## 2 Material and Methods
### Angular Distribution Models  
The general strategy to estimate instantaneous fluxes \(F\) from observed radiances \(I_{0}\) is to empirically learn--per scene type and per illumination geometry \(\theta_{s}\)--the directional intensity of upwelling radiances with respect to the prevalent upwelling flux. Therefore, the upward hemisphere (resolved by \(\theta_{v}\) and \(\phi\)) is discretized. Observations are sorted into angular bins. Once all bins of the upward hemisphere are filled, fluxes (\(\hat{F}\)) are estimated through the hemispheric integral of mean radiances \(\hat{I}\) from each bin. The last step is to infer on anisotropy \(R\) (Equation (4)). \[R(\theta_{s},\theta_{v},\phi)=\frac{\pi\hat{I}(\theta_{s},\theta_{v},\phi)}{F( \theta_{s})}=\frac{\pi\hat{I}(\theta_{s},\theta_{v},\phi)}{\int_{0}^{2\pi} \int_{0}^{\pi/2}\hat{I}(\theta_{s},\theta_{v},\phi)d\theta_{v}d\phi} \tag{4}\]  
A new radiance observation \(I_{o}\) can be converted into an instantaneous flux \(F\), as shown in Equation (5). \[F(\theta_{s})=\frac{\pi I_{o}(\theta_{s},\theta_{v},\phi)}{R(\theta_{s}, \theta_{v},\phi)} \tag{5}\]  
In CERES SSF Edition 4, liquid clouds over ocean were treated as a single scene type. To consider stark intensity changes of upwelling radiances with cloud fraction and cloud optical depth, collected radiances of each angular bin (resolved by \(2^{\circ}\) in \(\theta_{v}\) and \(\phi\)) served to produce a sigmoidal fit linking \(\hat{I}\) and \(\ln(f\hat{\tau})\). ADMs were therefore based on hemispheric integrals of sigmoidal fits. In effect, resulting anisotropies are a function of \(\ln(f\hat{\tau})\). In this study, \(f\approx 100\%\) and \(\hat{\tau}\approx 10\). Essentially, CERES TOA SW anisotropy--per angular bin and per discrete illumination geometry (resolved at \(2^{\circ}\))--remained constant. We produced new ADMs (hereafter referred to as \"refined ADMs\") by incorporating intensity changes of upwelling instantaneous reflectance \(\rho_{o}\) with cloud-top effective radius \(R_{e}\) and cloud-topped moisture \(CTWV\). The use of MODIS-retrieved \(R_{e}\) based on \(3.7\,\mathrm{\SIUnitSymbolMicro m}\) radiances (instead of \(1.6\,\mathrm{\SIUnitSymbolMicro m}\) or \(2.1\,\mathrm{\SIUnitSymbolMicro m}\)) ensured lower and less systematic uncertainties [28; 29]. We produced a linear model per angular bin and discrete illumination geometry of the following form (with least-square estimates \(a\), \(b\), and \(c\)):  
\[\ln\rho_{o}(\theta_{s},\theta_{v},\phi)=a+b\cdot\ln R_{e}+c\cdot CTWV \tag{6}\]  
An example is shown in Figure 4. Naturally, the spread in \(\hat{\tau}\) added variability in \(\rho_{o}\). We assumed that the linear fit was unaffected. This assumption required that \(\hat{\tau}\) was uncorrelated to \(\ln R_{e}\) and \(CTWV\), and we found no indication of such correlation. Incorporating all available data points could bias the least square estimate towards the bulk of the data and leave extremes (such as particularly small \(R_{e}\) and low \(CTWV\), and vice versa, as seen in Figure 4) unaccounted for. Angular bins containing few observations hindered strategies involving subsampling or computation of representatives such that the parameter space spanned by \(R_{e}\), \(CTWV\), and \(\rho_{o}\) was covered homogeneously. Similar to the CERES approach, we integrated TOA albedos as a function of linear models and inferred on TOA SW anisotropies. We produced a look-up table by applying linear models to \(R_{e}\) between 5 and 25 um (by steps of 1 um) as well as \(CTWV\) between 0 and 40 kg m\({}^{-2}\) (by increments of 2 kg m\({}^{-2}\)). In the case of insufficient CERES observations within a bin (less than 10 or samples with a spread in \(R_{e}\) of smaller than 10 um), we added reflectances from broadband radiative transfer simulations (introduced in Section 2.1) to complement CERES observations. This was generally necessary for \(\theta_{v}\) beyond 70\({}^{\circ}\) and occasionally for some bins at particularly low or high \(\theta_{s}\) (as listed in Table 2).