## 4 Results  
### Developing an Expression for the \(f_{L}\) Term  
Dynamics of \(f_{L}\) were explored in the synthetic data set. As found by [PERSON] et al. [76], dependencies on wavelength and \(\theta_{s}{{}^{\prime}}\) were most important, and an average spectral shape \(f_{L,\textit{arc}}(\lambda)\) scaled to solar zenith according to \(\sin(\theta_{s}{{}^{\prime}})\) was ultimately found to be most representative (Figure 2):  
\[f_{L}\big{(}\theta_{s}^{\prime},\lambda\big{)}=f_{L,\textit{arc}}(\lambda) \big{[}0.05959\sin\big{(}\theta_{s}{{}^{\prime}}\big{)}+0.9728\big{]}. \tag{30}\]  
Function \(f_{L,\textit{arc}}(\lambda)\) is provided in Appendix A, Table 10. Equation (30) was developed by minimizing residual errors between results from full radiative transfer and the ZTT model for each wavelength and each solar zenith angle for the full synthetic data set. Spectral shapes for \(f_{L}\) were relatively consistent in the results, so the term \(f_{L,\textit{arc}}(\lambda)\) was derived from an average for these results after spectral normalization. The scaling factor in brackets was a suitable function to minimize errors with respect to solar zenith angle. Only nadir viewing was again considered in the simulations for \(f_{L}\), and we again make the assumption from Section 2.1 that \(f_{L}\) for other viewing angles that define a specific in-water scattering angle \(\psi\) can be approximated by \(f_{L}\) observed at the nadir viewing geometry with equivalent \(\psi\). The potential caveats mentioned in Section 2.1 apply here as well. Restating Equation (30) in terms of \(\psi\), the following is obtained:  
\[f_{L}(\psi,\lambda)=f_{L,ave}(\lambda)[0.07762\sin(\psi)+1.0405]. \tag{31}\]  
Similarities between \(f_{L,ave}(\lambda)\) and a typical ocean absorption spectrum are noted (Figure 2), resulting from the effects of multiple scattering, where higher relative scattering (lower relative absorption) promotes a radiance field closer to that which would result from isotropic scattering, i.e., where \(f_{L}\) approaches unity.  
Figure 3A shows results deriving \(r_{rs}\) using the ZTT model for the synthetic data set, applying the approximation from Equation (31) for \(f_{L}\) and using all synthetic data IOPs as input. This match up is, thus, effectively assessing combined errors in the ZTT model from the approximations of \(f_{L},f_{b}\), \(K_{Lu}\), and \(\overline{\mu}_{d}\). We note \(f_{L}\) was specifically optimized to this data set, so in this respect Figure 3A shows what may be considered a best-case matchup. MAPE %\(\delta_{abs}\) for the full data set was relatively small at 2.68%. The largest errors were observed in three \(r_{rs}\) spectra, with the model showing overestimation bias centered around 570 nm. These spectra were associated with the highest [Chl] modeled in the data set and lowest \(\widetilde{b}_{bp}\).  
Figure 2: Spectral dependency of the \(f_{L}\) shape function at different \(\theta_{s}\)’. Thin lines are solved \(f_{L}\) functions for each \(\theta_{s}\)’ for the synthetic data set and stars are from the model approximation with constant shape described by \(f_{L,ave}(\lambda)\) in Equation (30).  
Figure 3: \(r_{rs}\) derived from the ZTT model with the synthetic data set compared to \(r_{rs}\) simulated using Hydrolight (HL): (**A**)\(f_{L}\) optimized spectrally (i.e., Equation (31) for the synthetic data set with Fournier-Forand phase functions individually determined for each [Chl]; (**B**)\(f_{L}\) optimized spectrally for the synthetic data set with constant \(P_{bb,ST}(\psi)\) (see text); and (**C**)\(f_{L}\) set to 1.05 with constant \(P_{bb,ST}(\psi)\). Colors represent wavelengths, from 400 to 700 nm with gray used for 350 \(<\lambda<\) 400 nm and 700 \(<\lambda<\) 800 nm.