## 3 Results and Discussion
### Overall Variability and Relationships Involving the Apparent Optical Properties
#### 3.2.3 Diffuse Attenuation Coefficients
For the red spectral region, the best fit parameters are still very close to the 1:1 line, specifically 0.9746 and 1.26 \(\times\)\(10^{-2}\) m\({}^{-1}\) for 650 nm and 0.9857 and 1.92 \(\times\)\(10^{-2}\) m\({}^{-1}\) for 683 nm (\(N\) = 62 for each wavelength). The \(r^{2}\) values at these wavelengths are only slightly lower than in the blue-green region and remained above 0.965. These results indicate that the magnitudes and behavior of \(K_{E}\) throughout the water column were consistent with those of \(K_{d}\) regardless of the spectral region including the long-wavelength portion of the spectrum where the effects of inelastic processes are strongest. Therefore, the use of a single relationship between \(K_{E}\) and \(K_{d}\) may be satisfactory regardless of wavelength. Such relationship determined from the combined data collected at wavelengths of 440, 550, and 650 nm is:  
\[K_{E}(z)=1.0093\;K_{d}(z)+7.88\times 10^{-4}\quad(r^{2}=0.996,\,RMSE=0.006\; \text{m}^{-1},\,N=194). \tag{8}\]  
This relationship is plotted in Figure 13b. It can be useful when \(K_{E}\) is required, for example for the analysis of [PERSON]'s equation, when \(E_{d}\) measurements are available but are not accompanied by \(E_{u}\) measurements. Similar to \(K_{E}\) vs. \(K_{d}\), the linear relationship between the \(K_{o}\) and \(K_{d}\) values measured throughout the water column is also quite robust regardless of light wavelength (Figure 13c). The plotted regression line corresponds to the best fit obtained from the combined data collected at 440, 550, and 650 nm:  
\[K_{d}(650)=0.4792\ \mathrm{exp}(-0.0451\ z)+0.0387\quad(r^{2}=0.925,\ RMSE=0.033\ \mathrm{m}^{-1},\ N=66) \tag{6a}\]0  
This type of relationship can be useful in the absence of scalar irradiance measurements throughout the water column which is a common situation, but when the measurement or estimate of spectral just below the surface is available. In such case, \(E_{o}\) can be propagated throughout the water column on the basis of vertical measurements of spectral \(E_{d}\) using the relationship between \(K_{o}\) and \(K_{d}\). The resulting depth profile of spectral \(E_{o}\) can, in turn, be subject to conversion to quantum units and spectral integration to obtain the depth-resolved PAR quantum scalar irradiance, \(E_{oPAR}\). Assuming the availability of \(E_{oPAR}\) just below the surface, e.g., [108], another relationship between the \(K\) coefficients can be useful for estimating \(E_{oPAR}\) at different depths from vertical measurements of spectral \(E_{d}\). This relationship of \(K_{oPAR}\) vs. \(K_{d}\) on the basis of data collected throughout the water column is shown in Figure 13d. In this analysis we ignored, however, the data collected at near-surface depths (\(<\)5 m) because \(K_{oPAR}\) decreases rapidly with depth within the near-surface layer owing to high absorption by water molecules within the long-wavelength portion of the spectrum [109; 10; 110; 111]. Figure 13d demonstrates that there is a reasonably good linear relationship between \(K_{oPAR}\) and \(K_{d}\) in the blue and green spectral regions, as illustrated for 440 nm, 490 nm, and 550 nm. The best-fit relationships are:  
\[K_{d}(650)=0.4792\ \mathrm{exp}(-0.0451\ z)+0.0387\quad(r^{2}=0.925,\ RMSE=0.033\ \mathrm{m}^{-1},\ N=66) \tag{6a}\]1 \[K_{d}(650)=0.4792\ \mathrm{exp}(-0.0451\ z)+0.0387\quad(r^{2}=0.925,\ RMSE=0.033\ \mathrm{m}^{-1},\ N=66) \tag{6a}\]2 (10b) \[K_{d}(650)=0.4792\ \mathrm{exp}(-0.0451\ z)+0.0387\quad(r^{2}=0.925,\ RMSE=0.033\ \mathrm{m}^{-1},\ N=66) \tag{6a}\]3  
A closer examination also showed that \(K_{PAR}(z)\approx[K_{d}(z,490)+K_{d}(z,550)]/2\):  
\[K_{d}(650)=0.4792\ \mathrm{exp}(-0.0451\ z)+0.0387\quad(r^{2}=0.925,\ RMSE=0.033\ \mathrm{m}^{-1},\ N=66) \tag{6a}\]4 \[K_{d}(650)=0.4792\ \mathrm{exp}(-0.0451\ z)+0.0387\quad(r^{2}=0.925,\ RMSE=0.033\ \mathrm{m}^{-1},\ N=66) \tag{6a}\]5  
Being very close to the 1:1 line, this relationship provides a convenient approximation to \(K_{oPAR}(z)\) based on \(K_{d}(z)\) and is plotted in Figure 13d.