# Measuring and Modeling the Polarized Upwelling Radiance Distribution in Clear and Coastal Waters
## 3 Results
### Comparison of \(DOLP\) Differences  
For the first quantitative comparison of the RT model and the data, we looked at scatter plots comparing the measured and modeled \(DOLP\) (Figure 6). For each cruise, a best linear fit line was calculated, with the \(y\)-intercept = 0. The fit was calculated for all of the data, as well as the data inside the Snell's circle; however, the calculated slopes were not significantly different, so we will discuss only the fit for the total image. The slopes were 0.94 (\(\pm\)0.002, \(r^{2}\) = 0.81), 0.88 (\(\pm\)0.006, \(r^{2}\) = 0.84), and 0.82 (\(\pm\)0.008, \(r^{2}\) = 0.76) for Hawaii, New York Bight, and Ligurian Sea, respectively. Because the \(DOLP\) was significantly overestimated in the Ligurian Sea data, the alternative normalized Mueller matrix Mod-V-F (\(\rho\) = 0.30) was tried, with all other parameters the same, resulting in a slope of 1.20 (\(\pm\) 0.009, \(r^{2}\) = 0.656); we concluded that the depolarization of 0.3 was too strong. The true average Mueller matrix for this location could be somewhere between the two we used. Figure 6: Scatter plots of \(DOLP\) for model versus data, separated by cruise. Also shown is the best fit line (with \(y\)-intercept = 0) (black line) and 1:1 line (red line). The crosses correspond to data outside the Snell’s circle (nadir angle larger than 48\({}^{\circ}\)), and the dots to points inside the Snell circle. V-F was used in the model for (**a–c**), Mod-V-F was used for (**d**). The correlation between \(DOLP_{diff}\) and several environmental parameters was investigated. In Figure 7, we show the \(DOLP_{diff}\) variation with solar zenith angle. In this figure, the mean \(DOLP_{diff}\) was calculated for each data image, along with the standard deviation (shown as error bars). There was no trend in the data; however, there does appear to have been a slight increase in the magnitude of the deviation at solar zenith angles greater than 70\({}^{\circ}\). In general though, in almost all cases, the mean \(DOLP_{diff}\) was within one standard deviation of zero. In Figure 8, we look at the \(DOLP_{diff}\) dependence on \(c_{t}\) and \(c_{t}/a_{t}\). While there was no clear trend in \(DOLP_{diff}\) with \(c_{t}\), a slight trend occurred as a function of \(c_{t}/a_{t}\). Low values of \(c_{t}/a_{t}\) (more absorption, less multiple scattering) had a slightly larger value of \(DOLP_{diff}\), while \(DOLP_{diff}\) tended towards 0 with larger values of \(c_{t}/a_{t}\) (more multiple scattering). However, the mean \(DOLP_{diff}\) was less than 5% for almost all cases, and this was within the measurement uncertainty of these instruments in the field. Finally, in Figure 9, we look at the \(DOLP_{diff}\) as a function of \(b_{bt}/b_{t}\) (\(b_{bt}/b_{t}\) is the fraction of total scattering which was in the backwards direction, i.e., scattering angles from 90\({}^{\circ}\) to 180\({}^{\circ}\)). As \(b_{bt}/b_{t}\) increased, \(DOLP_{diff}\) appeared slightly higher, indicating less agreement at higher values of \(b_{bt}/b_{t}\). Low values of \(b_{bt}/b_{t}\) were indicative of a low refractive index, phytoplankton-dominated environment, while higher values of \(b_{bt}/b_{t}\) indicated an environment with other higher refractive index particles, perhaps sediment. V-F was based on measurements in a variety of environments (clear ocean water to more turbid coastal water); however, measurements of the Mueller matrix of phytoplankton cultures also agreed with V-F. It has often been stated that the polarization resulting from higher index particles is less than V-F (i.e., larger _depolarization_ at a 90\({}^{\circ}\) scattering angle) [32]. It is probably the case in our data set, that as the \(b_{bt}/b_{t}\) ratio increases, the \(S_{12}\) and \(S_{21}\) elements of the Mueller matrix should also decrease in an absolute value sense, but not as much of a decrease as used in Mod-V-F, where the decrease in this matrix would have caused a significant negative \(DOLP_{diff}\). Figure 7: \(DOLP_{diff}\) vs solar zenith angle for each data image.