## 5 Discussion  
The ZTT analytical model for remote sensing reflectance is based on the restatement of the RT equation by [PERSON] [27,28] with the following refinements: (1) Full bidirectionality was retained in the final model (no assumption of nadir viewing), (2) the assumption of asymptotic diffuse attenuation for upwelling radiances at the surface was analytically described, (3) the entire model was formulated in terms of \(b_{b}\) and \(a\) to enable inversion, (4) the VSF was explicitly included with \(\beta_{p}(\psi)\)/\(b_{bp}\) set to a constant shape defined by \(P_{bb,ST}(\psi)\)[26], which showed the strongest performance of any phase function, including measured phase functions in disparate, diverse data sets, (5) water Raman effects were added, and (6) for \(r_{rs}\), an analytical model for \(\overline{\mu}_{d}\) was developed that included dependency on atmospheric visibility. IOP inputs to the ZTT model are \(b_{bp}\), \(a_{pg}\), \(\beta_{p}(\psi)\)/\(b_{bp}\), and \(\widetilde{b}_{pp}\). Moreover, the ZTT model enables systematic assessment of the effect of IOPs on the BRDF for the first time. ZTT showed strong performance in match ups with field data sets spanning a wide range of water types. In terms of MAPE, the model with constant phase function shape \(P_{bb,ST}(\psi)\) performed as well as full RT computations. Stronger performance was observed for these available data sets relative to current state-of-the-art LUT-configured models based on the first order approximation between \(r_{rs}\) and \(b_{b}/a\) or \(b_{b}/(a+b_{b})\), i.e., M02 and L11. MAPE for the ZTT model was slightly better using \(P_{bb,ST}(\psi)\) (\(\%\delta_{abs}\) of 16%) when compared to using directly measured backward phase functions for the validation data set (where \({}^{\%}\!\delta_{abs}\) was 17%). It may be argued, under certain conditions, the broad average \(P_{bb,ST}(\psi)\) may be more accurate than an isolated measurement because specific bias errors in any single VSF measurement at an individual angle may be avoided. This of course assumes such biases may be larger than the natural variability in the shape of the backward phase function. Another explanation is error assessments for the validation data set were made over a relatively few number of stations, 23. To roughly estimate uncertainty in this error metric, single stations were sequentially left out of the error determination for ZTT using \(P_{bb,ST}(\psi)\), which gave a total range in \({}^{\%}\!\delta_{abs}\) of 14.8 to 16.8% with standard deviation of 0.6%. A \(\sim\)1% difference in error may thus not be particularly meaningful. Figure 6: As in Figure 4, but for the NASA NOMAD data set containing \(a\), \(b_{b}\), \(r_{rs}\), and [Chl]. MAPE \({}^{\%}\!\delta_{abs}\) were 20%, 23%, 25%, and 26% in (**A–D**), respectively. The ZTT model is fully analytical in the sense it is described entirely by equations and \"almost fully analytical\" with respect to the inclusion of empiricism. Empiricism comes from optimizing the \(f_{L}\) parameter to our synthetic data set based on [Chl]-based bio-optical models, with MAPE improving by a significant \(\sim\)3% compared to setting\(f_{L}\) equal to a spectrally constant 1.05. Using the constant \(P_{bb,ST}(\psi)\) backward phase function shape also introduced empiricism. If sample-specific Fournier-Forand analytical phase functions, based on \(\widetilde{b}_{bp}\) were used, errors in the validation data set increased by 3-5% (Table 1). Particulate phase function shapes from [PERSON] et al. [7] were modeled using the T-matrix approach, assuming homogeneous spheroidal shapes with constant \(n_{p}\) of 1.06. Although the endmember phase function shapes were not consistent with \(P_{bb,ST}(\psi)\) and \(P_{bb,FF}(\psi\widetilde{b}_{bp})\) shapes, many of the blended phase functions were. If \(P_{bb,ST}(\psi)\) may be considered widely representative, increasing errors may be expected in applying the M02 model to the lowest and highest Chl conditions where the endmember phase functions would fully manifest.