## 3 Method
### Part 1: Retrieval Algorithm for Instantaneous TOA Albedo (Level-2)
#### 3.1.4 Angular Distribution Modeling  
Following the narrowband-to-broadband conversion, the Angular Distribution Models (ADMs) [35; 36] were used to determine the anisotropic factor (\(R\)) needed to convert directional reflectance to hemispherical albedo.  
The ADM scene type is a combination of _CERES surface type_ (Table 4) and cloud properties (cover, phase, optical thickness), and in the case of clear-sky water, also wind speed. The basic scene type selection was straightforward, by using look-up tables in which surface types, wind speed, and cloud parameters were classified in discretized bins, with each combination of bins leading to a single scene type. However, using discretized scene type criteria introduces a bias in the resulting anisotropic factor when the observed parameters deviate too much from their respective bin center values. Therefore, the wind speed bins (and hence the corresponding scene types) were weighted by considering the relative distance between the observed value and the lower and upper bin centers (linear interpolation). For cloud cover and cloud optical thickness this was done in a similar way, but here the interpolation was bilinear.  
Once the number of relevant scene types (\(n\)) and their respective relative weights (\(w_{j}\)) had been determined, for each scene type \(j\) the interpolated ADM radiance (\(\bar{I}_{j}\)) was derived by trilinearly interpolating viewing and illumination angular bins (\(\theta_{0},\theta,\phi\)), and the interpolated ADM flux (\(\bar{F}_{j}\)) was derived by linearly interpolating solar zenith angle bins (\(\theta_{0}\)). Then, using the scene type weights \(w_{j}\), the scene type weighted mean radiance (\(\bar{I}\)) and flux (\(\bar{F}\)) were calculated (Equation (4)), and from that the anisotropic factor (\(\bar{R}\)). The tilde (-) indicates quantities that had been determined from interpolated ADM values.  
\[\bar{R}=\frac{\pi\cdot\bar{I}}{\bar{F}}=\frac{\pi\cdot\sum\limits_{j=1}^{n} \left[w_{j}\cdot\bar{I}_{j}(\theta,\theta_{0},\phi)\right]}{\sum\limits_{j=1}^ {n}\left[w_{j}\cdot\bar{F}_{j}(\theta_{0})\right]}. \tag{4}\]  
The instantaneous shortwave TOA albedo \(\alpha_{SW}\) was then obtained by dividing the previously derived broadband reflectance (\(\rho_{SW}\)) by the scene type weighted mean anisotropic factor (\(\bar{R}\)). A potential source of bias is the linear interpolation between the angular bins (\(\theta_{0},\theta,\phi\)) in which the originally collected ADM radiances sometimes vary non-linearly [35]. This effect is minor for most of the angular bins. To remove it, an invariant but scene- and angular-dependent correction term (\(\delta\alpha_{SW}\)) was added to the instantaneous albedo:  
\[\alpha_{SW}=\frac{\rho_{SW}}{\bar{R}}+\delta\alpha_{SW}. \tag{5}\]  
The retrieval of input data (e.g., COT), needed for scene type identification (e.g., NTB, ADMs,...), requires a minimum amount of illumination. Therefore, the Level-2 albedo retrieval described in the previous Sections 3.1.1-3.1.4 was only performed for AVHRR pixels with a solar zenith angle lower than 84\({}^{\circ}\) (\"daytime pixels\").