## 3 Results  
Empirical ADMs allow us to estimate instantaneous TOA fluxes from satellite broadband radiometry. State-of-the-art shortwave ADMs (referred to as \"CERES SSF 4\" ADMs [(15)] over cloudy scenes are sensitive to cloud fraction \(f\), cloud optical depth \(\tilde{\tau}\), and cloud phase. In this study, we examined whether ADMs should also be sensitive to the cloud micro-physical structure and amount of absorbing atmospheric gas above the cloud layer. We generated alternative ADMs (referred to as \"refined ADMs\", Section 2.3), using CERES SSF 4 data (Section 2.1, years 2000-2005), including CERES-observed TOA SW reflectances as well as collocated MODIS-retrieved cloud-top effective radius and estimated cloud-topped water vapor (Section 2.2). As shown in Section 2.1, we restricted ourselves to fully overcast, horizontally homogeneous, low-level, and single-level clouds of optical thickness 10 and added simulated reflectances to angular bins in case of insufficient observations. moisture (1.3-6.4%). The uncertainty in anisotropy (based on reflectance residuals of linear models) was of similar order of magnitude (3.2-5.3%) for variations due to effective radii and variations due to cloud-topped moisture. By reproducing the CERES approach (exemplarily shown in Figure 4 as black dot) and extracting reflectance residuals, we approximated a corresponding CERES uncertainty. Compared to CERES ADMs, uncertainties of refined ADMs were smaller by a factor 1.2-1.7. Refined and CERES ADMs should produce different flux estimates, especially for departures from median conditions (e.g., small and large droplet size distributions). To quantify flux deviations, we took mean reflectances of refined models for three \(R_{e}\), three \(\theta_{s}\), and a fixed cloud-topped water vapor, and applied both refined and CERES SSF 4 ADMs. A radiance \(\hat{I}\) was produced from a reflectance through \(\hat{I}=\beta\)\(\cos\theta_{s}\), where \(S\) is the solar constant. Figure 5: TOA SW anisotropy of refined ADMs at various cloud-top effective radii (top, at a \(\theta_{s}\) of 29\({}^{\circ}\) and a steady cloud-topped water vapor of 4 kg m\({}^{-2}\)) and levels of cloud-topped water vapor (bottom, at a \(\theta_{s}\) of 32\({}^{\circ}\) and a steady cloud-top effective radius of 10 μm). Colors indicate respective scenarios. Black dots mark TOA SW anisotropy of CERES SSF 4. Error bars were based on reflectance residuals of linear models propagated into TOA albedo and TOA anisotropy. To obtain corresponding uncertainties for CERES (grey shade), we mimicked the CERES approach and obtained reflectance residuals. Grey dashed lines indicate the cloud glory position. \[\Delta F(\theta_{s})=\pi\hat{I}_{\text{refined}}(\theta_{s},\theta_{v},\phi,R_{e}, CTWV)\left(\frac{1}{R_{\text{refined}}(\theta_{s},\theta_{v},\phi,R_{e},CTWV)}- \frac{1}{R_{\text{CERES}}(\theta_{s},\theta_{v},\phi)}\right) \tag{7}\]  
Figure 6 shows flux deviations between ADMs (computed according to Equation (7)) of up to 20 W m\({}^{-2}\). For \(R_{e}\) of 5 and 20 um, these deviations were located in the forward scattering direction (\(\phi\) within 0-45\({}^{\circ}\)) and again in the direct backscatter (cloud bow and glory). For an \(R_{e}\) of 10 um, differences were mostly within 10 W m\({}^{-2}\) and showed a general positive bias. We suspect that employed simulations (predominantly at \(\theta_{v}\) of 70\({}^{\circ}\) and higher), which were free of horizontal heterogeneity or 3D effects, produced rather higher reflectances compared to observations. In effect, albedo was slightly overestimated and anisotropies at \(\theta_{v}\) lower than 70\({}^{\circ}\) were underestimated (by about \(-0.98\%\) at an \(\theta_{s}\) of 29\({}^{\circ}\) for \(\theta_{v}\) within 0-70\({}^{\circ}\), cloud-top \(R_{e}\) of 10 um, and \(CTWV\) of 4 kg m\({}^{-2}\)). To facilitate a fairer comparison and to avoid further analysis involving an anisotropy bias, we reproduced CERES SSF estimates (referred to as \"CERES-like\" ADMs).