## 3 Results
### Evaluation of the Modeled ERTPL Using Simulations  
Using the simulated irradiance and transmittance at different VZAs, the modeled ERTPL can be evaluated. It should be noted that the ERTPL model is based on the assumption that the surface is isotropic and that the transmittance is linearly related to the RTPL. However, in practice, these assumptions are not valid. First, for typical vegetation, both the reflectance and the SIF emission varies with the viewing and illumination directions [33]. Secondly, the relationship between the transmittance and RTPL can only be modeled by linear functions for a limited range of the view zenith angle. When the view zenith angle is close to 90\({}^{\circ}\), clearly this relation will become nonlinear (as Figure 4 shows). Therefore, in practice, the ERTPL will not be exactly as shown in Equation (9). Figure 5 shows the variation in the simulated up-welling radiance at top of canopy and atmospheric transmittance of typical vegetation (with an LAI of 4, chlorophyll content of 40 \(\upmu\)g/cm\({}^{2}\), and LIDF of spherical) inside the O\({}_{2}\)-A absorption band (761.1 nm) at different view zenith angles across the solar principal plane simulated by the SCOPE and MODTRAN 5 models. The directional characteristics of up-welling radiance at top of canopy is caused by the bidirectional reflectance effect of the canopy and the directional emission of SIF. The bidirectional reflectance effect of canopy has been widely studied, and has an obvious bowl-edge effect at the far-red band [33; 34]. The emission of SIF has also been proved to have similar directional distribution characteristics as reflectance for both observations and simulations [33]. This means that the up-welling radiance at top of canopy will increase as the view zenith angle increases, as shown in Figure 5. In contrast, at the oxygen absorption band, the atmospheric transmittance will fall as the view zenith angle (or RTPL) increases. In other words, the directional characteristics of the up-welling radiance and atmospheric transmittance have opposing influences on the observed upwelling radiance. Therefore, the errors caused by the two assumptions that were made in modeling the ERTPL of the hemispherical measurements will not offset each other to some extent (at least will not accumulate), and the accuracy of the modeled ERTPL value of 2H will be reasonable. Using the simulated values of \(L_{H}\) and \(L_{0}\) for different VZA and RAA, the irradiance observed by a hemispherical measurement system with cosine-corrected foreoptic at a height \(H\) (\(E_{cos}\)) above the canopy and at the top of canopy (\(E_{0}\)) can be calculated by integration using Equations (5) and (7). Hence, the equivalent atmospheric transmittance can be calculated as:  
Figure 5: The variation of the up-welling radiance at top of canopy and atmospheric transmittance at 761.1 nm with the view zenith angle simulated by SCOPE and MODTRAN 5 models (Cab = 40 \(\upmu\)g/cm\({}^{2}\), LAI = 4, LIDF is spherical). The spectral resolution is 0.3 nm, the view zenith angle is 0-89\({}^{\circ}\), the relative azimuth angle is 90\({}^{\circ}\) and the sensor height above the target surface is 20 m.  
\[\overline{T_{\uparrow}}=\frac{E_{\text{cos}}}{E_{0}}=\frac{\int_{0}^{2\pi}\int_{0}^{ \frac{\pi}{2}}L_{0}(\theta,\varphi)\cdot T_{\uparrow}(\theta)\cdot\cos\theta \cdot\sin\theta d\theta d\varphi}{\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}L_{0}( \theta,\varphi)\cdot\cos\theta\cdot\sin\theta d\theta d\varphi} \tag{10}\]  
For bands at atmospheric windows, the influence of atmospheric absorption is very weak. The purpose of this study was to analyze the effect of the atmospheric radiation transfer at the oxygen absorption bands on the SIF retrieval. Therefore, the O\({}_{2}\)-A band (centered at 761.1 nm in the simulated dataset), which is frequently used in SIF retrieval, was selected to evaluate the modeled ERTPL.