## 4 Recent Advances That Support and Build Upon Existing Approaches
### Semi-Physical Approach  
As evident in the evolution of CERES ADMs (Section 3.3), further segregation of scene type, by using additional scene parameters, may allow further reduction in the uncertainty of instantaneous irradiance estimates. Semi-physical ADMs that were recently developed for clouds over ocean ([PERSON] _et al._, 2017) include two additional atmospheric parameters beyond the traditionally used cloud fraction, phase, and optical depth:  
(1) MODIS cloud-top effective radius, \(R_{e}\): as known from Mie theory, the angular distribution of light scattered from a water droplet or ice crystal is related to the particle size, and therefore the particle size may affect top-of-atmosphere anisotropy of cloudy scenes as a whole.  
(2) Reanalysis-based above-cloud water vapor: gaseous absorption increases with absorber mass (here vertically integrated water vapor mixing ratio between cloud-top and top-of-atmosphere) and atmospheric path length, affecting greater viewing zenith angles more strongly and, thus, impacting top-of-atmosphere anisotropy.  
For training the semi-physical ADMs, the approach uses the same data as CERES-Terra/Aqua ADMs (see Section 3.3.3) plus collocated profiles from ERA-20C, and only mildly increases statistical complexity (this approach has six free parameters instead of five required for each sigmoidal curve). This is possible by using a physical framework that is then statistically optimized for each angular bin. It incorporates (1) the functional form of two-stream cloud albedo that essentially produces a sigmoidal curve and includes the asymmetry parameter \(g\), which is then functionally linked to MODIS-based \(R_{e}\). (2) The ocean reflectance anticipated in cloud-free portions of a footprint is described by the Cox-Munk model ([PERSON] _et al._, 2017). An approximate footprint albedo [i.e., the area-weighted mean of up to two cloud layers using (1) and of cloud-free portions using (2)] is then multiplied by an exponential term to resemble water vapor attenuation. Figure 11a shows an example of the variation in shortwave anisotropic factors as a function of \(R_{e}\). Like CERES-Terra/Aqua ADMs, the semi-physical approach discriminates between footprints containing clouds that consist of liquid, ice, or both. In contrast to CERES-Terra/Aqua ADMs, this approach allows to proportion ice and liquid fractions when both phases are present. Compared against CERES-Terra/Aqua ADMs and indicative of added explanatory power, radiance residuals are reduced in most angular bins outside the sun-glint region and mostly for footprints consisting of liquid, and both liquid and ice clouds. On the other hand, the glint region as well as geometries that are sparsely sampled remain best captured using a fully statistical approach.  
When applying the semi-physical ADMs to liquid clouds observed by CERES-MODIS and GERB-SEVIRI over the course of two months, the resulting instantaneous fluxes differed by up to 25 W m\({}^{-2}\) relative to those reproduced by CERES-Terra/Aqua ADMs [74]. As shown for selected \(R_{e}\) in Figure 11b-d, the sharpest corresponding anisotropy differences (equivalent to peak relative differences of \(\pm\)5%) are non-monotonic and occur mainly in the backscattering direction associated with single scattering features such as cloud bow and cloud glory (curved features in the top half of the polar plots) as well as the forward scattering direction (predominantly multi-scattering; broad features in the bottom half of the polar plots). These differences propagated into the daily and monthly mean of instantaneous irradiances at the 1\({}^{\circ}\)\(\times\) 1\({}^{\circ}\) scale (up to \(\pm\)10 and \(\pm\)5 W m\({}^{-2}\), respectively), predominantly in regions of persistently extreme values of \(R_{e}\) (smaller than 7 \(\upmu\)m or larger than 15 \(\upmu\)m) or above-cloud water vapor (not shown).