## 1 Introduction
Several studies used the NDVI to capture the urban vegetation cover condition [25; 26], but the complex urban geometry in compact urban areas affects the solar radiation and radiative transfer, i.e., it modifies the inherent spectro-directional signature of the observed targets, including the red and near-infrared reflectance applied to obtain the NDVI. However, previous studies have not elucidated the manner in which the urban 3D geometry and building materials affect the NDVI of mixed pixels. The applicability of multiple NDVI-based FVCestimation algorithms for diverse urban contexts has yet to be assessed, which is crucial to improve the urban FVC estimation accuracy. This study is based on numerical experiments with the discrete anisotropic radiative transfer (DART) model to construct urban simulation scenes with varying geometrical and material features, which allows us to explore the relationship between the 3D geometry and material reflectance and the NDVI of mixed pixels. The DART model is one of the most comprehensive physically based 3D models of Earth-atmosphere radiative transfer, covering the spectral domain from ultraviolet to thermal infrared wavelengths. It simulates the optical 3D radiative budget (RB) and can be applied to calculate the optical signals captured by proximal, aerial, and satellite imaging spectroradiometers and laser scanners for any urban and/or natural landscapes and for any experimental and instrumental configurations [27]. The DART model has been validated through a series of RAMI (RAdiation transfer Model Intercomparison) experiments [28; 29; 30; 31; 32], and its forward simulation of vegetation reflectance has also been rigorously verified by real measurements [33]. [PERSON] et al. [34] have conducted an extensive evaluation of the DART model's calibration accuracy for the iterative inversion of shortwave satellite images, elucidating its potential as an instrumental tool for obtaining spectral signature maps of urban materials. This means the DART model can handle the shortwave radiative transfer in urban areas. The red and near-infrared bands employed to calculate the key variable (i.e., NDVI) in our study are encompassed within the shortwave spectral domain. Likewise, the utility of the DART model has been substantiated across scientific realms, and its iterative enhancements and extensive applications have been demonstrated in detail by [PERSON] et al. [35]. These include the development of remote-sensing inversion techniques and the design of satellite sensors and novel vegetation indices. [PERSON] et al. [36] used DART to model the longwave radiation in a complex-relief terrain. [PERSON] et al. [16] used DART to model the urban radiative transfer to retrieve the effective emissivity. [PERSON] et al. [37] devised an innovative approach to produce detailed modeling of the mean radiant temperature (T\({}_{\text{mrt}}\)) spatial distribution using the DART model. The effectiveness of this methodology was substantiated through a case study in Singapore, a tropical city, where the DART-modeled T\({}_{\text{mrt}}\) was compared with field-estimated values. This means the DART model can handle the radiative transfer in any terrain; therefore, it is a promising candidate for developing and enhancing inversion methods in urban areas with vegetation and buildings. In this study, the research questions are as follows. (1) How does the urban geometry and materials affect the NDVI in urban areas? (2) How should the NDVI be applied to estimate the vegetation fractional abundance in urban areas? Section 2 describes the model used in this study, the simulation setup for the scenes, and the formulas used. Section 3 presents a compilation and analysis of the experimental results, assessing the effects of different factors on the NDVI and FVC, as well as the model validation. Comments on the results are articulated in Section 4 and conclusions will be presented in Section 5.