## 3 Results
### Accuracy of Three FVC Estimation Methods  
Figure 6 illustrates the FVC difference (diff\({}_{\text{FVC}}\)) between the predefined FVC (FVC\({}_{\text{true}}\)) and the FVC retrieved with the three algorithms (Equations (3)-(5)) for four types of scenes:  
1. Homogeneous building under shadow conditions (R\({}_{\text{wall}}\) = R\({}_{\text{roof}}\), 10:00 am)
2. Homogeneous building under sunlit conditions (R\({}_{\text{wall}}\) = R\({}_{\text{roof}}\), 12:00 pm)
3. Heterogeneous building under shadow conditions (R\({}_{\text{wall}}\) \(\
eq\) R\({}_{\text{roof}}\), 10:00 am)
4. Heterogeneous building under sunlit conditions (R\({}_{\text{wall}}\) \(\
eq\) R\({}_{\text{roof}}\), 12:00 pm)  
Figure 6: Histograms of the FVC difference (diff\({}_{\text{FVC}}\)) between the predefined FVC\({}_{\text{true}}\) and the FVC estimates by three algorithms: (**a**) FVC estimates based on Equation (3); (**b**) FVC estimates based on Equation (4); and (**c**) FVC estimates based on Equation (5). The findings indicate that Equation (3) gave the least accurate estimates of the FVC, with the FVC typically exceeding the \(\mathrm{FVC_{true}}\). The impact of the building material heterogeneity on the FVC estimation accuracy is inconsequential under sunlit conditions. Under shadow conditions, the \(\mathrm{dif_{FVC}}\_1\) for heterogeneous buildings is predominantly concentrated between 0.1 and 0.2, whereas the \(\mathrm{dif_{FVC}}\_1\) for homogeneous buildings exhibits greater dispersion. Likewise, the FVC estimation accuracy for scenes under shadow conditions is markedly inferior to that for scenes under sunlit conditions, indicating that shadows result in a considerable overestimation of the FVC (Figure 6a). The FVC estimation accuracy based on Equation (4) is the highest, with minimal differences among the four types of scenes, indicating that the method is more robust and less susceptible to fluctuations in accuracy due to the diverse urban scenes (Figure 6b). The FVC estimation accuracy based on Equation (5) is moderate, with the smallest differences among the four types of scenes and a general underestimation of the FVC. However, the FVC estimation accuracy for scenes under shadow conditions is slightly higher than that for scenes under sunlit conditions, and there is minimal difference in the FVC estimation accuracy for homogeneous and heterogeneous buildings (Figure 6c). Table 6 demonstrates that \(\mathrm{FVC}\_2\) performs best (0.060 \(\leq\) RMSE \(\leq\) 0.077, \(-\)0.057 \(\leq\) BIAS \(\leq\) 0.018), with the exception of scenes with heterogeneous buildings under sunlit conditions, where \(\mathrm{FVC}\_1\) is the best algorithm. There is a minimal difference in accuracy between \(\mathrm{FVC}\_1\) (0.074 \(\leq\) RMSE \(\leq\) 0.163, 0.048 \(\leq\) BIAS \(\leq\) 0.148) and \(\mathrm{FVC}\_3\) (0.095 \(\leq\) RMSE \(\leq\) 0.146, \(-\)0.135 \(\leq\) BIAS \(\leq\) \(-\)0.074), but the difference in accuracy among the four types of scenes for \(\mathrm{FVC}\_1\) is significantly greater. Figure 7 illustrates the absolute value of the difference between the predefined \(\mathrm{FVC_{true}}\) and the FVC estimates, i.e., the (\(\mathrm{dif_{FVC}}\_1\), \(\mathrm{dif_{FVC}}\_2\), \(\mathrm{dif_{FVC}}\_3\), and \(\mid\mathrm{dif_{FVC}}\_1\), \(\mid\mathrm{dif_{FVC}}\_2\)\(\mid\), \(\mid\mathrm{dif_{FVC}}\_3\)\(\mid\)) defined in Section 2.2. A smaller \(\mid\mathrm{dif_{FVC}}\)\(\mid\) indicates a higher FVC estimation accuracy. The building density of the scenes, from left to right, increases in a sequential manner across all the intervals with equal H/W. The specific parameters of scenes \"S1-S25\" are shown in Figure A1.