## 1 Introduction  
Radiative transfer (RT) approximations linking inherent optical properties (IOPs), such as spectral absorption \(a(\lambda)\) (m\({}^{-1}\)) and spectral backscattering \(b_{b}(\lambda)\) (m\({}^{-1}\)) to ocean color remote sensing reflectance \(R_{rs}(\lambda)\) are vital to interpreting \(R_{rs}\) because it is not possible to analytically invert the full RT equation [1; 2]. Once forward RT approximations are developed, inversions can then be explored to solve for IOPs from \(R_{rs}\) and subsequently ocean biogeochemical properties [3]. Ocean color \(R_{rs}(\lambda)\) (sr\({}^{-1}\)) here is defined as \(L_{w}(\lambda)/E_{d}(\lambda)\), or water leaving radiance (W m\({}^{-2}\) sr\({}^{-1}\) nm\({}^{-1}\)) normalized to above water downwelling irradiance (W m\({}^{-2}\) nm\({}^{-1}\)) (see Reference [4] for complete definitions of all optical parameters and Appendix A, Table A1 for notation). Ocean color expressions to date have almost exclusively relied on first order approximations of RT relating \(R_{rs}\) to \(b_{b}/a\) through a proportionality represented as \(f/Q\)[5; 6; 7; 8], or to \(b_{b}/(a+b_{b})\) with multi-term polynomial expressions based originally on [PERSON] et al. [9] with coefficients represented as \(l\) or \(G\) ([9; 10; 11; 12]; see review by [PERSON] et al. [13]). The coefficients describing the relationship between \(R_{rs}\) and IOPs are detailed in look-up tables (LUTs), or a neural network in the case of [8], with dependencies on geometry (i.e., solar zenith, viewing angle, relative azimuth) and in some cases wavelength, wind speed, atmospheric conditions, and/or chlorophyll concentration [Chl]. The LUTs are generated from full RT computations with so-called synthetic data sets where IOPs and their interrelationships are assumed and referenced to [Chl]. LUT coefficients are thus also implicitly dependent on IOPs. These ocean color relationships have been tremendously useful to the ocean color community for decades. [PERSON] [14] discusses the first order analytical relationship and the empiricism necessary to invert. Coefficients describing the relationship between \(R_{rs}\) and IOPs are dependent on the bidirectional reflectance distribution function (BRDF), which describes the transformation of downwelling irradiance for different solar zenith angles into the distribution of upwelling radiance. [PERSON] et al. [7] describe the current state-of-the-art BRDF model (herein referred to as M02), currently implemented operationally by the ocean color community (NASA Ocean Biology Processing Group (OBPG); [15]) to convert \(L_{w}\) measured at any viewing geometry to a conceptual \"exact normalized\" \(L_{w}\), \([L_{w}]_{N}^{ex}\), with Sun at zenith and nadir viewing in a non-attenuating atmosphere. This allows any measurement at any Sun-viewing geometry to be directly intercompared. Inversion algorithms to derive IOPs are then typically applied to \([L_{w}]_{N}^{ex}\) and are often based on the same type of first order \(R_{rs}\) to IOP approximation [13; 16]. The current semi-analytical algorithm (SAA) application to water-leaving radiances is thus a two-step process (after applying calibration coefficients, georeferencing, and atmospheric correction): (1) A BRDF correction to carry out the conversion to \([L_{w}]_{N}^{ex}\), followed by (2) application of an inversion algorithm to derive IOPs (see [13]). A key potential source of uncertainty in current SAA approaches is associated with the volume scattering function (VSF; \(\beta(\psi)\) m\({}^{-1}\) sr\({}^{-1}\), where \(\psi\) is scattering angle), and this uncertainty is ambiguously dispersed in both steps. The VSF dependency is of paramount importance; as [PERSON] and [PERSON] [5] note, the BRDF \"... is essentially controlled by the shape of the VSF... \" Historically, very few measurements of the VSF have been available to develop and test SAAs.