# Measuring and Modeling the Polarized Upwelling Radiance Distribution in Clear and Coastal Waters
## 3 Results
### Maximum DOLP and Nadir DOLP  
How large can the maximum _DOLP_ be in the upwelling radiance? Figure 4 shows this as a function of \(c_{t}/a_{t}\) and \(c_{t}\). We used the variation of the optical properties (\(c_{t}\), \(b_{t}\), \(b_{bt}\), and \(a_{t}\)) with the wavelength to fill in the data, thus this figure includes all wavelengths with their appropriate optical properties. The quantity \(c_{t}/a_{t}\) can be shown to equal the mean number of scattering events in the medium taken as a whole [(28)]. As the water gets more turbid, there is more multiple scattering, which will decrease the _DOLP_[(7; 42)]. The maximum _DOLP_ is often outside the Snell's cone, thus not retrievable from above the surface, so we also show the maximum value of the _DOLP_ inside the Snell's cone in Figure 4. As can be seen, this was less, sometimes significantly less, than the maximum _DOLP_ in the total upwelling field. Another point was that the maximum _DOLP_ in the total upwelling field occurred at an azimuth nearly perpendicular to the principal plane, as this position was always at a 90\({}^{\circ}\) scattering angle to the sun, and had the largest component of single scattered light due to a  
Figure 3: Similar to Figure 2, but for a case with more turbid water. The data were taken on 22 March 2009, at 9:40 UTC in the Ligurian Sea. The conditions were: SZA = 48\({}^{\circ}\), 550 nm, \(c_{t}\) = 0.44 m\({}^{-1}\), \(\omega_{0}\) = 0.82, and clear skies. The figure geometry is the same as Figure 2.  
combination of light attenuation with depth and the relative smoothness of the scattering function in the backward direction. For the maximum _DOLP_ inside the Snell's circle, as can be seen in Figures 2 and 3, this azimuthal angle will move towards the principal plane, and hence got closer to the position where, above the surface, it will be in the region of the solar glitter pattern.  
Another interesting parameter is the _DOLP_ at the nadir angle, as this was the direction that many in situ radiometers make their measurement (Figure 5). Here the RT model results are presented rather than the data, as in our definition of the reference frame, the nadir direction is a point of singularity, and the data become excessively noisy at that point.  
Figure 4: Maximum _DOLP_ in upwelling radiance distribution vs \(c_{t}\) (**a**) and \(c_{t}/a_{t}\) (**b**). The symbol legend is the same for both graphs. Filled circles represent the maximum _DOLP_ in the total upwelling field, while the open circles represent the maximum _DOLP_ in the portion of the upwelling light field inside the Snell’s circle (nadir angle less than 48\({}^{\circ}\)).  
Figure 5: Nadir _DOLP_ as a function of solar zenith angle and \(c_{t}/a_{t}\) computed from the RT model. Contours are lines of constant _DOLP_.  
Note that the nadir light field could be significantly polarized and the \(DOLP\) could be greater than 20%. Also, as can be seen, the largest values (22%) occurred at large solar zenith angles (when the scattering angle decreased to values closer to 90\({}^{\circ}\)), but there was also a trend of decreasing \(DOLP\) as \(c_{t}/a_{t}\) increased at a constant solar zenith angle due to increasing multiple scattering. The plane of polarization in all cases was perpendicular to the principal plane.