## 2 Materials and Methods
### Basic Irradiance Reflectance Model  
To estimate the irradiance reflectance from the backscattering albedo we use the Albert and Mobley algebraic radiance model valid for the infinitely deep medium case. The model is based on a careful analysis of solutions of the radiative transfer equation by the Hydrolight code for over 177,000 cases that encompass the full range of parameters for optically complex waters [1]. \[R_{\infty}=p_{1}x\Big{(}1+p_{2}x+p_{3}x^{2}+p_{4}x^{3}\Big{)}\Big{(}1+p_{5} \frac{1}{\cos\theta_{s}}\Big{)}\, \tag{15}\]  
In this formula \(R_{\infty}\) is the irradiance reflectance over waters deep enough that there is no contribution from the bottom reflectance. \(\theta_{s}\) is the sun angle just below the surface of the scattering medium. Standard irradiance reflectance measurements are carried out by comparing the signal  
Figure 2: Wet to dry angularly averaged Fresnel reflectivity factors as a function of wavelength for important components of materials and vegetation: crystalline quartz, calcite and cellulose. from a high-quality diffuse reflecting surface (Spectralon) that fills the field of view of the portable spectrometer to the signal from the substance to be measured under the same illumination conditions. Therefore, we must normalize the original Albert and Mobley expression to have \(R_{\infty}=1\) when \(x=1\). \[R_{spc-\infty}=\frac{p_{1}x\big{(}1+p_{2}x+p_{3}x^{2}+p_{4}x^{3}\big{)}}{p_{1} (1+p_{2}+p_{3}+p_{4})}\, \tag{16}\]  
We have used the notation \(R_{spc-\infty}\) in Equation (16) to clearly note that we are referring to the calibrated Spectralon normalized irradiance reflectance but from now on we will simply assume that all reflectances have been properly normalized. Table 2 gives the coefficients of Equations (15) and (16). The expression that [PERSON] and [PERSON] use is based on an extensive empirical survey carried out with an exact radiative code. The results of this survey are fitted as a fourth order polynomial which is an inconvenient form to use if we need in some cases to reverse the process and, for instance, estimate \(x\) from \(R_{spc-\infty}\). [PERSON] [5] developed a two-stream radiative model and obtained approximate solutions for the irradiance reflectance from an infinite depth medium. We found that we could closely match the result of [PERSON] and [PERSON] by parametrizing the formulas given by [PERSON]. This approach yields simpler more general formulas that can easily be inverted as desired. \[R_{spc-\infty}=\frac{\big{(}1+f^{2}\big{)}-\sqrt{\big{(}1+f^{2}\big{)}^{2}-4f ^{2}x^{2}}}{2\,f^{2}x}, \tag{17}\]  
\(f\) is an empirical parameter that varies from 0 to 1. A very close fit to the results of [PERSON] and [PERSON] is obtained with \(=0.79\). We have used the notation \(R_{spc-\infty}\) in Equations (16) and (17) to clearly note that we are referring to the calibrated Spectralon normalized irradiance reflectance but from now on we will simply assume that all reflectance have been properly normalized to unity. Formula (17) is easily inverted to obtain \(x\) as a function of \(R\) if required. \[x=\bigg{(}\frac{1+f^{2}}{f}\bigg{)}\bigg{(}\frac{fR}{1+fR^{2}}\bigg{)}, \tag{18}\]  
The value of \(f\) is correlated with but not equal to the mean cosine of the total scattering function \(<\)cos\(>\) which is defined as follows. \[<\cos>\ =2\pi\int_{0}^{\pi}p(\theta,\lambda)\cos\theta\sin\theta\ d\theta\, \tag{19}\]  
when \(f=0\), the single scattering is nearly isotropic and the irradiance reflectance is equal to the backscattering albedo. When \(f\) approaches 1 the single scattering becomes highly forward peaked. Note that \(p(\theta,\lambda)\) is the total scattering function and it includes both the reflected and the transmitted part of the radiation. The transmitted part is controlled by refraction and diffraction which dominate scattering in the forward hemisphere for grains or cells much larger that the wavelength.