## Materials and Methods
### Two-Leaf PRI Model
#### 2.3.2 Estimations of P\({}_{t}\) and P\({}_{s}\)  
According to Equation (8), the sum of the fractions of sunlit and shaded leaves and sunlit and shaded backgrounds is unit, thus the fraction of shaded leaves (\(P_{S}\)) was:  
\[P_{S}=1-P_{T}-P_{VG} \tag{10}\]  
Estimation of canopy \(P_{T}\) was based on both canopy and leaf reflectances presented by [PERSON] et al. [73]. The ratio of the observed canopy reflectance (R\({}_{can}\)) to leaf reflectance (R\({}_{leaf}\)) at 670 nm (the multi-scattering and the contribution from background in this band is the lowest), defined as R\({}_{SL}\) was taken to be approximately equal to the degree to which the canopy was exposed to radiation. Determination of the observed \(P_{T}\) at each view angle was tested on two different strategies of whether or not considering the contribution to canopy reflectance from shaded leaves, which were only exposed to diffuse radiation. A hybrid geometric-optical model (GOST) with ray tracing method was also tested to estimate \(P_{T}\), which was also treated as a reference to exam \(P_{T}\) calculated from the other two methods, as GOST was a novel model to simulate the fractions of the four components of a canopy presented in equation (8). _Model I:_ Treating \(P_{T}\) as the fraction of the canopy exposed to total radiation. According to the definition of sunlit leaf, a substantial portion of penumbra region was treated as sunlit leaves. Hypothesizing this part of penumbra region suffered the same level of light stress on photosynthesis as sunlit leaves, without separating direct and diffuse radiation, \(P_{T}\) in this model (\(P_{T,I}\)) was equal to \(R_{\mathrm{SL}}\):  
\[P_{T,I}=R_{SL} \tag{11}\]  
This model could represent the total degree of leaves exposed to radiation. It satisfied the assumption that photosynthesis often saturates under high radiation while it was supposed to be non-saturate under diffuse radiation. However, as the canopy reflectance was composed with reflectances of both sunlit and shaded leaves, \(R_{\mathrm{SL}}\) might include the contribution from shaded leaves of a part of penumbra region. Therefore, this model was hypothesized overestimating the real \(P_{T}\). _Model II:_ Treating \(P_{T}\) as the fraction of the canopy exposed to total radiation subtracted by the fraction of shaded leaves exposed to diffuse radiation. Hypothesizing a substantial portion of penumbra region suffered a much lower level of light stress than sunlit leaves and similar to shaded leaves, we further hypothesized that the direct radiation contributes to the light stress while the diffuse radiation cause none light stress considering the penumbra effect. Therefore, \(P_{T}\) in this model (\(P_{T,II}\)) was the degree of the canopy exposed to direct radiation, and could be treated as the ratio of canopy reflectance from direct incident radiation (instead of total incident radiation) to leaf reflectance. With the ratio of diffuse radiation to total radiation (\(R_{\mathrm{{\it dif}}}\)), the fraction of shaded leaves was subtracted from \(R_{\mathrm{SL}}\) to calculate \(P_{T,II}\). The canopy reflectance come from direct and diffuse radiation, can be expressed as:  
This model also satisfied the definition that sunlit leaves absorbed both direct and diffuse radiation while shaded leaves only absorbed diffuse radiation. However, as the real sunlit and shaded leaves were not always easy to distinguish, this model treated a substantial portion of penumbra region as shaded leaves, hence we hypothesized that it may over-calculate the fraction of shaded leaves and hence underestimate the real \(P_{T}\). _Model III:_ Modeling \(P_{T}\) using a hybrid geometric-optical model with ray tracing method for sloping terrains (GOST) [71; 72; 79]. In the geometric-optical model for sloping terrains (GOST), the shape of trees was simplified as adiactinic cylinder and cone, and there was no penumbra region within the canopy. It assumed a Neyman distribution, also called double-Poisson distribution, of the trees.