# Measuring and Modeling the Polarized Upwelling Radiance Distribution in Clear and Coastal Waters
## 3 Results
### Overall View of the Upwelling Polarization Signal  
Figure 2 shows an example image for clear water. In the right column, the data parameters are shown (_I_, \(Q\), \(U\), _DOLP_, and \(\chi\)), while in the center column, the results from the RT model are shown, where the model inputs are based on the measured parameters for the data (\(b_{p}\), \(a_{pg}\), solar zenith angle). As mentioned earlier, during processing of this data, the symmetry about the principal plane was used to average the left and right side of the data (with proper handling of the sign of \(U/I\)). Thus, to make these images, this symmetry was used to generate the left side of the data images. The left column is a single scattering calculation, where the Mueller matrix was assumed to be V-F, and the solar zenith angle was the value appropriate for the data. As can be seen, the same broad patterns were visible in the single scattering model, the RT model and data. The \(I\) component of the Stokes vector was largest near the horizon in the direction towards the sun. The \(Q/I\) and \(U/I\) patterns were very similar in all three cases. For the _DOLP_, the single scattering model was significantly different than the data and the RT model. With single scattering, the maximum _DOLP_ was both larger than in the RT model and the data, and occurred for all nadir angles at the scattering angle matching the minimum in the \(S_{12}\) and \(S_{21}\) elements (90-100\({}^{\circ}\)). Interestingly, while it was a subtle effect in the clear water case, the maximum _DOLP_ in the model and data occurred at the same scattering angle, but at an azimuth of 90\({}^{\circ}\) relative to the principal plane (the plane containing the anti-solar position and nadir). This can be contrasted with the case of the downwelling sky radiance distribution, where the maximum _DOLP_ occurred in the principal plane, with a decrease in _DOLP_ towards the horizon (for example, Liu and Voss [38]). The same cause was in effect in both cases: for downwelling radiance, the horizon had more multiple scattering than the principal plane at the 90\({}^{\circ}\) scattering angle, while for the upwelling radiance, in the water, there was more multiple scattering for the principal plane at a 90\({}^{\circ}\) scattering angle than for nadir angles closer to the horizon. The other feature in the data for this clear water case was the separation of the neutral points, areas of zero _DOLP_, on either side of the principal plane, as has been discussed previously [39; 40]. Finally, the angle of the polarization plane, \(\chi\), was very similar in all three cases, perhaps less well-defined for the data in the area of the low polarization, around the anti-solar position. An example with more turbid water is shown in Figure 3. Once again, the patterns were generally the same, however in this case the data are noisier. Note that the data shown in this case has not been averaged using the left/right symmetry across the principle plane, thus appears less symmetric than the example in Figure 2. As has been pointed out before [41], the most stable parameter was \(\chi\), with the exception of the area with very low _DOLP_, in which case the plane of polarization was not well defined. In general, the images appeared much noisier in the data from the two more turbid sites (New York Bight and Ligurian Sea) than in the clear water off Hawaii. The pattern of \(Q/I\) and \(U/I\) tended to shift more between each image that goes into these data averages. In the scale of \(Q/I\) and \(U/I\) ranging from \(-1\) to \(1\), the standard deviation obtained during averaging the data was approximately 0.05 for the New York Bight and Ligurian Sea data, while it was 0.012 for the clearer Hawaii data. One can also