# Modeling Shadow with Voxel-Based Trees for Sentinel-2 Reflectance Simulation in Tropical Rainforest
## 2 Methodology
### Description of Canopy-Level Shadow and Reflectance Simulator  
Various simulation models have been developed to understand the relationships between the reflectance which the satellite sensor observes and forest structures (e.g., the Discrete Anisotropic Radiative Transfer Model (DART) [27], Forest Light Environmental Simulator (FLiES) [28], and Radiosity-Graphics combined Method (RGM) [29]). While the accuracy of simulated reflectance obtained from those simulators is high, it is necessary to provide parameters obtained through field measurements. The simulator required in this study considers the shadow fraction caused by canopy shapes of the virtual forest. Then, we focused on the simulation model which uses shadows caused by trees as the shielding ratios of direct and diffuse solar irradiance reaching the forest developed by [PERSON] and [PERSON] (2021) [23]. Therefore, we developed CSRS ([[https://github.com/Takumi-Fuji693](https://github.com/Takumi-Fuji693)]([https://github.com/Takumi-Fuji693](https://github.com/Takumi-Fuji693)) 6/CSRS.git, accessed on 29 June 2022), which incorporates that simulation model into a virtual forest. CSRS has the following three advantages. (1) The canopy shape can be freely changed as long as it is geometrically representable, (2) compared to existing simulators, CSRS requires fewer parameters, and (3) the irradiance reaching the virtual forest at an arbitrary point in time can be calculated at the voxel level by considering the shadows of the trees.  
The scale of the virtual forest was 50 m \(\times\) 50 m, as shown in Figure 5. Individual trees in the virtual forest were represented by voxel models with a size of 0.5 m. After the virtual forest's creation, Cast Shadow (CS) and Self Cast Shadow (SCS), which correspond to the shielding ratios of direct and diffuse solar irradiance, were calculated for each voxel. The details on the calculation methods of CS and SCS can be found in [PERSON] and Takeuchi [30]. Reflectance was then calculated for each voxel according to Equations (1)-(3). Since the lateral boundaries of the forest scene were not taken into account, voxels within a central 40 m \(\times\) 40 m area were used for computing reflectance. With a target pixel size of 10 m, 16 pixels could be obtained from a single forest scene. In this study, three virtual forest scenes were created by changing the tree positions randomly, so 48 pixels of reflectance were calculated per canopy shape.  
\[I(\lambda)=\{F_{dir}(\lambda)(1-CS)+F_{dif}(\lambda)(1-SCS)\}\frac{\rho( \lambda)}{\pi} \tag{1}\]  
\[I_{sensor}(\lambda)=\frac{\int I(\lambda)SRF(\lambda)d\lambda}{\int SRF( \lambda)d\lambda} \tag{2}\]  
\[r(\lambda)=\frac{\pi I_{sensor}}{F_{dir}(\lambda)+F_{dif}(\lambda)} \tag{3}\]  
where \(I(\lambda)\) is radiance from a voxel considering \(CS\) and \(SCS\) (W/m\({}^{2}\)/sr/\(\upmu\)m); \(\lambda\) is the wavelength; \(F_{dir}(\lambda)\) is direct horizontal irradiance (W/m\({}^{2}\)/\(\upmu\)m); \(F_{dif}(\lambda)\) is diffuse horizontal  
Figure 4: Canopy shapes used for the virtual forest. The canopy structures are tree height (TH), canopy length (CL), canopy radius (CR) and diameter at breast height (DBH).  
irradiance (W/m\({}^{2}\)/\(\upmu\)m); \(\rho(\lambda)\) is spectral reflectance of the over- and understory; \(I_{sensor}(\lambda)\) is radiance which the sensor observes (W/m\({}^{2}\)/sr/\(\upmu\)m); \(SRF(\lambda)\) is the spectral response function of the target sensor; \(r(\lambda)\) is reflectance observed by the satellite.