## 2 Forward Model Development  
[PERSON] [27,28] derived an exact restatement of the RT equation assuming a plane-parallel, optically deep water column in terms of upwelling radiance \(L_{u}\) in viewing direction \(\theta_{v}\):  
[PERSON]frac{L_{u}(\theta_{s},\theta_{v},\phi)}{E_{od}}=\frac{f_{b}(\theta_{s}, \theta_{v},\phi)\frac{b_{h}}{2\pi}}{-\cos(\theta_{v})K_{L_{u}}(\theta_{s}, \theta_{v},\phi)+c-f_{L}(\theta_{s},\theta_{v},\phi)b_{f}}, \tag{1}\]  
where \(E_{od}\) is the scalar downwelling irradiance, \(K_{Lu}\) is the attenuation coefficient for upwelling radiance, \(b_{f}\) is forward scattering, and \(c\) is the attenuation coefficient. Parameters \(f_{b}\) and \(f_{L}\) are light field shape factors representing the path radiance term of the RT equation [27,28]. Factor \(f_{b}\) describes the redirection of downwelling radiance into the upwelling viewing angle for a given VSF and is normalized to the redirection that would be observed if the VSF was isotropic. Factor\(f_{b}\) is thus directly linked to the shape of the VSF in the backward direction. The factor \(f_{L}\) is defined similarly but describes the redirection  
Figure 1: Refracted, in-water scattering angles made between the solar zenith and viewing angle, simulated for the upcoming NASA PACE satellite imager through a complete polar orbit (solid blue). Scattering angle distributions for SeaWiFS were similar. Data courtesy [PERSON] (NASA CSFC). Angular weighting functions of commercial backscattering sensors WET Labs ECO-BB, ECO-NTU, and MCOMS and IMO-SC6 (see text) are overlaid after scaling by \(5\times 10^{6}\). of upwelling radiance into the upwelling viewing angle and is thus directly linked to the shape of the near forward VSF. These shape factors are expected to vary within a relatively narrow range near unity. All terms are a function of depth \(z\) and wavelength \(\lambda\). The Zaneveld expression does not account for any inelastic processes, such as molecular Raman scattering or fluorescence [30; 31; 32; 33]. Using physically reasoned approximations for \(f_{b}\) including the assumption of single scattering, the following was obtained by [PERSON] ([28]; his Equation (14)):  
\[\frac{L_{u}(\theta_{s},\theta_{v},\phi)}{E_{od}}=\frac{\beta(\psi)}{-\cos( \theta_{v})K_{Lu}(\theta_{s},\theta_{v},\phi)+c-f_{L}(\theta_{s},\theta_{v}, \phi)b_{f}}, \tag{2}\]  
where \(\beta\) is the VSF (including water) and \(\psi\) is the in-water scattering angle formed between peak incident sunlight at solar zenith angle \(\theta_{s}\) and scattered light traveling in viewing direction (\(\theta_{v}\),\(\Phi\)), where \(\cos(\psi)=\cos(\theta_{s})\cos(\theta_{v})-\sin(\theta_{s})\sin(\theta_{v}) \cos(\phi)\). Azimuth \(\phi\) is relative to the Sun's direction. Zenith angles \(\theta\) are in-water, refracted through the air-sea interface and determined from a vertically downward direction. For nadir viewing, \(\psi=\pi-\theta_{s}\) and \(-\cos(\theta_{v})=1\) in the denominator. Note approximations of Equation (2) provided in Zaneveld [28] assumed only nadir viewing while we retain the full BRDF functionality here. In theoretical analyses, [PERSON] et al. [34] showed \(b_{b}\) retrievals based on the Zaneveld expression had errors as large as \(-20\%\) and \(+40\%\). However, an extraordinarily wide range of VSF shapes was applied in their simulated data, including VSF shapes for specific particulate subcomponents, such as bacteria, minerals, and phytoplankton. These populations were considered separately as quasi-monodispersions with phase function shapes computed from Lorenz-Mie theory, i.e., assuming homogeneous spheres. These phase function shapes thus had large oscillations with respect to angle, structure that we now know is extremely unrealistic for VSFs representative of bulk in situ particle populations (e.g., [35; 36; 37; 26]). [PERSON] et al.