# Effect of the Partitioning of Diffuse and Direct APAR on GPP Estimation
## 2 Materials and Methods
### Diffuse and Direct APAR (DDA)-Based Method for Half-Hourly GPP Estimation
\(\tau_{l_{i}}\) is the canopy directional transmittance at different times within a day per 30 min, and \(\tau_{l_{0}}\) is the transmittance at 10:30 a.m., which can be determined using the gap fraction model [54; 55; 56]:  
\[\tau_{t_{i}}=e^{-k\times G(\theta)\times LAI\times\frac{CI}{cos(\theta_{t_{i}})}} \tag{19}\]  
\[\tau_{l_{0}}=e^{-k\times G(\theta)\times LAI\times\frac{CI}{cos(\theta_{t_{0}})}} \tag{20}\]  
where \(k\) is the leaf extinction coefficient, \(G(\theta)\) is the projection of unit foliage area on the plane perpendicular to the sun incident direction, \(LAI\) is the leaf area index, \(CI\) is the clumping index, and \(\theta_{t_{i}}\) is the solar zenith angle (SZA) at different times within a day per 30 min, and \(\theta_{t_{0}}\) is the SZA value at 10:30 a.m. Here, \(G(\theta)\) was set using a measured leaf inclination angle (LIA) dataset [57] matched with average leaf inclination angle (ALA) of different leaf inclination angle distribution functions (LADF) [58] for calculating G values towards different vegetation types in the selected FLUXNET sites (Table 4). Then, according to the matched LADF, the G functions can be calculated as follows [59; 60]:  
\[G(\theta)=\int_{0}^{\pi/2}\varnothing(\theta,\theta_{L})f(\theta_{L})d\theta_ {L} \tag{21}\]  
\[\varnothing(\theta,\theta_{L})=\left\{\begin{array}{c}cos\theta cos\theta_{L},\ \ \ \ \ \  
Statistical metrics containing the determination coefficient (R2), root mean square error (RMSE), and a corrected RMSE (RMSE*) were calculated as follows:  
\[R^{2}=\left(\frac{\sum_{l=1}^{n}\bigl{(}GPP_{FLUX}-\overline{GPP_{FLUX}}\bigr{)} \bigl{(}GPP_{cal}-\overline{GPP_{cal}}\bigr{)}}{\sqrt{\sum_{i=1}^{n}\bigl{(}GPP _{FLUX}(i)-\overline{GPP_{FLUX}}\bigr{)}^{2}}\sqrt{\sum_{i=1}^{n}\bigl{(}GPP_{ cal}(i)-\overline{GPP_{cal}}\bigr{)}^{2}}}\right)^{2} \tag{23}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\bigl{(}GPP_{cal}(i)-GPP_{FLUX}(i)\bigr{)} ^{2}} \tag{24}\]  
\[RMSE^{*}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\bigl{(}(GPP_{cal}(i)-b)/a-GPP_{FLUX }(i)\bigr{)}^{2}} \tag{25}\]  
where \(GPP_{cal}\) is the half-hourly GPP retrieved using either the DDA method or the total APAR-based method; \(GPP_{FLUX}\) is the half-hourly GPP observations derived from the FLUXNET2015; the over-bars denote the mean values of different GPP items; and n is the number of samples. Constant a and b are the slope and intercept of the linear function of the estimated GPP and observed GPP, a corrected RMSE value (RMSE*) used here is intended for removing the influence of the scale difference between satellite data and in-situ data as well as the systematic error of LUE-based model in order to accurately express the performance of partitioning of diffuse and direct APAR in GPP estimations. In addition, a t test for the half-hourly GPP_DDA and GPP_TA was applied to evaluate the difference in partitioning the diffuse and direct APAR in the GPP estimation:  
\[Rt=\frac{\overline{x}-\overline{y}}{\sqrt{\frac{s_{x}^{2}}{n}+\frac{s_{y}^{2}} {m}}} \tag{26}\]  
where \(\overline{x}\) and \(\overline{y}\) are the mean value of samples; \(s_{x}\) and \(s_{y}\) are the standard deviations of the samples; \(n\) and \(m\) are number of samples. The \(t\) test returns a \(p\)-value to denote the probability that the observed test statistic is as extreme as or more extreme than the observed value under the null hypothesis, which represents the discrepancy between different data sets. The significance level was set as 0.05.