## 2 Forward Model Development
### Diffuse Attenuation of Upwelling Radiance \(K_{Lu}\)  
For the term \(K_{Lu}\), [PERSON] [28] suggested an assumption of equivalency to \(K_{\infty}\), the diffuse attenuation coefficient in the asymptotic regime. Asymptotic theory is based on the principle that the shape of the light field with depth gradually transforms from being dependent on the incident surface light field to being constant, azimuthally symmetric (so \(L\) is only a function of \(\theta\)), and dependent only on IOPs. Attenuation coefficients for all aspects of the light field, i.e., for all radiances and therefore all irradiances, are equivalent in the asymptotic regime and are also IOPs. This is a critical assumption for the purposes of model development since \(K\) can then be described only in terms of IOPs. [PERSON] the assumption, even though the omnidirectional light field in surface waters is far from asymptotic, due to the decoupling of upwelling radiances to downwelling radiance distributions. Measurements of upwelling radiance fields from the 1960s and 1970s showed a near constant shape and attenuation rate with depth. Recently, [PERSON] and [PERSON] [40] confirmed this assumption in RT simulations with no more than 3% error when the sun was at solar zenith and the following relationship held for _above water_ solar zenith \({\theta_{s}}^{\prime}\) up to 75\({}^{\circ}\):  
\[\frac{K_{Lu}-K_{\infty}}{K_{\infty}}=F\big{(}{\theta_{s}}^{\prime}\big{)}=f_{1 }{\theta_{s}}^{\prime}+f_{2}{\theta_{s}}^{\prime}+f_{3}{\theta_{s}}^{\prime} ^{2}+f_{4}{\theta_{s}}^{\prime}+f_{5}, \tag{3}\]  
which included a full range of possible natural water types. Coefficients \(f\) are provided in Appendix A, Table 11. Angle \({\theta_{s}}^{\prime}\) is related to \(\theta_{s}\) by Snell's Law, i.e., \({\theta_{s}}^{\prime}=\sin^{-1}(1.34\sin(\theta_{s}))\). Only nadir viewing was considered in the simulations for \(K_{Lu}\) in Equation (3), and we make the assumption that \(K_{Lu}\) for other viewing angles that define a specific scattering angle \(\psi\) can be approximated by Equation (3) with nadir viewing geometry after assigning a \({\theta_{s}}^{\prime}\) that provides an equivalent in-water \(\psi\). For example, for \({\theta_{s}}^{\prime}=60^{\circ}\), \(\theta_{s}\) will be 40.3\({}^{\circ}\) and \(\psi\) will be 139.7\({}^{\circ}\) for nadir viewing; we assume the resulting \(F(60^{\circ})\) will also be applicable to any off-nadir viewing angle with \(\psi=139.7^{\circ}\). This assumption has been verified using Hydrolight RT simulations (methodology addressed in Section 3.2) to no worse than 2% in the solar plane and no worse than 5% within the upwelling hemisphere for in-water scattering angles consistent with remote sensing (i.e., Figure 1). This assumption implies a rotational reference frame, where the first order determinant of radiance field shape in the model, i.e., \(\psi\), is preserved. Two potential drawbacks of this assumption are (1) the influence of skylight may be skewed in the rotated reference frame, and (2) the range of viewing angles is restricted since the smallest \(\psi\) is \(\sim\)134\({}^{\circ}\) for underwater nadir viewing. The range \(\psi>134^{\circ}\), however, comprises \(>\)95% of the expected scattering angles that will be measured by the PACE imager (Figure 1). Further work with field measurements is needed to verify this assumption. Reformulating Equation (3) in terms of dependency on the in-water scattering angle, we obtain:  
\[F(\psi)=f_{A1}\psi^{A}+f_{A2}\psi^{3}+f_{A3}\psi^{2}+f_{A4}\psi+f_{A5}. \tag{4}\]  
Coefficients \(f_{A}\) are provided in Appendix A, Table 11.  
From Gershun's Law we can set \(K_{\infty}=a/\overline{\mu}_{\infty}\), where \(\overline{\mu}_{\infty}\) is the average cosine of the asymptotic light field.