## 3 Results and Discussion
### Overall Variability and Relationships Involving the Apparent Optical Properties
#### 3.2.1 Average Cosines
The increased role of inelastic processes, especially Raman scattering, with increasing depth is illustrated for the average cosines in the red part of the spectrum in Figure 9. This effect was particularly well-pronounced for the downwelling light field in terms of a well-behaved trend of a decrease in \(\overline{\mu}_{d}\) with increasing depth towards \(\overline{\mu}_{d}\) = 0.5, which is clearly seen in our data at the two examined wavelengths in the red, 650 nm and 683 nm (Figure 9a). The best-fit exponential functions to the data are:  
\[\overline{\mu}_{d}(650) = 0.5058\ \exp(-0.0460\ z)\ +\ 0.5140\ (r^{2}=\ 0.828,RMSE=\ 0.061,N=\ 65); \tag{3a}\] \[\overline{\mu}_{d}(683) = 0.6512\ \exp(-0.0715\ z)\ +\ 0.4659\ (r^{2}=\ 0.855,RMSE=\ 0.067,N=\ 65), \tag{3b}\]  
where \(r^{2}\) is the determination coefficient and \(N\) the number of observations. A distinct group of outlying data marked in Figure 9a was collected at significantly larger solar zenith angle (\(\theta_{s}\) = 57.1\({}^{\circ}\)than the remaining data. These outliers were not included in the regression analysis. Whereas at 650 nm Raman scattering is expected to have a dominant effect, at 683 nm chlorophyll-_a_ fluorescence can have an additional significant or even dominant effect at certain depths, depending on chlorophyll-_a_ concentration and its variation within the water column. The combined effects of Raman scattering and chlorophyll-_a_ fluorescence at 683 nm result in faster decrease of \(\overline{\mu}_{d}(683)\) with depth from the near surface through the lower portion of the \(Chl_{max}\) layer compared with \(\overline{\mu}_{d}(650)\). In addition, in our dataset the interplay of vertically variable chlorophyll-_a_ fluorescence and Raman scattering throughout the water column does not seem to have a noticeable deteriorating effect on the strength of the relationship \(\overline{\mu}_{d}(683)\) vs. \(z\) as compared with \(\overline{\mu}_{d}(650)\) vs. \(z\). Figure 8: **(a)** Relationship between \(\overline{\mu}_{d}\) at selected light wavelengths as indicated and solar zenith angle \(\theta_{s}\). **(b)** Same as **(a)** but for \(\overline{\mu}_{u}\). The presented data were collected at all stations within the top 10 m layer. For comparison, solid lines in **(a,b)** represent the relationship reported by [PERSON] et al. [29] and dashed lines by [PERSON] and [PERSON] [94]. Dotted line in **(a)** represents \(\overline{\mu}_{d}=\cos(\theta_{sw})\) where \(\theta_{sw}\) is the zenith angle of the refracted solar beam just beneath the ocean surface obtained from [PERSON]’s law. Figure 9: Average cosines in the red spectral bands (650 and 683 nm) plotted as a function of depth \(z\) using data from all stations and depths. **(a)**\(\overline{\mu}_{d}\)(_z_, 650) (crosses) and \(\overline{\mu}_{d}\) (_z_, 683) (open circles) vs. \(z\). **(b)** Same as **(a)** but for \(\overline{\mu}_{u}\). Data points surrounded by dotted lines in **(a)** are for the station in the Farallon Basin with the solar zenith angle significantly larger (-57\({}^{\circ}\)) than at other stations. Solid lines show the best-fit regression functions for the average cosines vs. depth. In contrast to \(\overline{\mu}_{d}\), \(\overline{\mu}_{u}\) in the red did not exhibit such strong dependence on depth although it is clear that \(\overline{\mu}_{u}\) assumed generally lower values near the surface (Figure 9b). The \(r^{2}\) values for the \(\overline{\mu}_{u}\) vs. \(z\) data dropped to 0.488 at 650 nm and 0.147 at 683 nm. The differences in the patterns of depth dependencies for \(\overline{\mu}_{d}\) and \(\overline{\mu}_{u}\) can be attributed to relatively smaller role played by inelastic processes at near-surface depths for the downwelling light field compared with the upwelling light field in the red spectral region. It is also important to point out that in comparison with our dataset the dependence of \(\overline{\mu}_{d}\) on depth is expected to be reduced for datasets collected in turbid waters, at large solar zenith angles, and/or heavily overcast skies.