## 3 Results
### Validation
[49] revealed that the information that vegetation indices can provide is at the pixel scale, and the use of sub-pixel information (e.g., the area of shaded regions in a pixel) was found to greatly improve the accuracy of biomass estimates. Consequently, there may be a promising approach to enhance the reliability of the NDVI in satellite imagery for applications in urban areas, which entails the construction of a model that can determine the area of shaded regions and buildings by utilizing higher spatial resolution building density and height data for a given solar altitude angle and observation angle. This would then allow for the implementation of different levels of adjustment to the NDVI in urban areas for high-accuracy inversion of the FVC. Interestingly, our findings indicate that the NDVI is affected by the reflectance of building materials when shadow components are present within a mixed scene comprising heterogeneous buildings (Section 3.2). [PERSON] et al. [50] provided compelling evidence supporting our results by presenting the data of numerical experiments using the radiative transfer model Amartis v2 [51]. Their scenario was designed to reflect a heterogeneous and shaded building with walls and a roof, a configuration similar to some of our own scenario setups. In our scenes, we gave additional consideration to the absence of shadows and found that the heterogeneous building had essentially negligible effects on the NDVI under sunlit condition. However, further verification of this observation with data derived by other models, field measurements, and satellite data is required. The relationships between the urban geometry and background reflectance and the NDVI and FVC are proposed in this study. The relationships mentioned in our study are derived from a simple symmetric urban scene, assuming that the 3D urban geometry and background reflectance are the main factors controlling the NDVI. Indeed, the urban geometry within mixed pixels in actual urban scenes is often more complex. This study considered only one atmospheric condition. However, atmospheric conditions are variable, and the signals received by the sensors are often mixed with atmospheric effects and signals from adjacent pixels, which remain to be investigated. Likewise, the magnitudes of the building heights and densities selected in this paper are based on the general characteristics of typical urban scenes, with a maximum building height of 40 m. In some special urban clusters with ultra-high building heights, such as Hong Kong, the building heights generally exceed 100 m, with very large H/W values, which may necessitate further research. Despite the limitations of our experiments, our study has yielded valuable conclusions. By quantifying the geometry and background reflectance of a typical urban area by applying the DART model, we have explored the general pattern of the NDVI in urban areas and provided some useful suggestions for addressing the scientific challenge of \"how to estimate FVC using NDVI in urban areas\" (Section 3.3). This will assist future research in achieving a deeper understanding of the impact of the urban 3D environment on red and near-infrared bands' reflectance, as well as in the selection of urban FVC inversion methods when using space-borne multi-spectral data. Likewise, we found that the results obtained from the simulation data based on the DART model overlapped with many previous studies and were also the same as the satellite data validation results (Section 3.4), which further demonstrated the effectiveness of the model in simulating urban areas. We also discovered that many conclusions that are challenging to ascertain from satellite data can be derived from simulated data. The radiance of the mixed pixels received by a remote-imaging radiometer is influenced by a multitude of factors, which can distort the relationships between them. However, this issue can be mitigated to some extent by the use of simulated data, i.e., better controlled experiments.