## 5 Discussion
### Assessing Residual Bias in the Model  
The assumption of single scattering in the simplification of \(f_{b}\) (Section 2, Equation (2)) is not expected to invoke significant errors at turbidity levels found for most oceanic and coastal waters [5; 13]. As shown by [PERSON] [23], secondary scattering events for light redirected toward the backward direction, i.e., downwelling radiance scattered into the upwelling radiance field, has a negligible effect on the shape of the upwelling radiance distribution because the vast majority of these scattering events are in the near forward direction. The full effects of multiple scattering were included in relationships for other terms in the model as they were determined using full RT modeling. MASCOT VSF measurements used to determine \(P_{bb,ST}(\psi)\) only extended to \(170^{\circ}\), so the application of this function for \(\psi>170^{\circ}\) in the model is uncertain. There were no \(\psi>170^{\circ}\) in the data sets tested here, although such scattering angles may be expected for some viewing geometries for the PACE imager (Figure 1). There is a need for a better understanding of phase function shapes for \(\psi>170^{\circ}\) and this extends to other applications, such as lidar [92]. Only nadir viewing was considered for the analytical relationships for \(K_{Lu}\) and \(f_{L}\) and we suggest these relationships may be transferable to other viewing geometries as long as \(\psi\) is preserved. This assumption has been verified in Hydrolight simulations to have small errors for representative cases, but cannot be assessed with the field data sets available here, which all have nadir viewing. This is being considered in future work, and, moreover, emphasizes the need for routine measurements of full upwelling radiance distributions in ocean color validation work as validation only with nadir radiance \(L_{u}(\theta_{s}\rho_{v}=\pi)\) neglects the BRDF effects that are essential to any algorithm. The optimization of \(f_{L}\) was based on the [Chl]-based synthetic data set. A significant deviation from the bio-optical model assumptions of that synthetic data set could lead to a discrepancy in match ups, however it worth pointing out (1) \(f_{L}\) has a relatively weak dependency on IOPs except for the VSF shape in the backward direction (which is effectively accounted for in the solar zenith dependency in Equation (30)), and (2) other state-of-the-art models (i.e., M02 and L11) are constructed entirely around such synthetic data sets. A benefit of the ZTT model is that an analytical relationship for a specific term of the model may be replaced later if a more optimal approach is demonstrated. As shown in Zaneveld [27; 28], \(L_{u}\) normalized to \(E_{od}\) is more closely linked to the RT Equation than \(L_{u}/E_{d}\), where an additional \(\overline{\mu}_{d}\) term must be added to the model (Section 2.6). Adding this term is expected to increase uncertainty in the RT model and inversion. Scalar irradiance has the additional advantage of being less dependent on solar zenith angle and sky radiance distributions. As discussed in Section 2.6, sensor technology is available to measure \(E_{od}\) and inclusion of these measurements in field efforts focused on algorithm development and validation should be encouraged. When the \(\overline{\mu}_{d}\) term must be included in association with \(E_{d}\) measurements, AOT is also a useful validation measurement that may be made in the field [93] and used to derive visibility \(V\)[58] for input in Equation (16) or full RT computations. Lastly, potential uncertainties in the full RT computations must be considered, as this was used to parameterize several ZTT terms. Polarization is not considered in the ZTT model or in Hydrolight, which is likely important [29; 33]. There is a need in the research community for commercially available RT code with a focus on the ocean that includes full polarization, along with concomitant measurements of polarized radiance in the field.