## 5 Discussion
Phase functions in the Chl range 1 to 10 mg m\({}^{-3}\) were in closest agreement to \(P_{bb,ST}(\psi)\). Applying phase functions derived using the [PERSON] et al. [7] approach in the ZTT model increased errors relative to using a constant \(P_{bb,ST}(\psi)\) by a significant 7% for the validation data set of [29] (from 16% to 23%) (Table 1). This error was also larger than the 21% error observed for M02 (addressed further below). Since L11 used a Fournier-Forand phase function where \(\widetilde{b}_{bp}=1\%\) and the average Petzold phase function as endmembers [shown in 26, their Figure 3], many blended shapes from L11 should be relatively similar in the backward to the \(P_{bb,ST}(\psi)\) used in ZTT (although the Petzold shape does exhibit systematic discrepancies). This particular \(P_{bb,FF}\) with \(\widetilde{b}_{bp}=1\%\) is actually a very close match to \(P_{bb,ST}\). L11 was developed through least-squares fitting with respect to Hydrolight results, so all the blended phase functions are weighted in the results. A \"typical marine atmosphere\" and associated incident sky radiance distribution were used in the Hydrolight RT computations for the synthetic data set, the effects of which enter the ZTT model through spectral optimization of the upwelling radiance shape factor\(f_{L}\) (Section 4.1). The parameters \(\Psi_{K_{Lu}}\) and \(\overline{\mu}_{\infty}\) were also solved with Hydrolight RT simulations assuming the same atmosphere. The M02 and L11 models also assumed a single \"typical marine atmosphere\" in their simulations using Hydrolight based on Reference [58], so this aspect of the models should be consistent. The \(\overline{\mu}_{d}\) term in the ZTT model has added flexibility in accommodating varying atmospheric visibility (closely related to AOT; [58]) inasmuch as \(V\) is an explicit term for \(\overline{\mu}_{d}\). One question arising from this work is, if the phase function shape \(P_{bb,ST}\) may indeed be considered highly representative, how would a LUT-based model following the approach of M02 but using \(P_{bb,ST}\) perform? To test this, we developed a LUT based on the first order \(r_{rs}\) proportionality to \(b_{b}/a\), using the synthetic data set described in Section 3.1. The particle phase function was constant, consisting of \(P_{bb,ST}\) in the backward and a Fournier-Forand phase function with \(\widetilde{b}_{bp}=0.006\) for the forward direction (this had the best performance for the ZTT model in the validation data set [29], with MAPE = 17%). The LUT approach resulted in a MAPE of 22% for the validation data set and 26% for the NOMAD data. This performance is very close to M02 (21% and 25%, respectively) and L11 (21% and 26%, respectively). An insight we can take from this analysis is that a component, such as phase function shape in the different models cannot be considered in a vacuum, i.e., each model has a large set of assumptions that ultimately influence final performance. Furthermore, although not always explicitly stated in the literature, field validation efforts are likely to have had a role in updating assumptions through the course of development and testing of some models to ensure the model was well aligned with available data. For example, in our case, the \(f_{L}\) expression (Section 4.1) was developed from an assumed synthetic data set and the \(\widetilde{b}_{bp}\) value of 0.006 was recommended as it ultimately provided the best results with the field validation data sets available. Recent work by [PERSON] et al. [84] has suggested the possibility of substantial variability in phase function shape in the backward direction, including up to a 40% increase for \(\beta_{p}(\psi)/b_{bp}\) in the near backward relative to \(\beta_{p}(120^{\circ})/b_{bp}\). These results are in stark contrast to the comprehensive study of [26] and other published phase functions in the literature (e.g., [35; 85]).