## 1 Introduction  
Gross primary production (GPP) is defined as the total amount of carbon dioxide that is fixed by plants into organic matter through the process of photosynthesis over a given period, as part of the energy radiating to the canopy [1, 2]. The light use efficiency (LUE) model is one of the major models frequently employed to estimate GPP [3]. The fraction of absorbed photosynthetically active radiation (FAPAR) is a fundamental input variable for the LUE model, which is used to calculate GPP [2]. Consequently, precise estimation of FAPAR is crucial for the accurate simulation of GPP [4]. FAPAR can be obtained through direct measurements using various instruments. One such method involves calculating the photosynthetically active radiation (PAR) flux by measuring it in four directions using PAR sensors, such as those carried by AccuPAR and SunScan [5, 6, 7]. An alternative approach involves calculating the total light transmission or gap fraction using optical instruments like LAI-2000, TRAC, DEMON, AccuPAR, and SunScan [8]. These instruments have been used to calculate canopy light interception, with measurements obtained above and below the canopy using fisheye optical sensors or probes. Implementing such methods on a large regional scale is challenging, and they are predominantly used for validating regional scale FAPAR products. Large-scale regional FAPAR is primarily obtained by remote sensing techniques, with the predominant methods including the inversion of a vegetation radiative transfer model [2], statistical model based on vegetation indices [9, 10, 11, 12], and the [PERSON]'s law calculation model based on leaf area index (LAI) [13, 14, 15, 16, 17]. The vegetation radiative transfer model inversion method establishes parameters such as observation geometry, leaf optical properties, and canopy structure to simulate the propagation process of solar radiation by modeling the reflectance of the vegetation canopy [18, 19], and ultimately determines the FAPAR. The Moderate Resolution Imaging Spectroradiometer (MODIS) LAI/FPAR products, most widely used by LUE-GPP models, are based on the vegetation radiative transfer model [2]. However, previous ground-based validation results have shown that the MODIS LAI/FPAR products have an overestimation problem, especially at the beginning and end of the growth period [20, 21, 22, 23, 24, 25, 26]. Statistical models based on vegetation indices are also commonly used to calculate the FAPAR. These models are characterized by ease of operation, fewer parameters and variables, and efficiency [27, 28, 29, 30]. However, statistical models require a substantial number of in situ measured samples, and there are inherent limitations to the extrapolation capability of statistical models, which restricts their application to large regional scales. Furthermore, some vegetation indices are influenced by the soil background or become saturated under high fractional vegetation coverage with some vegetation types [31, 32, 33, 34], which also affects the application of statistical models. According to the Beer-Lambert's law, the FAPAR can be calculated using the LAI [35]. Consequently, FAPAR can be calculated based on the MODIS LAI products using the Beer-Lambert's law [36]. FAPAR calculated by this method can effectively improve the estimation of GPP based on the LUE-GPP model compared to the MODIS GPP products, especially for agricultural and grassland ecosystems [37, 38]. A growing body of research has adopted this approach, employing the [PERSON]'s law to directly estimate GPP from LAI datasets, bypassing the use of MODIS FPAR products [14, 15, 16, 17]. However, for the [PERSON]'s law to calculate FAPAR via LAI, the underlying assumption is the homogeneous turbid medium assumption of the vegetation canopy [39, 40, 41]. Given that this assumption delineates the propagation of uncollided photons in a homogeneous turbid medium, its application to the transmission of light through a heterogeneous vegetation gap is precluded.