## 3 Results
### Specific Properties of Vegetation  
Absorption in vegetation is controlled by the absorption of the chlorophyll-a filled chloroplasts in the cell. As the concentration of chlorophyll-a and/or the size of the chloroplasts increases the absorption through the cell increases until the chloroplast absorbs more of the light at a given wavelength until in the limit of large concentrations and/or size it becomes a dark spot masking all the light its surface intercepts at this wavelength. This absorption saturation effect was first extensively studied by [PERSON] and [PERSON] (2008) who called it the package effect. This is the factor that dominantes the variability in the absorption spectrum for different types of vegetation. To compute this effect first we need formulas for the absorption efficiency \(Q_{a}\) of the chloroplasts. These are derived in Appendix B for both the original model that asssumed a spherical shape for the chloroplasts and for a new model that assumes disk-like chloroplasts. As mentioned in Appendix B, the exact formulas can be approximated to a sufficent accuracy by a simpler exponential model. From now on we will use the more realistic disk-like shape to model the absorption saturation effect. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Substance** & \(\alpha_{o}\) & \(\
u\) & \(\lambda_{o}\) & \(\sigma_{r}(\%)\) \\ \hline California Sand & 0.01055 & 0.708 & 0.346 & 5.0 \\ Hawaii Sand & 0.04631 & 1.438 & 0.102 & 4.8 \\ Greenland Sand & 0.03323 & 0.453 & 0.350 & 4.9 \\ Limestone (Trenton) & 0.08401 & 1.217 & 0.102 & 3.8 \\ Limestone (Fossil) & 0.01662 & 0.688 & 0.348 & 6.6 \\ Clay & 0.02180 & 1.273 & 0.350 & 5.9 \\ Sandy Loam & 0.01087 & 1.882 & 0.195 & 4.1 \\ Gray Silty Loam & 0.00850 & 1.636 & 0.298 & 3.3 \\ Brown Loam & 0.01292 & 1.595 & 0.286 & 3.4 \\ Dark Loam & 0.02345 & 1.395 & 0.350 & 4.5 \\ Granite & 0.02102 & 0.737 & 0.304 & 8.6 \\ Schist & 0.11558 & 0.134 & 0.338 & 0.9 \\ Shale & 0.02988 & 0.738 & 0.113 & 1.5 \\ Shale & 0.04426 & 1.165 & 0.101 & 6.5 \\ Shale & 0.02364 & 0.219 & 0.344 & 2.8 \\ Shale & 0.11558 & 0.368 & 0.350 & 3.3 \\ Silstone & 0.02669 & 0.229 & 0.345 & 3.9 \\ Silstone & 0.01803 & 0.377 & 0.343 & 4.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: This is a table of the functional fits to various minerals according to the formula. \[a_{v}^{*}(\lambda)=\bigg{(}\frac{1}{2\,u_{cp}}\bigg{)}\Big{(}1-e^{-a_{o}^{*}( \lambda)2\;u_{cp}}\Big{)}\, \tag{26}\]  
With:  
\[u_{cp}=\rho_{cp}\tau_{cp}\, \tag{27}\]  
\(a_{o}^{*}(\lambda)\) is the specific mass absorption coefficient of chlorophyll-a at low concentration in units of m\({}^{2}\) gr\({}^{-1}\). \(\rho_{cp}\) is the chlorophyll-a mass density inside the chloroplast in gr m\({}^{-3}\) and \(\tau_{cp}\) is the thickness of the chloroplast disk in meters. The mean thickness of a randomly oriented set of disks is 2 \(\tau_{cp}\) which explains the factor of 2 seen in Equation (26). We verified the validity of this model by first comparing the theory for disks given by Equation (26) with the [PERSON] et al. [9] empirical formula for chlorophyll-a absorption in type I waters which is based on in-depth analysis of a compilation of most of the available datasets. To do this we rewrite Equation (26) as a specific absorption gain function:  
\[\frac{a_{v}^{*}(\lambda)}{a_{o}^{*}(\lambda)}=\bigg{(}\frac{1}{a_{o}^{*}( \lambda)\;2\;u_{cp}}\bigg{)}\Big{(}1-e^{-a_{o}^{*}(\lambda)2\;u_{cp}}\Big{)}\, \tag{28}\]  
As can be seen in Figure 4, the overall behavior of the absorption is captured by the gain formula and this over three orders of magnitude in chlorophyll-a density. We note that the hysteresis seen in the empirical curves is due to an additional wavelength shift as a function of chlorophyll-a density. This effect was in fact observed by [PERSON] [16]. The results shown in Figure 4 are a strong indication that the dominant effect in the spectral variation as a function of chlorophyll-a density is the absorption saturation effect.