Cloud property retrievals were obtained from the Collection 6.1 MODIS Level-3 (L3) Atmosphere Daily Global Product (MOD08_D3 and MYD08_D3 for Terra and Aqua, respectively; Platnick et al., 2017a, b). The Level-3 MODIS Atmosphere Daily Global Product is a daily global spatial aggregation of the parameters generated from the Level-2 MODIS Atmosphere Products: Aerosol (MOD04_L2, MYD04_L2), Water Vapor (MOD05_L2, MYD05_L2), Cloud (MOD06_L2, MYD06_L2), and Atmosphere Profile (MOD07_L2, MYD07_L2). The L3 statistics are summarized over a 1° equal-angle latitude-longitude grid. The MODIS L3 parameters primarily considered in this study are CF, defined here as the MODIS data field: Cloud Retrieval Fraction Liquid – Liquid Water Clouds from Successful Optical Properties Retrievals, and LWP. Ice cloud fraction (ICF) was additionally used for filtering scenes with overlying ice cloud.

Daily gridded Nd estimates from MODIS were obtained from Gryspeerdt et al. (2022a). Nd retrievals in this dataset are based on the Level-2 Collection-6 MODIS Cloud Product (MOD06_L2, MYD06_L2) and follow several sampling strategies. This work considers the strategy named G18; proposed by Grosvenor et al. (2018), it balances data quantity with accuracy, accounting for several known biases in the retrieval (Gryspeerdt et al., 2022a).

Albedo retrievals were derived from the CERES Daily Time-Interpolated TOA/Surface Fluxes, Clouds, and Aerosols (SSF1deg-Day) product (NASA/LARC/SD/ASDC, 2015b, a) and calculated as the ratio of reflected shortwave radiation to incident solar radiation: 1αCERES=FSWTOAFsolarTOA where αCERES represents the observed CERES albedo, FSWTOA is the observed TOA shortwave flux, and FsolarTOA is the observed TOA solar insolation flux. CERES and MODIS retrievals were paired by satellite platform (Aqua–Aqua and Terra–Terra) and analysed jointly.

Finally, for investigating albedo–cloud sensitivity in stratocumulus region, the ECMWF ERA5 reanalysis (temperature at 700 hPa) was used in order to calculate the estimated inversion strength (EIS), with the formula: 2EIS=LTS-Γm850z700-LCL where LTS is the lower-tropospheric stability, Γm850 is the moist adiabat at 850 hPa, z700 is the height of p = 700 hPa surface, and LCL is the lifting condensation level. EIS is known to be a predictor of stratus cloud amount (Wood and Bretherton, 2006), and this study assesses the relationship between EIS and differences in albedo estimates to help explain stratocumulus-to-shallow-cumulus transition.

In order to reduce the overall impact of surface albedo variations, the study area was geographically limited to ocean and latitudes between 60° S and 60° N. Only cases with a maximum solar zenith angle (SZA_max⁡) not exceeding 65° were considered, following the threshold used in the G18 sampling strategy for estimating Nd. All data were filtered through cloud fraction of ice clouds (ICF < 0.01) to minimise their contribution to the scene albedo, which restricted the analysis primarily to low-altitude clouds (with cloud top pressure usually above ∼ 680 hPa; not shown). The resulting subset of cases considered in this study is pictured in Fig. 1.

The main method of the study is a 3D joint histogram (kernel) constructed from global observational data, where mean albedo is calculated for discrete bins of CF, LWP, and Nd. Once constructed, this kernel functions as a reference distribution that maps combinations of these three cloud properties to expected albedo values, without requiring additional variables beyond CF, LWP, and Nd. This differs from radiative kernels such as that in Duran et al. (2025), which diagnose the TOA shortwave radiative response rather than albedo. Our approach is purely diagnostic and conceptually closer to the cloud-feedback kernels of Zelinka et al. (2012), as it is derived from observational relationships without requiring model perturbations or radiative transfer calculations.

The kernel was constructed as follows. The daily gridded 1 × 1° data were binned into discrete intervals based on CF, LWP, and Nd. Each dimension was divided into a predefined set of ranges, varying from 0 to 1 (linear scale) with 50 bins for CF, from 1 to 1000 g m⁻² (logarithmic scale) with 40 bins for LWP, and from 1 to 300 cm⁻³ (logarithmic scale) with 30 bins for Nd. For each bin, the average albedo (αavg) was then calculated as a multi-year mean value of all 1 × 1° daily datapoints across the globe that fall into the same bin of CF, LWP, and Nd.

For each daily gridded MODIS and CERES observation, for each 1 × 1° gridbox, the difference between the average albedo in the given CF–LWP–Nd bin and the CERES albedo at the given location and time, expressed as Δα, was calculated as: 3Δα=αavg-αCERES The relative difference in estimated albedo was calculated as: 4Δαrel=αavg-αCERESαavg⋅100

The percentage of correct albedo estimates depends on the adopted accuracy threshold, which in the present analysis is expressed both in terms of the absolute (Fig. 2a) and the relative (Fig. 2b) value of |Δα|. The results show that when the accuracy threshold is set to |Δα| = 0.05, the reconstruction of albedo is correct in more than 80 % of the cases. However, if the stricter threshold of 0.02 is applied, this fraction decreases to 40.91 %. Although a deviation of 0.02 (corresponding to 2 percentage points) may be considered relatively small, the analysis presented in Fig. 2b demonstrates that such a change in absolute terms translates into approximately 10 % in relative terms.

The biases in albedo estimates are not distributed symmetrically around zero. On the contrary, their distribution shows a distinct left-skewed asymmetry (Fig. 3). The majority of discrepancies are concentrated around Δα ≈ 0.02, which corresponds to about 10 % relative change. At the same time, cases of underestimate exceeding 50 % relative change are clearly observed, whereas such extreme overestimates are practically absent.

The spatial distribution of the percentage of underestimated and overestimated albedo values, based on the threshold of 0.02, is illustrated in Fig. 4a–b. Both maps reveal well-marked zonal structures. Underestimates of Δα < -0.02 are particularly frequent at latitudes poleward of ∼ 40° in both hemispheres, whereas they occur relatively rarely in tropical regions. The opposite tendency is observed for overestimates (Δα > 0.02). A clear underestimate of albedo is visible over the regions dominated by marine stratocumulus clouds, with the most distinct example on the west coast of South America highlighted by a red rectangle in all panels of Fig. 4. Underestimates are also apparent in mid-latitudes, within regions influenced by the storm tracks of extratropical cyclones. In contrast, overestimates are most commonly observed along the Intertropical Convergence Zone (ITCZ) over the Pacific and Atlantic oceans, and to some extent over the Indian Ocean. Figure 4c presents the geographical distribution of absolute albedo bias (cases with either underestimated or overestimated albedo by more than 0.02). Generally, albedo biases are mostly evenly distributed spatially, regardless of the number of datapoints at a given location (Fig. 1). In addition to the features of a probable meteorological background, the maps show the presence of some artifacts, which may be related either to the geometry of the observations, such as the solar zenith angle or to the specific features of the retrieval algorithms.

The zonal patterns shown in Fig. 4 become even more apparent when cases of under- and overestimates are separated into individual 5° latitude bands (Fig. 5). On average, the global percentage of underestimates over the analysed multi-year period amounted to 27.7 %. This value varies significantly with latitude: from less than 10 % around the equator to more than 50 % at latitudes higher than 50° in both hemispheres. Similarly, overestimates occurred in 32.1 % of cases globally; although the latitudinal variability was slightly less pronounced than in the case of underestimates, the values still ranged from 15 %–20 % at high latitudes to around 50 % in the equatorial zone.

The biases observed over marine stratocumulus regions were examined in greater detail in the context of the transition from thick stratocumulus decks close to the South American coastline towards shallow cumulus clouds occurring further westward over the ocean (the region marked in red in Fig. 4). To investigate this, the relationship between EIS and Δα was analysed for the test period 2007–2019 using Aqua satellite data. Figure 6a presents the joint EIS–Δα histogram, normalised by each EIS column. The results show a clear dispersion of Δα values along a characteristic curve: negative values dominate for EIS below 0 K, positive values are typical for EIS between approximately 0 and 15 K, and negative values appear again for EIS above 15 K. EIS values exceeding 15 K most likely then correspond to cases of thick stratocumulus, which in Fig. 4a are observed by CERES to be brighter than calculated from the CF–LWP–Nd values using the kernel approach; clouds with lower EIS are instead associated with shallow cumulus, which often had an overestimated albedo (Fig. 4b). EIS values below 0 K might represent either situations with very limited cloudiness or cases of convective clouds, that is, conditions without a well-defined temperature inversion.

Figure 6b shows the joint histogram of SZA_max⁡ and Δα, normalised by the explanatory variable in each case. The dependence between these two quantities is particularly distinct and appears to explain, to a large extent, the distribution of albedo biases presented in Fig. 3. Numerous cases of overestimates around Δα ≈ 0.02 are typical for SZA_max⁡ below around 35°, whereas above ∼ 40° a clear relationship between SZA_max⁡ and Δα exists, with Δα decreasing significantly as SZA_max⁡ increases. For solar zenith angles above 50°, differences in albedo estimates reach values as high as 0.05–0.07. This explains the significant number of strong underestimates also visible in Fig. 3, and might also account for the underestimates observed at high latitudes in Fig. 4a.

There is a clear relationship with the geometry of observations, as well as expected differences resulting from the variable properties of clouds. These deviations are not fully captured by the mean cloud properties. In this part of the study several possible methodological adjustments were tested with an aim of improving the accuracy of albedo estimates (Table 1).

As a first step, modifications related to cloud fraction were considered. Since it can be expected that for low cloud fraction the varying brightness of the underlying ocean surface (resulting e.g. from the presence of atmospheric aerosol, waves, or the angle of solar rays) may have a larger influence on the albedo, only cases with CF greater than 0.95 were selected for analysis (methodological modification no. I). Methodological modification no. II involved performing an estimation using a much larger number of bins (1000) for CF, in order to verify that a smaller number of bins (50) was sufficient to capture the characteristic U-shaped distribution of CF (with very small or nearly complete cloud cover occurring most frequently, while intermediate values appear relatively rarely).

Two methodological adjustments related to the SZA_max⁡ were then tested. In the first case, all pixels with SZA_max⁡ ≥ 40° were excluded in order to eliminate situations where the relationship between SZA_max⁡ and Δα becomes approximately linear (modification no. III). The second correction consisted of determining the mean Δα value within each 1° SZA_max⁡ bin, and subsequently applying this correction to every estimated albedo case (modification no. IV).

The next group of modifications was aimed at testing possible corrections related to specific local conditions not captured by a global-mean analysis. These included: calculating αavg separately for each 5° latitude band (modification no. V), calculating αavg separately for each 5° latitude–longitude grid cell (modification no. VI), and calculating αavg separately for each 1° latitude–longitude grid cell (modification no. VIIa). Additionally, a combined adjustment of both filtering out the SZA_max⁡ ≥ 40° pixels and computing αavg separately for 1° grid cell (modification no. VIII) was tested.

In all of the above modifications, the kernel was constructed using CF, Nd, and LWP values from the full annual time series. The last two modifications were related to the temporal aggregation used in the kernel construction. In modification IX, the kernel is instead constructed from monthly-averaged CF, Nd, and LWP, while in modification X it is constructed from daily-averaged values of CF, Nd, and LWP. When evaluating the difference between estimated and observed albedo, the reconstructed albedo is compared against observations corresponding to the same month (modification IX) or the same day (modification X), respectively. Table 1 shows the ratio of under- and overestimated cases of albedo across the globe for each tested methodological modification.

Computing αavg within 5° latitude bands yielded only a modest reduction in errors, and the total share of incorrect estimates decreased only from 59.1 % to 50.0 %. At the 1° grid (pixel) level, the error rate dropped nearly sixfold, to 11.2 %. Since the kernel was built for the ocean, the underlying surface albedo is most likely not affecting this improvement, and the other sources of variability that were not considered by the CF–LWP–Nd–α function seem to impact the estimates – possibly other cloud properties and regimes, such as cloud morphology, that are more location-dependent. Both modifications accounting for the SZA_max⁡ – the exclusion from the analysis of pixels with SZA_max⁡ ≥ 40° (modification III), as well as implementing a correction of mean Δα within each SZA_max⁡ interval (modification IV) – resulted in the improvement that was greater for overestimates than for underestimates. Figure 7a–b shows the histogram of Δα after applying modification no. IV. Many of the corrected biases correspond to overestimates previously identified in the ITCZ region (not shown).

Restricting the analysis to cases with CF > 0.95 did not improve the estimates; instead, it substantially increased the number of overestimates. Increasing the number of CF bins had a very limited effect on the results – the improvement was negligible, showing that 50 bins is already sufficient to represent the non-linearity in the CF-albedo relationship.

The most effective improvements in the estimates were achieved when αavg was calculated separately for individual grid cells (modification VIIa). However, the results of this modification were significantly affected by bin sparsity, which in many cases have only been filled once. As a result, the reconstructed albedo is effectively drawn from the same datapoint, that is later used in the comparison, meaning that the resulting bias estimates do not represent an independent test of a time-averaged kernel. In order to reduce strong undersampling of the kernel, for modification VIIa two alternative bin configurations were examined. While modification VIIa retained the original 50 × 40 × 30 bin configuration, modifications VIIb and VIIc used coarser bin structures (10 × 10 × 10 and 5x5x5, respectively) to improve bin population. As shown in Table 1, these coarser bin configurations reduced the sparsity problem but yielded only marginally better accuracy than the global kernel method, suggesting that simply widening bins does not fully resolve the trade-off between spatial resolution and kernel robustness.

Modification VIII – a combination of the two methods, IV and VIIa – produced the most accurate results: the overall error percentage decreased to 5.4 %, i.e. more than tenfold compared to the original value; moreover, the share of under- and overestimates became nearly equal (Fig. 7c–d).

When applying modifications IX and X, an improvement in the method is observed, with the ratio of correct albedo reconstructions increasing from 40.9 % for no temporal aggregation (original method without modifications) to 57.7 % when the kernel is constructed from the monthly-averaged CF, Nd, and LWP and to 60.6 % when it is constructed from the daily-averaged CF, Nd, and LWP (Table 1). However, it should be noted that the daily-averaged version (modification X) may also be subject to bin undersampling, as the number of datapoints available on individual days is considerably smaller than over longer averaging periods. Therefore, the daily results should be interpreted with similar caution as the finest spatial resolution case. Generally, the tests demonstrated that among the investigated modifications, only those accounting for regional and seasonal variability of albedo provided a substantial improvement in the estimates.