## 2 Materials and Methods  
### Basic Model  
The key parameter in any radiative transfer model of reflectance is a parameter we will refer to in this paper as the backscattering albedo \(x(\lambda)\). This defined as the ratio of the total scattering in the back hemisphere to the sum of the absorption and total backscattering. \[x(\lambda)=\frac{b_{b}(\lambda)}{a(\lambda)+b_{b}(\lambda)}\, \tag{1}\]  
In the above expression \(a(\lambda)\) is the absorption coefficient while \(b_{b}(\lambda)\) is the backscattering coefficient. The backscattering coefficient is defined in standard form by the following expression. \[b_{b}(\lambda)=2\pi\int_{\pi/2}^{\pi}p(\theta,\lambda)\sin\theta\ d\theta\, \tag{2}\]  
In the expression above \(p(\theta,\lambda)\) is the scattering phase function. The main aim of our work from now on is to obtain expressions for the various contributions to both \(b_{b}(\lambda)\) and \(a(\lambda)\). Figure 1 shows graphically the various mechanisms discussed above and will serve as a guide in this task. The materials of interest to us, minerals and vegetation, absorb little per grain or cell and the light ray will encounter many inner surfaces before being scattered back out. In our evaluation of \(b_{b}(\lambda)\) we will treat the reflection from the first surface boundary as similar to that of the inner deeper boundaries. The formulas we will use for the backscattering from the inner surfaces of minerals are those which describe reflection from rough surface elements with random orientation. For vegetation we will use the formulas for backscattering from smooth surfaces which is a more appropriate representation. This model has been used recently to describe the backscattering from complex naturally occurring structures such as coccoliths [6; 7]. The backscattering cross-section function for the reflection from a randomly oriented set of uniform (Lambertian) diffusers is given by:  
Figure 1: This figure is a schematic of the microstructure elements relevant to scattering and absorption for both minerals (grains) and vegetation (cells). The incident light rays (1) are reflected (2) and transmitted at the first surface (3). The rays transmitted through the first surface are subsequently both reflected back from the inner surfaces of the grains (4) and absorbed. The rays that penetrate deeper (5) are multiply scattered before coming back to the surface and have a near Lambertian (uniform) scattering distribution. \[\sigma_{b}(\lambda)=\sigma_{g}\ \omega_{t}(n)2\pi\int_{\pi/2}^{\pi}\frac{4}{3\pi}( \sin\theta-\theta\cos\theta)\sin\theta d\theta=\frac{5}{6}\ \sigma_{g}\ \omega_{t}(n)\, \tag{3}\]  
\(\sigma_{g}\) is the geometric cross-section of an individual scattering structure and \(\omega_{t}(n)\) is the Fresnel reflectance integrated over a set of randomly oriented surfaces of relative index of refraction \(n\) that together compose the surface of those structures. The total backscattering coefficient of the ensemble of the scattering elements is by definition:  
\[b_{b}(\lambda)=\frac{5}{6}\ \omega_{t}(n)\ N\ \sigma_{g}\, \tag{4}\]  
Assuming that the number density of the scattering structures \(N\) is such that the sum of their geometric cross-sections is equal to the area of the material normal to the impinging light, we obtain the following formula for the backscattering coefficient of the material. \[b_{b}(\lambda)=\frac{5}{6}\frac{\omega_{t}(n)}{d}\, \tag{5}\]  
where \(<\!\!d\!\!>\) is the mean diameter of the scattering structures. \(\omega(n)\) for unpolarised light is given by the following formulas [3,4].