts with much lower albedo values around 0.66 compared to the reference observation and Eq. (), which display sea ice albedo of similar magnitude of around 0.77. While the tuned PW79 reaches its minimum in July at 0.42, Eq. () reveals its minimum at 0.39, which is closer to the reference observation with 0.27. During the freeze-up period in August and September, the tuned PW79 demonstrates a rapid increase in sea ice albedo, similar in magnitude to the winter period. In contrast, the reference observation and Eq. () depict a more gradual increase, with sea ice albedo not recovering as rapidly to winter magnitude in August.

While Eq. () remains gradually increasing in September, the reference observation shows a decrease of sea ice albedo, which contradicts the expected freeze-up behaviour of Arctic sea ice . showed that the quality and accuracy of the VIIRS albedo product decrease with increasing solar zenith angle in September. Despite being trained on the reference observation, Eq. () provides a more physically reasonable prediction for September, likely due to the sparse observational data available for that month, as reduced sunlight over the pole limits September data availability (see Appendix ), resulting in less weight being given to these observations during training. Therefore, PySR relies more heavily on the complete data from March–August (see Fig. ). As a result, PySR implicitly captures seasonal patterns, particularly temperature-driven trends (see Sect. ), which extend naturally into September. In doing so, Eq. () effectively corrects for potential measurement artefacts in the September data by leveraging the functional relationships between ΔT* and α learnt from better sampled months.

Both tuned and untuned PW79 exhibit very low standard deviations during the winter period and September, with higher values around 0.13 during the melting season for the tuned PW79. Conversely, the reference observation shows high standard deviations with a maximum of 0.19, which are attributed to the spatial variability of sea ice albedo and measurement, data processing, and retrieval errors as already discussed in Sect. . Equation () reveals lower standard deviations with a maximum of 0.16, potentially eliminating errors, and are attributed solely to the spatial variability of the albedo. The low standard deviations in the tuned and untuned PW79 stem from its simplistic nature, relying on constant albedo values based on snow cover and surface temperature, where each constant represents a sea ice surface type. This results in PW79 perceiving the sea ice as highly uniform during the winter and September, whereas the tuned PW79 captures more variability during the melting season.

Overall, Eq. () presents a clear improvement over PW79, aligning with observed sea ice albedo variations by capturing both the seasonal progression and its magnitude. The untuned PW79 does not capture the observed albedo seasonality, maintaining a high value of 0.81 year-round. It should be noted that the sea ice albedo is calculated for each of these methods with observed melting conditions, which could differ from the conditions in FESIM.

Although the tuned PW79 better captures sea ice albedo seasonality, it significantly deviates in magnitude, inaccurately reflecting albedo changes and showing an earlier, quicker freeze-up than the reference observation. Both the reference observation and Eq. () align with previous field campaigns from SHEBA and MOSAiC , identifying five phases of Arctic sea ice: dry snow (March–Aril), melting snow (May), pond formation (June), pond development (July), and freeze-up (August–September). It is noticeable that these studies were based on highly localised measurements, whereas this study investigates spatially averaged data over a 25 km resolution.

Figure e reveals regional differences in model performance for Eq. (), suggesting that the global optimisation approach on the entire training set is not able to capture the underlying patterns uniformly across all regions. This finding motivates us to explore spatial and temporal variations in model performance by conducting optimisations on regional and monthly subsets. As Eq. () provides physically meaningful coefficients, as demonstrated in Sect. , we are able to gain insights into the underlying physical mechanisms governing model performance. In contrast, optimising NNs on subsets would not offer the same level of interpretability due to their inherent black-box nature. To ensure consistency across all subset optimisations, we divide our training set into monthly and regional subsets, utilising 20 000 data samples for each region and 105 data points for each month.

Both monthly (MSE=0.0122) and regional (MSE=0.0117) optimisation strategies outperform global optimisation (MSE=0.0156) in terms of reducing overall MSE (Fig. a–c) albeit making the coefficients depend on the region or month greatly increases the complexity of Eq. (). This improvement is likely due to the ability of Eq. () to capture regional and monthly variations in the data. The regional optimisation approach leads to significant reductions in MSE for certain regions, such as the Barents Sea (from 0.0460–0.0220), Kara Sea (from 0.0409–0.0263), and East Greenland Sea (from 0.0335–0.0148) (Fig. e). However, these regions, which border the North Atlantic, continue to exhibit high MSEs across all optimisation strategies, suggesting that they may be influenced by physical processes not well-represented by Eq. (), such as Atlantic Oceanic heat transport or strong winds prevailing in these regions . Another potential reason is that the underlying physics operate on a time scale smaller than our data, which are on a daily basis due to the temporal resolution of the satellite data.

In terms of the magnitude of improvement, regional optimisation yields higher proportional improvements compared to monthly optimisation (Fig. d and e). The greatest proportional improvements in the reduction of MSE are observed for the Chuckshi Sea with regional optimisation (75 %), September with monthly optimisation (69 %), and the Beaufort Sea (59 %). However, the Central Arctic shows little improvement with regional optimisation, likely due to its dominant representation in the dataset (64 % of the entire dataset). As a result, the global optimisation is already greatly influenced by the Central Arctic data, and the optimal coefficients are likely biased towards this region, leaving little room for improvement with regional optimisation.

One limitation of the regional optimisation approach is that it produces sharp borders in the error map (Fig. c), reflecting the regional focus of the optimisation process, which leads to a lack of a smooth error transitions between regions. Additionally, some instances of overfitting are observed, where regional or monthly optimisation results in high MSE values compared to global optimisation. For example, the MSE of July with monthly optimisation (0.0088) is higher than with global optimisation (0.0080), and similar patterns are seen for Hudson Bay with regional optimisation (0.0138 vs. 0.00116), and Laptev Sea with regional optimisation (0.0185 vs. 0.0172).

The optimised coefficients for each region and month resulting in the analysis in Sect.  are displayed in Appendix  and Appendix , respectively. Physical interpretation of each region and month goes beyond the scope of this study. Instead, we focus on the Barents Sea as a case study. This region exhibits the highest MSE from global optimisation and significant improvement with regional optimisation. For a direct comparison, the coefficients are standardised to their unitless form (Table ).

In the Barents Sea, the effect of hsnow becomes significantly smaller (p̃snow, std=0.03) compared to the pan-Arctic region (p̃snow=0.85), while hice becomes more important (p̃ice,std=0.64) than in the pan-Arctic region (p̃ice,std=0.09). These differences reflect the distinct physical conditions in these regions. The Barents Sea experiences high seasonality, with thin sea ice prevalent and little to no snow present, compared to the whole pan-Arctic region . Consequently, variations in thin sea ice play a more significant role in α in the Barents Sea, whereas variations in thin snow influence α in the pan-Arctic region.

In both cases, T0m has the highest weight (of around 2). However, T2m is less significant in the Barents Sea (p̃T2m,std=0.34) than in the pan-Arctic region (p̃T2m,std=0.99). In the pan-Arctic region (c̃=0.95), smaller ΔT* are required to trigger melting, while in the Barents Sea, the lower value of c̃=-0.04 triggers melting conditions already at higher ΔT*. The shift of the transition to higher ΔT* implies other heat sources affecting sea ice optical properties which are not considered in Eq. (), as already discussed in .

Our findings indicate that the pan-Arctic region represents a stable ice regime, in which snow and small ΔT* modulate α, while the Barents Sea represents a fragile ice regime, where ice properties and temperatures already at higher ΔT* regulate α sensitivity. The Barents Sea is one of the most rapidly changing regions, becoming ice-free in summer and contributing to approximately one-quarter of the Arctic sea ice loss in winter. This change is associated with surface warming in the Gulf Stream and the increase of the Atlantic oceanic heat transport passing the Barents Sea Opening .

This case study gives a first insight on how Eq. () can be transferred to different ice regimes, specifically from a stable, pan-Arctic regime dominated by multiyear ice in the Central Arctic to a fragile ice regime characteristic of the Barents Sea. Although we acknowledge that the MSE in the Barents Sea remains relatively high compared to other regions after fine-tuning (see Sect. ), the functional form of Eq. () remains physically reasonable. This enables a comparison between the coefficients obtained from global and regional optimisations demonstrating that the optimal coefficients are state-dependent.