## 3 Results and Discussion
### Overall Variability and Relationships Involving the Apparent Optical Properties
#### 3.2.3 Diffuse Attenuation Coefficients
The best-fit exponential functions to the data in Figure 12 are:  
\[K_{d}(650)=0.4792\ \mathrm{exp}(-0.0451\ z)+0.0387\quad(r^{2}=0.925,\ RMSE=0.033\ \mathrm{m}^{-1},\ N=66) \tag{6a}\] \[K_{d}(683)=0.7229\ \mathrm{exp}(-0.0947\ z)+0.0846\quad(r^{2}=0.898,\ RMSE=0.050\ \mathrm{m}^{-1},\ N=66)\] (6b) \[K_{u}(650)=0.5614\ \mathrm{exp}(-0.2258\ z)+0.0730\quad(r^{2}=0.806,\ RMSE=0.032\ \mathrm{m}^{-1},\ N=64)\] (6c) \[K_{u}(683)=1.0000\ \mathrm{exp}(-0.4711\ z)+0.0641\quad(r^{2}=0.391,\ RMSE=0.062\ \mathrm{m}^{-1},\ N=64). \tag{6d}\]  
The relationship for \(K_{u}(683)\) is greatly inferior because at this wavelength the effect of Raman scattering is reinforced by chlorophyll-\(a\) fluorescence emission, which resulted in a very steep decrease of \(K_{u}(683)\) in the top 20 m (Figure 12b). In contrast to \(K_{d}\), other \(K\) coefficients are poorly documented in the literature. It is therefore useful to examine the relationships between \(K_{d}\) and other \(K\) coefficients using our dataset containing depth-resolved measurements within the euphotic layer. The results from this analysis are shown in Figure 13. Not shown in the figure are results that indicated a very good linear relationship with close agreement between the \(K_{d}\) and \(K_{od}\) values (\(r^{2}=0.985\)) and between \(K_{u}\) and \(K_{ou}\) (\(r^{2}=0.953\)). This agreement was generally observed regardless of light wavelength and depth of measurement within our dataset. The relationship between \(K_{u}\) and \(K_{d}\) is reasonably good in the blue and green spectral regions (Figure 13a). For example, at 440, 490, and 550 nm the best-fit linear relationships are:  
\[K_{u}(z,\,440)=1.0115\;K_{d}(z,\,440)+9.47\times 10^{-3}\quad(r^{2}=0.828,\,RMSE =0.023\;\text{m}^{-1},\,N=66) \tag{7a}\] \[K_{u}(z,\,490)=1.0220\;K_{d}(z,\,490)+1.71\times 10^{-3}\quad(r^{2}=0.691, \,RMSE=0.021\;\text{m}^{-1},\,N=66)\] (7b) \[K_{u}(z,\,550)=1.4699\;K_{d}(z,\,550)-4.83\times 10^{-2}\quad(r^{2}=0.6 72,\,RMSE=0.023\;\text{m}^{-1},\,N=66). \tag{7c}\]  
As the wavelength increases into the red spectral region the relationship weakens considerably or nearly vanishes as suggested by the large scatter of data points for 650 nm and 683 nm in Figure 13a. For these wavelengths, the \(r^{2}\) values drop to 0.444 and 0.079, respectively. This result can be attributed to increased effects of inelastic processes with increasing wavelength, which act differentially on the downwelling and upwelling light attenuation. It is noteworthy that one possible application of the relationships between \(K_{u}\) and \(K_{d}\) in the blue and green is to use the \(K_{d}\) measurements for extrapolating \(E_{u}\) measurements to deeper depths where upwelling light is no longer detectable with the \(E_{u}\) sensor but is still detectable with the \(E_{d}\) sensor. However, a special caution must be exercised for situations with large solar zenith angles, as evidenced by one outlying data point in our dataset which was obtained for relatively large \(\theta_{s}\) of 57.1\({}^{\circ}\) compared with the remaining data. While it is known that at large \(\theta_{s}\)\(K_{d}\) tends to increase significantly in the surface layer [11; 17], the effect of solar angle is expected to be less pronounced for \(K_{u}\). Such result is seen in our dataset in Figure 6e,f for the near-surface measurements taken at the example station in the Farallon Basin. Our analysis shows that the linear relationship can be also used for estimating the diffuse attenuation coefficient of net irradiance, \(K_{E}\), from \(K_{d}\) using the depth-resolved data in the euphotic layer (Figure 13b). For the blue-green spectral region the best-fit relationship is nearly a perfect 1:1 line with the correlation coefficient \(r^{2}>0.998\). Specifically, the slope and offset parameters of the linear fit are: 1.0001 and \(-1.67\times 10^{-4}\) m\({}^{-1}\) for 440 nm, 0.9999 and 1.66 \(\times\)\(10^{-5}\) m\({}^{-1}\) for 490 nm, and 1.0015 and 1.78 \(\times\)\(10^{-5}\) m\({}^{-1}\) for 550 nm (\(N\) = 66 for each wavelength).