## 2 Materials and Methods
### Methods
#### 2.7.2. Canopy Structure and Area Solar Radiation Metrics  
The leaf area index (_LAI_) quantitatively characterizes plant canopy layers and is often used to model forest canopy structure. It is defined as the one-sided leaf area per unit ground surface area for broadleaf or half of the total needle surface area per unit ground surface area for conifer forests [31]. \[LAI=\frac{leaf\ area(\text{m}^{2})}{ground\ area\ (\text{m}^{2})} \tag{1}\]  
_LAI_ is also a key variable for regional and global models of biosphere-atmosphere exchanges of energy, carbon dioxide, water vapor and other materials [32]. Canopy photosynthesis and its equivalent gross primary productivity (GPP) should theoretically reach a maximum as leaf area (_LAI_) increases to a value where spectral radiation in the 0.4 to 0.7 \(\upmu\)m visible wavelength is totally intercepted [33]. It can be challenging to quantify _LAI_ accurately due to significant spatio-temporal variability and measurement limitations inherent in current methodologies [26]. These methods utilize \"direct\" approaches employing field sampling or \"indirect\" approaches involving optical instruments combined with modeling [26]. Optical instruments used to estimate _LAI_ rely on the measurement of light transmittance and, because they are based on [PERSON]'s law, the resultant measures are termed \"effective _LAI_\" [26]. _LAI_ was derived in ERDAS Imagine Spatial Modeler by first deriving a Normalized Difference Vegetation Index (_NDVI_) from the multispectral image and then using a linear model to determine _LAI_[34]. _NDVI_ is calculated from the visible red and near-infrared light (_NIR_) [35]. \[NDVI=\frac{NIR-Red}{NIR+Red} \tag{2}\]  
_NDVI_ uses normalization to minimize effects of variable irradiance and is commonly used to indicate the amount and vigor of vegetation and to differentiate vegetated and non-vegetated areas in an image. Plants appear relatively dark in the PAR and relatively bright in the _NIR_[35]. The biophysical interpretation of _NDVI_ for Figure 4 is the fraction of absorbed photosynthetically active radiation (_fPAR_). Although derived _NDVI_ values can range from \(-1.0\) to \(+1.0\), values less than zero do not have any biological meaning. The image data is classified for ease of interpreting _NDVI_ and _LAI_ values. Figure 4: Classified _NDVI_ map centered on the study area landscape in 2 m resolution (Source: DigitalGlobe). The empirical conversion from the \(NDVI\) described by [PERSON] et al. [34] was then used to derive the canopy leaf area in the ERDAS Spatial Modeler tool (Figure 5) [24]. \[LAI=(17.35*NDVI)-9.01 \tag{3}\]  
Photosynthetically available radiation (_PAR_) and its relationship to \(LAI\) is well established in the literature [26]. The theoretical basis for the absorption of light as it passes through a forest canopy is described by the Beer-Lambert law:  
\[t(x)=\exp[-K(LAI(x))] \tag{4}\]  
where:  
* \(t\) is the proportion of _PAR_ incident at the top of the canopy that is transmitted to a given point \(x\) within the canopy;
* \(LAI(x)\) is the total leaf area above point \(x\);
* \(K\) is the extinction coefficient. The coefficient indicates that light intensity decreases exponentially as it passes through each canopy layer [4,33]. Known as the attenuation coefficient, it describes the extent to which the radiant flux of a beam is reduced as it passes through a specific material, in this case the vegetation canopy. When a narrow (collimated) beam passes through canopy strata, the beam will lose intensity due to two processes: absorption and scattering. A detector can be used to measure a beam's directional path, or conversely using a non-narrow beam, one can measure how much of the lost radiant flux was scattered and how much was absorbed. The extinction coefficient is, therefore, the sum of the absorption coefficient and the scattering coefficient [36].