## 1 Introduction  
FPAR is the fraction of the incoming absorbed photosynthetically-active radiation (PAR) that a plant canopy absorbs for photosynthesis and growth in the 0.4-0.7 nm spectral range [1]. FPAR is a very important parameter in calculating gross primary productivity (GPP) using the light-use efficiency model. It is also used to estimate the assimilation of carbon dioxide by vegetation. FPAR can also be used as an indicator of the state and evolution of vegetation cover; this function can advantageously replace the normalized Difference Vegetation Index (NDVI). The FPAR inversion methods generally use the relationship between FPAR and NDVI [2] or between FPAR and the Enhanced Vegetation Index (EVI) [3]. In a certain range, there is a linear relationship between FPAR and NDVI [4, 5]. In the Carnegie-Ames-Stanford approach (CASA) model, for example, FPAR is calculated using the NDVI linear stretching model [5, 6]. [PERSON] _et al._ used an empirical relationship based on NDVI as a backup algorithm for MODIS FPAR product derivation (MOD15) [1]. In the vegetation photosynthesis model (VPM) [7, 8], FPAR is a function of EVI, but the difference between EVI and NDVI increases the blue band reflectivity in the EVI calculation. These empirical functions are mostly based on measured data or from the regression of large amounts of historical data. This may possibly indicate that a function gives good results only at a local scale. It has been a difficult issue to find the relationship for large-area FPAR inversion using remote sensing data. [PERSON] _et al._ studied the relationship of FPAR changes with soil, vegetation canopy and atmospheric parameters [6]. The relationship of FPAR and NDVI is independent of pixel heterogeneity, but sensitive to background characteristics and atmospheric and bidirectional reflectance. If the study focusses on the nadir, the influence of atmospheric and bidirectional reflectance can be ignored, even if the soil is moderately reflective and the background impact is also negligible. Therefore, [PERSON] _et al._ believe that the conditions for a linear relationship between FPAR and NDVI are: a solar zenith angle less than 60\({}^{\circ}\), a focus on the nadir, a soil background of medium brightness and an atmospheric optical thickness at 550 nm of less than 0.65 [9]. In addition, NDVI is also affected by chlorophyll content. [PERSON] _et al._ found that NDVI is sensitive to the chlorophyll content of the canopy and the bottom leaves [10]. Therefore, establishing the conditions of the relationships between FPAR and all kinds of vegetation indices is very challenging. This kind of method is not universal and, coupled with the NDVI saturation phenomenon, will reduce FPAR inversion accuracy. On the other hand, some researchers have been trying to find out an FPAR expression based on the radiative transfer mechanism. The MODIS FPAR algorithm describes canopy spectral and directional characteristics based on the three-dimensional radiative transfer model [6]. [PERSON] _et al._ also derived an expression between FPAR, surface albedo, canopy gap probability and LAI based on radiative transfer theory [11]. [PERSON] obtained FPAR expressions combining the radiative transfer model and a geometrical optics model [12]. [PERSON] _et al._ developed a new FAPAR-P model, which considers the effects of canopy structures, ambient radiation and multiple scattering between soil and vegetation [13]. This kind of method has a clear physical mechanism and is more universal than empirical models. PAR on the top of the vegetation canopy can be classified into direct radiation and diffuse radiation. These two parts have different irradiation paths and intensities. Moreover, the canopy absorption of the two parts is not the same, which affects vegetation canopy GPP.