## 3 Results and Discussion
### Overall Variability and Relationships Involving the Apparent Optical Properties
#### 3.2.1 Average Cosines
Under such conditions the near-surface values of \(\overline{\mu}_{d}\) are expected to be lower (i.e., closer to 0.5) compared with most of our near-surface data, and hence the expected changes of \(\overline{\mu}_{d}\) with depth would not be as well pronounced. Thus, the strong sensitivity of near surface \(\overline{\mu}_{d}\) to illumination conditions at the sea surface suggests that no single relationship can describe a change of \(\overline{\mu}_{d}\) with depth. This is illustrated in Figure 9a which includes an outlying case of data points collected at significantly larger \(\theta_{s}\) than the remaining data, emphasizing a need to pay particular attention to the possible effect of solar zenith angle. In addition, it is important to emphasize that in the blue-green spectral region where the effects of inelastic processes are weaker, the average cosines exhibited much less distinct dependency on depth (see Figures 5e-g and 7a-c). Our analysis also shows that regardless of light wavelength, none of the average cosines were related significantly to _Chl_ in our dataset. The highest \(r^{2}\) of 0.42 was found between \(\overline{\mu}_{d}(z,\,650)\) and _Chl_(2) (not shown). For \(\overline{\mu}_{d}\) in the blue-green where inelastic processes are less important than in the red, and for \(\overline{\mu}_{u}\) at all examined spectral bands, the relation was much weaker or essentially indiscernible. This lack of relation with _Chl_ is not surprising because the average cosine within the water column, if not dominated by inelastic processes, depends on the interplay of the inherent absorption and scattering properties of seawater in rather complex way [85], and _Chl_ cannot adequately represent such effects. Similar patterns observed in our data of \(\overline{\mu}\) and \(\overline{\mu}_{d}\) (e.g., Figure 5) indicate that these two average cosines were well correlated. The relationships between \(\overline{\mu}\) and \(\overline{\mu}_{d}\) obtained with our data are presented in Figure 10a. Specifically, the linear relationship for the blue-green spectral region as determined from the data collected at various depths and two wavelengths, 440 nm and 550 nm, is:  
\[\overline{\mu}=0.996\ \overline{\mu}_{d}\ -\ 0.0352\quad\Big{(}r^{2}=\ 0.978,RMSE=\ 0.042,N=\ 158\Big{)}.\] (4a) On the basis of data at two wavelengths in the red, 650 and 683 nm, the relationship is: \[\overline{\mu}=1.51\ \overline{\mu}_{d}\ -\ 0.472\quad\Big{(}r^{2}=\ 0.929,RMSE=\ 0.167, N=\ 134\Big{)}. \tag{4b}\]  
These relationships indicate that reasonably good estimates of \(\overline{\mu}\) can be obtained from downwelling irradiance measurements. As seen, \(\overline{\mu}\) and \(\overline{\mu}_{d}\) in the blue-green were typically close to one another within the examined euphotic layer. In the red, the correlation is marginally lower and still very strong but \(\overline{\mu}\) tends to be somewhat smaller than \(\overline{\mu}_{d}\) over a nearly complete data range with the exception of the highest values. In contrast to \(\overline{\mu}\) vs. \(\overline{\mu}_{d}\) relationship, \(\overline{\mu}_{u}\) and \(\overline{\mu}_{d}\) were not correlated or very weakly correlated (Figure 10b). The \(r^{2}\) values for the blue-green and red spectral regions are 0.026 and 0.109, respectively. Therefore, \(\overline{\mu}_{u}\) cannot be estimated from \(\overline{\mu}_{d}\) or vice versa, which means that a complete irradiance quartet or radiance angular distribution is required for investigating both \(\overline{\mu}_{u}\) and \(\overline{\mu}_{d}\). Figure 10b also reveals that the \(\overline{\mu}_{u}\) data in the blue-green appeared to be grouped in at least two distinct clusters, one with \(\overline{\mu}_{u}>\) -0.3 and the other with \(\overline{\mu}_{u}<\) -0.3. A large number of \(\overline{\mu}_{u}\) measurements falls in the range from 0.2 to -0.3, suggesting that much of the upwelling light in these cases propagated relatively close (within -10\({}^{\circ}\)-20\({}^{\circ}\)) to the horizontal direction.