# Sensitivity of Common Vegetation Indices to the Canopy Structure of Field Crops
## Materials and Methods
### Study Area and Field Measurements
The possibility of light transmitted through the canopy at polar angle \(\theta\) can be modeled for a random spatial distribution element canopy using an exponential model:  
\[P(\theta)=\exp\left[-\frac{k(\theta,\chi)\text{ LAI }\sqrt{a}}{\cos\theta}\right], \tag{1}\]  
where \(a\) is the leaf absorptivity for light set to 0.85 in this study (SunScan SSI user manual) and \(k(\theta,\chi)\) is the canopy directional light extinction coefficient depending on LAD. The canopy transmittance model used by SunScan uses an ellipsoidal LAD model depending on the single parameter \(\chi\) to characterize LAD. Commonly, \(\chi\) is assumed to have equal unity corresponding to random distribution of leaf normals due to a lack of information on the actual LAD. In this study, we calculated \(\chi\) from the measured MTA using the following function derived from Equation (16) in [36]:  
\[\chi=3+\left(\frac{\text{MTA}}{553}\right)^{-0.6061}. \tag{2}\]  
For an ellipsoidal leaf angle distribution model, \(k(\theta,\chi)\) can now be calculated as [13]:  
\[k(\theta,\chi)=\frac{\sqrt{\chi^{2}+\tan^{2}(\theta)}}{\chi+1.774(\chi+1.182)^{ -0.733}}. \tag{3}\]  
The calculations were performed as follows. First, we determined MTA from the measured leaf angles. Next, we calculated the species-specific ellipsoidal leaf angle distribution parameter \(\chi\) from Equation (2) and two extinction coefficient values from Equation (3): \(k(\theta_{S},\chi)\) corresponding to the solar zenith angle \(\theta_{S}\) at the time of the SunScan measurements, and \(k(0,\chi)\) for the nadir direction. To obtain the LAI, we solved Equation (1) for LAI and used the SunScan-measured \(P(\theta_{S})\) and \(k(\theta_{S},\chi)\). Finally, the LAI for each plot was averaged from 4-5 sequential SunScan readings. We used the LAI value for each plot to calculate the fraction of the canopy cover as:  
\[\text{Fcover}=1-P(0), \tag{4}\]  
where the light transmission in the nadir direction, \(P(0)\), was calculated from Equation (2) with \(\theta=0\). Fcover can also be directly related to SunScan measurements. After making all the substitutions listed above, we can write an equation relating Fcover and SunScan-measured transmittance measurements:  
\[\text{Fcover}=1-P(\theta_{S})^{\frac{k(0,\chi)\cos\theta_{S}}{k(\theta_{S}, \chi)}}. \tag{5}\]  
A portable spectroradiometer (ASD Handheld, Boulder, CO, USA) with spectral range between 325 and 1075 nm and a spectral resolution of 3 nm was used to measure the bare soil reflectance in four sample plots in a harvested area on 7 October 2011. Bare soil reflected radiation was collected at a distance approximately 40 cm above the soil surface and a field of view of 25\({}^{\circ}\). Soil reflectance was calculated as the ratio of soil-reflected radiance to that measured above a calibrated white Spectralon panel (Labsphere Inc., Sutton, NH, USA). Due to the different solar illumination conditions between soil spectral and airborne spectral measurements, the measured soil reflectance was corrected with a bidirectional reflectance distribution function (BRDF) model developed by [PERSON] et al. [37] and modified by [PERSON] and [PERSON] [38]. The soil spectrum used for PROSAIL model simulation is presented in Figure 1.