## 3 Results
### Specific Properties of Vegetation
There was a large amount of variability in the original experimental data sets on which the empirical formulas are based so the discrepancies are not surprising. However, in the case that concerns us, which is the absorption in vegetation itself, the number density of phytoplankton which is the main uncontrolled empirical variable becomes severely constrained. The bulk chlorophyll-a mass density \(\rho_{cll}\) is given by:  
\[\rho_{cll}=n_{c}\;n_{cp}\;\rho_{cp}V_{cp}\, \tag{29}\]  
Figure 4: Graph of the ratio of the specific absorption gain to the unsaturated absorption gain. The dotted lines are from the empirical formula of [PERSON] et al. [9] for chlorophyll-a. The solid lines are from Equation (21) with the parameters noted. The number density of cells is \(n_{c}\), the number of chloroplasts per cell is \(n_{cp}\) and the volume of each chloroplast is \(V_{cp}\). In the open ocean, \(n_{c}\) may be weakly correlated with the other parameters while in a continuous block of cells as is the case in vegetation there is a very strongly constrained relationship. We can see this as follows. For a volume of vegetation \(V_{m}\) filled by cells with a volume \(V_{c}\) we have:  
\[\rho_{chl}V_{m}=\left(\frac{V_{m}}{V_{c}}\right)\rho_{cp}\;n_{cp}\;V_{cp}=V_{ m}\;\rho_{cp}\bigg{(}\frac{n_{cp}\;V_{cp}}{V_{c}}\bigg{)}\, \tag{30}\]  
\[\rho_{chl}=\;\rho_{cp}\bigg{(}\frac{n_{cp}\;V_{cp}}{V_{c}}\bigg{)}\, \tag{31}\]  
We expect the ratio of the total volume of chloroplasts \(n_{cp}\;V_{cp}\) to the cell volume \(V_{c}\) to be almost constant. The variability induced by \(n_{c}\) for open water has disappeared and the bulk chlorophyll-a concentration is now as expected simply proportional to the chlorophyll-a concentration inside the chloroplasts that we use to estimate the absorption saturation. The main implication of the discussion above is that we expect to be able to model the spectral shape of the absorption spectrum of chlorophyll-a with a single fitting parameter \(u_{cp}\). Before this becomes feasible, there are, however, several significant hurdles which have to be overcome. First, the Bricaud formula for chlorophyll-a absorption can be scaled to any concentration no matter how small even for ranges that lie well outside the zone of the data used for the original fit. This creates a problem when trying to determine \(a_{o}^{*}(\lambda)\) as a limiting value for low chlorophyll-a concentrations as we can extrapolate back to unphysically small values of concentration. [PERSON] et al. [10] used a different approach to model the chlorophyll-a absorption from naturally occurring populations of organisms. They determine the absorption spectra for two limiting populations of organisms, the nano population and micro population. For low concentrations of chlorophyll-a the absorption spectrum of the nano population applies while for high concentrations the spectrum of the micro population is the appropriate one to use. As the concentration of chlorophyll-a increases, the spectrum evolves as a linear combination of both these extreme cases. We first attempted to use the nano population spectrum from [PERSON] et al. [10] as the limiting case \(a_{o}^{*}(\lambda)\) for low chlorophyll-a concentrations. Unsurprisingly, we found that the difference between our model and the Bricaud form diverged significantly at the higher concentrations. This is problematic since for vegetation, which is the case of interest for us, the chlorophyll-a concentrations are expected to be large. In fact, they exceed the range of validity of the Bricaud formulas. To handle these extreme cases with a reasonable expectation of accuracy we decided to take a different approach. The technique is based on using a Bricaud spectrum at a given reference value with sufficiently high chlorophyll-a concentration but in a zone where the fit is still valid and extending the range from that point using the gain saturation equations.