## 3 Results and Discussion
### Overall Variability and Relationships Involving the Apparent Optical Properties
#### 3.2.1 Average Cosines  
As indicated earlier the average cosines serve as simple proxies of the angular structure of the light field and they have or have not been treated as AOPs by different investigators in the past [9; 10; 46; 48]. Based on our measurements of irradiance quartet the average cosine of underwater light field, \(\overline{\mu}\), ranged from about 0.38 to 0.88 in the blue and green spectral regions. In the red, the range was much larger, from 0.07 to 0.94, owing to strong effects of inelastic processes. The average cosine for downwelling light field, \(\overline{\mu}_{d}\), varied from 0.41 to 0.91 in the blue and green and 0.37 to 0.98 in the red. The approximate range for the average cosine for the upwelling light field, \(\overline{\mu}_{u}\), was 0.2-0.5 in the blue and green and 0.27-0.75 in the red. Our data are consistent but generally exhibit a broader range of values compared with relatively few experimental data of average cosines reported in literature. [PERSON] [18] reported on measurements of \(E_{o}\), \(E_{d}\), and \(E_{u}\) in the Mediterranean Sea and Norwegian fjord. Using his data, we calculated that \(\overline{\mu}\) was in the range of 0.47-0.95 in the blue-green spectral region (427, 477, and 532 nm) between the sea surface and 70 m depth, and 0.7-0.97 in the red (633 nm) within top 20 m layer. [PERSON] and [PERSON] [52] reported on measurements of \(\overline{\mu}\) in the range 0.7-0.9 at wavelengths shorter than 600 nm and depths between 24 and 55 m. [PERSON] et al. [29] measured the full angular distribution of radiance at a single wavelength of 555 nm in surface waters of the Pacific Ocean off Hawaii Islands, Santa Barbara Channel, and in Bedford Basin, and used these measurements to determine the ranges for \(\overline{\mu}\), \(\overline{\mu}_{d}\), and \(\overline{\mu}_{u}\), which were approximately 0.41-0.82, 0.5-0.94, and 0.2-0.48, respectively. Our determinations in the blue and green are generally consistent with the data from literature. The comparison for the red spectral region is, however, more limited because only few fragmentary data in the red have been previously reported. Our measurements of the average cosines in the red exhibited a very wide range of variability within the water column, providing a more comprehensive characterization of the average cosines that are strongly affected by inelastic processes in this spectral region. It has long been recognized that at near-surface depths in relatively clear ocean waters, the downwelling radiance distribution, and hence \(\overline{\mu}_{d}\), are strongly dependent on the sun position in clear sky or more generally on sky conditions, for example clear skies vs. overcast skies [11]. Such sensitivity of the downwelling average cosine to surface illumination conditions cautions against its indiscriminate use as a general descriptor (in the sense of AOPs) of water bodies, especially in the surface layers, and emphasizes a need for specifying surface boundary conditions. Our data collected within the top 10 m layer in the Gulf of California support the general trend of a decrease in near-surface \(\overline{\mu}_{d}\) with increasing solar zenith angle regardless of light wavelength (Figure 8a). For comparison, the best fit lines to experimental data of [PERSON] and [PERSON] [94] obtained at a 5 m depth (Mediterranean Sea) in the blue (465 and 474 nm) and experimental data of [PERSON] et al. [29] at a 5 m depth (open ocean waters in the Pacific) in the green (555 nm) are also shown in Figure 8a. In contrast, our data of \(\overline{\mu}_{u}\) at near-surface depths did not exhibit a clear dependence on the solar zenith angle (Figure 8b). We note, however, that [PERSON] and [PERSON] [94] and [PERSON] et al. [29] suggested a very weak dependence based on their data, as indicted in Figure 8b.