# Measuring and Modeling the Polarized Upwelling Radiance Distribution in Clear and Coastal Waters
## 2 Methods
### Modeling the Polarized Upwelling Radiance Distribution  
A Monte Carlo model was used to solve the vector radiative transfer equation (Equation (5)) for the propagation of the Stokes vector, \(I\), at wavelength \(\lambda\),  
\[cos(\theta)\frac{dI(\lambda,\tau,\theta,\phi)}{d\tau}=-I(\lambda,\tau,\theta, \phi)+\omega_{0}\int_{4\pi}R(\alpha)P\big{(}\lambda,\tau,\theta^{\prime},\phi ^{\prime}\rightarrow\theta,\phi\big{)}R^{\prime}(\alpha^{\prime})I(\lambda, \tau,\theta^{\prime},\phi^{\prime})d\Omega^{\prime} \tag{5}\]  
The four components of the column vector \(I=[I,Q,U,V]^{T}\) were defined in Equation (2). The single-scattering albedo, \(\omega_{0}\), the optical depth of the medium, \(\tau\), the scattering phase matrix, P, and the rotation matrix, R, are defined below. The right-handed coordinate system in Equation (5) has its origin at the top of the medium, \(z\)-axis pointed downward, and \(x\)-axis directed away from the sun. The nadir angle, \(\theta\), relative to the \(z\)-axis, and the azimuth angle, \(\phi\), in the \(x\)-\(y\) plane, define the direction of photon propagation. Photons originate in the solar beam, i.e., with \(\theta=\theta_{0}\), the solar zenith angle, and \(\phi_{0}=0\). The model simulated a horizontally homogenous, two-layer system with a non-absorbing, Rayleigh-scattering atmosphere over an ocean comprised of Rayleigh-scattering water molecules plus scattering and absorbing hydrosols (\"particles\"). Fresnel reflectance at the air-water interface used the boundary condition defined in [PERSON] et al. [8]. The single-scattering albedo, \(\omega_{0}\), and the optical depth of the medium, \(\tau\), were defined as per usual (e.g., [PERSON] [25]):  
\[\omega_{0}=\frac{b_{t}(\lambda)}{c_{t}(\lambda)}, \tag{6}\]  
\[\text{and }\tau(\lambda)=\int_{0}^{z}c_{t}(\lambda)dz. \tag{7}\]  
where the parameters \(a_{t}(\lambda)+b_{t}(\lambda)=c_{t}(\lambda)\) are the total absorption, scattering, and attenuation coefficients (in m\({}^{-1}\)), respectively. For the atmospheric layer, \(a_{t}(\lambda)=0\), therefore \(\omega_{0}=1\), and \(\tau\) was taken from [PERSON] [26] assuming standard atmospheric conditions. For the oceanic layer, absorption (\(a_{w}\)) and scattering (\(b_{w}\)) coefficients of seawater were interpolated to PolRADS and DPOL spectral bands from Table 1.1 in [PERSON] et al. [27]. In the Ligurian Sea and New York Bight experiments, the absorption coefficient of dissolved and particulate constituents (\(a_{pg}\)) and the particle scattering (\(b_{p}\)) coefficients were measured in situ, depth-weighted, and interpolated as necessary to the PolRADS and DPOL spectral bands as described by [PERSON] et al. [14]. No in situ measurements were available from Hawaii, therefore for that dataset, \(a_{pg}\) and \(b_{p}\) were derived from Morel and Gentili [28] (Equations (7) and (8) in reference [28]) using an estimated total chlorophyll concentration of 0.1 g/m\({}^{3}\), the long-term average value at the measurement site. Summing the water and particle contributions gave the total coefficients required for Equations (6) and (7): \(a_{t}=a_{\omega}(\lambda)+a_{pg}(\lambda)\) and \(b_{t}=b_{\omega}(\lambda)+b_{p}(\lambda)\). For each image data set, the model was run at the corresponding solar zenith angle. Scattering events in Equation (5) are represented by a 4 \(\times\) 4 matrix called the Mueller matrix, \(M\). For example, a photon travelling in direction \(\zeta^{\prime}\) with Stokes vector \(I^{\prime}\) would be scattered in a new direction \(\zeta\) with Stokes vector \(I\) by the linear transformation \(I(\zeta)=MI^{\prime}(\zeta^{\prime})\). Note that \(M\) is defined relative to the scattering plane (as is traditional), the plane defined by vectors \(\zeta^{\prime}\) and \(\zeta\), but we have defined \(I(\theta,\phi)\) relative to the viewing direction and the nadir direction.