## 3 Results
### Evaluation of the Modeled ERTPL Using Simulations
According to Equation (10), the equivalent atmospheric transmittance of the spectral observations with cosine-corrected foreoptic at a height above the canopy of 20 m is 0.924, and the corresponding ERTPL is 37.7 m (~1.89H), which is close to the modeled ERTPL of 2H. The accuracy of atmospheric correction of the hemispherical observation of irradiance at 761.1 nm (within the O\({}_{2}\)-A absorption band) at a height of H above the canopy with the ERTPL of 1.89H or 2H was evaluated by comparing with the simulated reference irradiance at top of canopy, as shown in Figure 6. The scatters of both corrected irradiance with ERTPL of 1.89H and 2H locate close to the 1:1 line, and the RRMSEs are 0.16% and 0.55%, respectively. The difference between the performance of ERTPL of 1.89H and 2H is quite tiny. The results indicate that the ERTPL of 2H modeled in this study is efficient for the atmospheric correction for tower-based hemispherical observation of up-welling irradiance. Performance of the Atmospheric Correction Using the Modeled ERTPL for SIF Retrieval from the Simulated Dataset  
As the fundamental objective of the analysis of the ERTPL of the hemispherical spectral observations was the retrieval of the SIF from tower-based observations, the retrieved SIF from the simulated spectral dataset with 80 different vegetation conditions (as described in Section 2.1) were employed to evaluate the accuracy of the modelled ERTPL. The Fraunhofer Line Discrimination (FLD) principle [35] is the mainly used methodology for SIF retrieval at canopy level. Several different FLD-based algorithms have been proposed and applied, such as the standard FLD [35], the 3-bands FLD (3 FLD) [36], the improved FLD (iFLD) [37], the principal components analysis based FLD (pFLD) [38], etc. Besides, some spectral fitting methods (SFM) were also proposed and have been proved to be more reliable for SIF retrieval from spectral data with relatively low spectral resolution and signal-to-noise ratio. According to the study by [PERSON] et al. [39], the  
Figure 6: Comparison of irradiance at 761.1 nm corrected using the ERTPL of 1.89H or 2H and the simulated reference irradiance at top of canopy. 3 FLD algorithm is robust and simple for SIF retrieval from data with spectral resolution of 0.3 nm, and only three spectral samples are needed. So the 3 FLD method was selected for the SIF retrieval from the simulated dataset (with no noise) in this study. Using the 3 FLD method, the SIF can be calculated as:  
\[SIF_{in}=\frac{\Big{(}I_{left}w_{left}+I_{right}w_{right}\Big{)}L_{in}-I_{in} \Big{(}L_{left}w_{left}+L_{right}w_{right}\Big{)}}{\Big{(}I_{left}w_{left}+I_{ right}w_{right}\Big{)}-I_{in}} \tag{11}\]  
where \(w\) is the weight of the band and is inversely proportion to the distance between the left-hand/right-hand band and the inner band; \(I\) is the downwelling irradiance arriving at the TOC; \(L\) is the total upwelling radiance at the TOC; and the subscripts \"_in_\", \"_left_\" and \"_right_\" refer to the bands inside, at the left of and at the right of the absorption band, respectively. SIF retrieved using radiation simulations at the top of the canopy (SIF\({}_{\text{TOC}}\)), tower-based SIF retrieved using original spectral simulations at a height of H (20 m) without atmospheric correction (SIF\({}_{\text{H}}\)), and tower-based SIF retrieved using atmospherically corrected spectra and RTPL values of H and 2H (SIF\({}_{\text{corr\_H}}\) and SIF\({}_{\text{corr\_2H}}\)) were compared with the reference values of the simulated SIF, as shown in Figure 7.  
As shown in Figure 7, all the SIF\({}_{\text{H}}\) values are negative and located far from the 1:1 line (with a relative root mean square error (RRMSE) of 293.79%), which means that the atmospheric correction was necessary for the tower-based SIF observations although the height of sensor was only 20 m. Compared to the SIF\({}_{\text{H}}\) values, the SIF\({}_{\text{corr\_H}}\) values are located much closer to the 1:1 line but the errors are still high (RRMSE = 133.71%).