## 3 Results
### Validation  
Table 7 and Figure 8 illustrate three FVC algorithms, which collectively demonstrate a significant improvement in the FVC estimation accuracy through the application of Equation (9) for correction. The accuracy rankings of the three FVC algorithms in our study area are as follows: FVC_3, FVC_2, and FVC_1 (Table 7). To validate the results in Section 3.3, the \"true FVC\" values (refer to Section 2.4) were classified into four intervals: 0-0.1, 0.1-0.2, 0.2-0.3, and greater than 0.3, respectively (Table 8). Regardless of the algorithm employed to calculate the \"true FVC\", a greater than 50% pixel count within the \"true FVC\" range of 0-0.1 indicates low overall vegetation cover in the study area. This is the same as the simulation results in Section 3.3, \"The accuracy of the three algorithms performs as FVC_3 > FVC_2 > FVC_1 when 0 < FVC \(\leq\) 0.1\" (Figure 7a). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & FVC\_1 & FVC\_2 & FVC\_3 \\ \hline \multirow{2}{*}{Before Correction} & RMSE & 0.249 & 0.183 & 0.169 \\  & BIAS & 0.205 & 0.144 & 0.098 \\ \multirow{2}{*}{After Correction} & RMSE & 0.164 & 0.114 & 0.087 \\  & BIAS & 0.119 & 0.080 & 0.040 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Accuracy of different FVC estimation algorithms before and after correction using Equation (9). (FVC), leaf area index (LAI), chlorophyll content, gross primary productivity (GPP), and fraction of absorbed photosynthetically active radiation (FAPAR) [46]. It is affected by the background reflectance, solar radiation and observation geometry, since these factors alter the reflectance, and by the bidirectional reflectance distribution function (BRDF) [46]. The urban geometry and materials are highly heterogeneous and may also affect the reflectance in different bands. To address this issue, it is important to consider the effects of the urban geometry and background reflectance on the NDVI. In our study, we examined the impact of the urban geometry and background materials on the NDVI and its application in estimating the vegetation fractional abundance in urban areas. Experiments were carried out by applying the DART model to a large set of synthetic urban scenes to calculate the NDVI and FVC separately. We attempted to interpret the results in Section 3.1 by sampling pixels (Figure 9) and comparing the NDVI values of the components (Figure 10) within mixed scenes. Figure 9c shows a schematic of the sampled pixels, which were randomly selected from a mixed scene. The reflectance in the red and near-infrared bands was taken at 10 am (i.e., shaded) and 12 pm (i.e., unshaded) for these sampling pixels. The findings indicate that shadows have a considerable impact on the NDVI, as evidenced by the presence of elevated NDVI values in the shaded regions. Sunlit areas mainly receive direct solar radiance, while shaded areas receive diffuse solar radiation from the sky and scattered radiation from adjacent pixels. Due to the absence of direct solar radiance in shaded areas, the reflectance in the red and near-infrared bands decreases significantly, the reflectance in the red band decreases faster (Figure 9a), and the ratio of \((\rho_{\text{NIR}}-\rho_{\text{Red}})/(\rho_{\text{NIR}}+\rho_{\text{Red}})\) increases (Figure 9b). The variable k in Figure 9b represents the slope of the line connecting each data point with the origin and the NDVI value of the pixel. A straightforward calculation reveals that for pixels observed from 12 pm to 10 am, the average \(\rho_{\text{NIR}}+\rho_{\text{Red}}\) value decreases from 0.597 to 0.139 and the average \(\rho_{\text{NIR}}\)-\(\rho_{\text{Red}}\) value decreases from 0.206 to 0.094 (Figure 9b). The sum value of the two bands decreases at a faster rate than the difference value. The NDVI values for pixels at 12 pm range from 0.216 to 0.906, while for pixels at 10 am, they range from 0.288 to 0.921.