# Measuring and Modeling the Polarized Upwelling Radiance Distribution in Clear and Coastal Waters
## 2 Methods
### Modeling the Polarized Upwelling Radiance Distribution
Pre- and post-multiplication by the rotation matrix \(R(\alpha)\) is required to account for changes in reference frames: \(I(\theta,\phi)=R(\alpha)MR(\alpha^{\prime})I^{\prime}(\theta^{\prime},\phi^{ \prime})\), where  
\[R(\alpha)=\left( \begin{array}{cccc}1&0&0&0\\ 0&cos(2\alpha)&sin(2\alpha)&0\\ 0&-sin(2\alpha)&cos(2\alpha)&0\\ 0&0&0&1\end{array} \right) \tag{8}\]  
The rotation angle \(\alpha=cos^{-1}(\hat{l}_{l}\cdot\hat{l}_{r})\) is measured clockwise from the vector \(\hat{l}_{l}\) in the initial reference frame to the vector \(\hat{l}_{r}\) in the rotated reference frame [8]. The scattering phase matrix, \(P\), is a Mueller matrix normalized to the integral of the \(M_{11}\) element over all solid angles:  
\[P=M/b \tag{9}\]  
where  
\[b=2\pi\int_{0}^{\pi}M_{11}sin(\theta)d\theta \tag{10}\]  
\(M_{ij}\) represents the \(i\)th row and \(j\)th column of \(M\), and \(\Theta\) is the scattering angle. It is advantageous, when investigating the polarization effects of the Muller matrix, to normalize in a different way, specifically to the \(M_{11}\) element at each angle \(\Theta\) (Equation (11)). \[S_{ij}(\Theta)=M_{ij}(\Theta)/M_{11}(\Theta) \tag{11}\]  
Then,  
\[P=\widetilde{\beta}S \tag{12}\]  
where \(\widetilde{\beta}=M_{11}/b\) is the scattering phase function typically used in scalar radiance transfer models. For our modeling, \(P\) in the atmosphere and \(P_{vv}\), the seawater component of the ocean, were both set to \(P_{r}\), given by Rayleigh scattering:  
\[P_{r}=\frac{3}{16\pi}\left( \begin{array}{cccc}1+cos^{2}(\Theta)&-sin^{2}( \Theta)&0&0\\ -sin^{2}(\Theta)&1+cos^{2}(\Theta)&0&0\\ 0&0&2 cos(\Theta)&0\\ 0&0&0&2 cos(\Theta)\end{array} \right) \tag{13}\]  
(We examined a few cases using a code that included the (small) depolarization of Rayleigh scattering in the atmosphere and water by molecular anisotropy and concluded that its omission would not significantly affect the results.) Particle scattering in the ocean used \(\widetilde{\beta}\)[25; 29] and two alternative parameterizations of \(S\). One set of model runs used \(S\) from [PERSON] [30], which was experimentally determined from samples of seawater (referred to here after as V-F). Note that [PERSON] and [PERSON] found all off-diagonal elements of \(S\) equal to zero, within experimental error, except for \(S_{12}\) and \(S_{21}\). Furthermore, [PERSON] and [PERSON] found \(S_{12}\approx S_{21}\). Subsequent laboratory measurements of the Mueller matrix elements for phytoplankton [31; 32; 33] have been similar to V-F. Specifically, the \(S_{12}(\Theta)\) element of phytoplankton samples tends to have a minimum value of between \(-0.6\) to \(-0.8\) at an angle between \(90^{\circ}\) and \(100^{\circ}\). In contrast, experiments with solutions of suspended marine sediment have revealed smaller absolute values for \(S_{12}(\Theta)\) element near \(\Theta=90^{\circ}\)[32]. Because of this, we decided to carry out a second set of runs with a particle phase matrix having a smaller minimum in \(S_{12}\) near \(90^{\circ}\) (Figure 1). This was accomplished by replacing the V-F \(S_{12}(\Theta)\) with that for Rayleigh scattering with a depolarization factor \(\rho\). \(\rho\) is the ratio of light scattered with polarization parallel to the plane of incidence to that scattered perpendicular to the plane of incidence, when the incident radiation is polarized perpendicular to the plane of incidence. The element \(S_{12}(\Theta)\) as a function of \(\rho\) is given by Equation (14) [34]. The other elements of \(S\) remained unchanged for this Mueller matrix (referred to as Mod-V-F). Setting \(\rho=0.3\) resulted in a minimum value of \(S_{12}(90^{\circ})=-0.37\), which is within the range of \(-0.38\) to \(-0.25\) observed by [PERSON] et al. [32]. \[S_{12}(\Theta)=S_{21}(\Theta)=\frac{(1-\rho)\Big{(}cos(\Theta)^{2}-1\Big{)}}{ \Big{(}1+cos(\Theta)^{2}\Big{)}+\Big{(}3-cos(\Theta)^{2}\Big{)}\rho} \tag{14}\]  
[PERSON] et al. [35] presented four inequalities, originally derived by [PERSON] and [PERSON] [36] and [PERSON] et al.