## 2 Forward Model Development
### Average Cosine of the Downwelling Light Field \(\overline{\mu}_{d}\)  
A model for \(\overline{\mu}_{d}\) is now required in the expression from Equation (12). If the sky is ignored and we assume a negligible fraction of the incident solar beam is scattered in the near-surface, a reasonable first-order approximation of \(\overline{\mu}_{d}\) should be \(\mu_{w}\equiv\cos(\theta_{s})\). Adding in a cardioidal radiance distribution for skylight, [PERSON] and [PERSON] [57] obtained:  
\[\frac{1}{\overline{\mu}_{d}}\approx\frac{0.6}{\mu_{w}}+\frac{0.4}{0.859}, \tag{13}\]and noted that for sun angles between 8\({}^{\circ}\) and 62\({}^{\circ}\), \(\overline{\mu}_{d}\) varied only from 0.79 to 0.94. Thus, even without knowledge of solar zenith angle, a median value could be used with an accuracy better than 10%. Numerator values of 0.6 and 0.4 in Equation (13) represent the fractions of direct (\(E_{dd}\)/\(E_{d}\)) and diffuse (\(H\) = \(E_{ds}\)/\(E_{d}\)) downwelling light, respectively, which together equal unity. These values primarily depend on \(\theta_{s}\)' and horizontal visibility \(V\), the latter of which depends on aerosol optical thickness (AOT). For 20\({}^{\circ}\leq\theta_{s}\)' \(\leq\) 60\({}^{\circ}\) and for \(H\) between 0.2 and 0.5, the variability of \(\overline{\mu}_{d}\) is ~7% for a fixed \(\theta_{s}\)'. The term \(\overline{\mu}_{d}\) can be factorized in two parts, one part dependent on the IOPs, the other dependent on the atmospheric conditions and geometry:  
\[\overline{\mu}_{d}\bigg{(}\theta_{s}^{\prime},V,\frac{b_{b}}{a},\eta_{bb} \bigg{)}\approx M_{d}^{+}\big{(}\theta_{s}^{\prime},V\big{)}\times M_{d}^{*} \bigg{(}\frac{b_{b}}{a},\eta_{bb}\bigg{)}. \tag{14}\]  
The atmospheric component can be represented as:  
\[M_{d}^{+}\big{(}\theta_{s}^{\prime},V\big{)}=\frac{\Big{[}\frac{1-H(\theta_{s }^{\prime},V)}{\mu_{w}}+\frac{H(\theta_{s}^{\prime},V)}{0.859}\Big{]}^{-1}}{P^ {3}[\cos(\theta_{s}^{\prime})]}. \tag{15}\]  
As mentioned, [PERSON] and [PERSON] [57] assumed \(H\) = 0.4. [PERSON] and [PERSON] [58] later provided a relationship for \(H\) as a function of \(\theta_{s}\)' and \(V\) that included skylight. Specifically, the results of ([58]; their Figure 4) can be fit as follows:  
\[ \begin{array}{ll}H(\theta_{s}^{\prime},V)=&\big{[}e_{1}V^{2}+e_{2}V+e_{3} \big{]}\theta_{s}^{{}^{\prime}5}+\big{[}e_{4}V^{2}+e_{5}V+e_{6}\big{]}\theta_ {s}^{\prime 4}\\ &+\big{[}e_{7}V^{2}+e_{8}V+e_{9}\big{]}\theta_{s}^{{}^{\prime}3}+\big{[}e_{10 }V^{2}+e_{11}V+e_{12}\big{]}\theta_{s}^{{}^{\prime}2}\\ &+\big{[}e_{13}V^{2}+e_{14}V+e_{15}\big{]}\theta_{s}^{\prime}+e_{16}V^{2}+e_{ 17}V+e_{18}\,.\end{array}  \tag{16}\]  
Coefficients \(e\) are provided in Appendix A, Table 11. Note the typical default value used in Hydrolight is \(V\) = 15 km. The \(P^{3}\) term in the \(M_{d}^{+}\) relationship is a 3 rd order polynomial in \(\cos(\theta_{s}\)') to correct where [PERSON]'s original approximation deviates from the [PERSON] and [PERSON] relationship at large \(\theta_{s}\)'. The \(P^{3}\) term is:  
\[P^{3}\big{[}\cos(\theta_{s}\)')\big{]}=0.7792\cos^{3}\big{(}\theta_{s}\)' \(-1.7366\cos^{2}\big{(}\theta_{s}\)')+1.1551\cos\big{(}\theta_{s}\)')+0.7842. The IOP-dependent component \(M_{d}^{*}\) was modeled using the approach in [40] with the extended \(b_{b}\)/\(a\) range and \(\eta_{bb}\) resolution discussed in Section 2.2. The final fitted relationship is:  
\[ \begin{array}{ll}M_{d}^{*}\Big{(}\frac{b_{b}}{a},\eta_{bb}\Big{)}=&\Big{[} m_{d,1}^{*}\log\eta_{bb}+m_{d,2}^{*}\Big{]}\Big{(}\log\frac{b_{b}}{a}\Big{)}^{ 3}+\Big{[}m_{d,3}^{*}\log\eta_{bb}+m_{d,4}^{*}\Big{]}\Big{(}\log\frac{b_{b}} {a}\Big{)}^{2}\\ &\qquad\qquad+\Big{[}m_{d,5}^{*}\log\eta_{bb}+m_{d,6}^{*}\Big{]}\log\frac{b_{ b}}{a}+m_{d,7}^{*}\log\eta_{bb}+m_{d,8}^{*}.\end{array}  \tag{17}\]  
Coefficients \(m_{d}^{*}\) are provided in Appendix A, Table 11. Absolute percent error with this relationship relative to computations using full radiative transfer (Hydrolight, see Section 3.2) varies from 0.06% to 0.4% across all \(\theta_{s}\)'.