## 2 Materials and Methods
### ERTPL Modeling Based on Theoretical Derivation  
The RTPL for hemispherical observations made with cosine-corrected foreoptic is more complex than for conical observations made using a bare fiber. When the FOV is as large as \(180^{\circ}\), the ERTPL can no longer be estimated as being equal to the height of the sensor above the target surface. For spectral measurements made at the atmospheric windows, the influence of atmospheric radiation transfer on tower-based observations can be ignored as the height of the sensor above the target surface is usually only tens of meters. However, SIF observations are made at the atmospheric absorption bands and these are very sensitive to the RTPL. Therefore, for hemispherical observations, accurate modeling of the ERTPL is important. According to Equation (3), the upwelling hemispherical irradiance observed with the cosine-corrected foreoptic can be expressed as:  
\[E_{cos}=\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}L_{0}(\theta,\varphi)\cdot T_{ \uparrow}(\theta)\cdot\cos\theta\cdot\sin\theta d\theta d\varphi \tag{5}\]  
Assuming that the surface is homogeneous and isotropic, and ignoring the bidirectional reflectance effect of the surface and the directional characteristics of SIF emission (i.e., \(L_{0}\) is independent of \(\theta\) and \(\varphi\)), the observed upwelling irradiance can be expressed as:  
\[E_{cos}=L_{0}\cdot\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}T_{\uparrow}(\theta) \cdot cos(\theta)\cdot sin(\theta)d\theta d\varphi=2\pi L_{0}\cdot\int_{0}^{ \frac{\pi}{2}}T_{\uparrow}(\theta)\cdot cos(\theta)\cdot sin(\theta)d\theta \tag{6}\]  
On the other hand, the observed upwelling irradiance can be expressed as the product of the upwelling irradiance at top of canopy (\(E_{0}\)) and the equivalent transmittance between the canopy and the sensor (\(\overline{T_{\uparrow}}\)):  
Figure 2: Fractional contribution of the signal coming from each \(1^{\circ}\) interval of the view zenith angle to the hemispherical irradiance (red diamonds) together with the accumulated contribution of the signal within the FOV (blue circles). \[E_{\text{cos}}=E_{0}\cdot\overline{T_{\uparrow}}=\overline{T_{\uparrow}}\cdot\int_{0 }^{2\pi}\int_{0}^{\frac{\pi}{2}}L_{0}(\theta,\varphi)\cdot\cos\theta\cdot\sin \theta d\theta d\varphi=\overline{T_{\uparrow}}\cdot\pi L_{0} \tag{7}\]  
Combining Equations (6) and (7), the equivalent transmittance can be calculated as:  
\[\overline{T_{\uparrow}}=\frac{2\pi L_{0}\cdot\int_{0}^{\frac{\pi}{2}}T_{ \uparrow}(\theta)\cdot cos(\theta)\cdot sin(\theta)d\theta}{\pi L_{0}}=2\cdot \int_{0}^{\frac{\pi}{2}}T_{\uparrow}(\theta)\cdot cos(\theta)\cdot sin(\theta )d\theta \tag{8}\]  
For the oxygen absorption bands, the transmittance is mainly related to the RTPL. Figure 3 shows the variation of the transmittance of the up-welling radiation with RTPL, based on simulations made by MODTRAN 5 with different aerosol optical depth (AOD\({}_{550}\) is set as 0.1, 0.3, and 0.5). As there is no absorption by water and ozone at the wavelength of 761.1 nm we studied, the water vapor column is fixed as 3 g/cm\({}^{2}\), and the ozone column is fixed as 300 DU. The view zenith angle is set to the range 0-72\({}^{\circ}\) with a sensor height of 20 m above the target surface, and the corresponding range of the RTPL is about 20-65 m. The results show that the transmittance is approximately linearly related to the RTPL within the range we tested (with R\({}^{2}\) > 0.99). As shown in Figure 2, the signal within an FOV of 72\({}^{\circ}\) contributes more than 90% of the total hemispherical irradiance. So when estimating the equivalent transmittance for hemispherical observations, it is reasonable to assume a linear relation between \(T_{\uparrow}\) and RTPL when the atmospheric condition is fixed.