## 1 Introduction  
Accurate estimation of the amount of carbon dioxide fixed by vegetation photosynthesis is a key component of the global carbon cycle and is also important for studies on ecosystem-climate interactions and ecosystem responses to extreme climate events [1; 2]. There have been many approaches to estimating GPP (gross primary production) including the eddy covariance (EC) technique [3; 4; 5; 6], process-based models [7; 8; 9] and LUE-models (light use efficiency models)based on the absorbed photosynthetically active radiation (APAR) absorbed by the canopy [10; 11] or by green leaves [12; 13]. However, the solar-induced chlorophyll fluorescence (SIF) emitted from vegetation chloroplasts is a more promising proxy for GPP than vegetation indices derived from reflectance data [14; 15]. Recent instrumental developments in hyperspectral remote sensing make it possible to measure SIF using field spectrometers at ground level [16], airborne imaging spectrometers [17], and also satellite sensors including the high-spectral-resolution Japanese Greenhouse gases Observing SATellite (GOSAT) [18; 19; 20; 21] and Orbiting Carbon Observatory 2 (OCO-2) [22] as well as the Scanning Imaging Absorption spectrometer for Atmospheric CHartograhY (SCIAMACHY) [23; 24] and the Global Ozone Monitoring Experiment 2 (GOME-2) [25; 26]. The simple linear response of the SIF to GPP at the large scale has been considered by many researchers. Based on the satellite-based Vegetation Photosynthesis Model (VPM) proposed by [PERSON] et al. [12], which includes the light use efficiency (LUE, \(\varepsilon_{P}\)), the GPP can be calculated using Equation (1):  
\[\mathrm{GPP}\ =\ \mathrm{PAR}\times\mathrm{fPAR}_{chl}\times\varepsilon_{P}\ =\ \mathrm{APAR}_{chl}\times\varepsilon_{P} \tag{1}\]  
where PAR is the incident irradiance, \(\mathrm{fPAR}_{chl}\) is the fraction of absorbed photosynthetically active radiation absorbed by chlorophyll, \(\varepsilon_{P}\) is the light use efficiency and \(\mathrm{APAR}_{chl}\) is the absorbed photosynthetically active radiation absorbed by chlorophyll [12; 27; 28]. Similarly, the total SIF originally emitted at the photosystem level (\(\mathrm{SIF}_{PS}\)), a by-product of photosynthesis, can be expressed as  
\[\mathrm{SIF}_{PS}\ =\ \mathrm{APAR}_{chl}\times\varepsilon_{F} \tag{2}\]  
where \(\varepsilon_{F}\) is defined as the quantum yield for fluorescence (i.e., the fraction of APAR photons that are re-emitted from the photosystems as SIF photons). In this context, based on Equations (1) and (2), the GPP can be directly estimated using a linear model based on \(\mathrm{SIF}_{PS}\). However, the remotely sensed SIF is only part of total SIF emitted from the chloroplasts within the leaves and is also not isotropic due to the upward radiation transfer path. As demonstrated by numerous observations and simulation results, the remotely sensed SIF at the canopy level (\(\mathrm{SIF}_{canopy}\)) has a noticeable directional variation similar to that of the canopy reflectance in the solar principal plane [29]. In addition, the remotely sensed \(\mathrm{SIF}_{canopy}\) is strongly influenced by the effects of leaf and canopy structure, which affects the fluorescence reabsorption and the light penetration within the plant canopy [30]. Therefore, the remotely sensed \(\mathrm{SIF}_{canopy}\) can be similarly expressed as [14; 28]  
\[\mathrm{SIF}_{canopy}\ =\ \mathrm{APAR}_{chl}\times\varepsilon_{F}\times f_{ esc} \tag{3}\]  
where \(f_{esc}\) is a parameter related to the optical properties of the canopy, which accounts for the fraction of SIF photons escaping from the photosynthesis level to the canopy level. It is determined by the canopy structure, the pigments that are present, the soil background, the illumination and the imaging geometries. Therefore, if the ratio of \(\varepsilon_{p}\) to \(\varepsilon_{F}\) is constant, there should be a strong linear relationship between GPP and \(\text{SIF}_{canopy}\) for a given vegetation type.