## 2 Materials and Methods
### Methods
#### 2.2.1 Temporal Expansion Model
By combining with sinusoidal model (see Equations (8) and (9)), the theoretical \(R_{nd}\) (\(R_{nd\_theoretical}\)) could be expressed as follows:  
\[\begin{split} R_{nd\_theoretical}&=\frac{1}{24}[LDt \cdot R_{nD}+(24-LDt)\cdot R_{nN}]\\ &=\frac{1}{24}\left[LDt\cdot\frac{\int_{t_{\text{rise}}}^{t_{ \text{rise}}}R_{n}(t)dt}{\int_{t_{\text{rise}}}^{t_{\text{set}}}dt}+(24-LDt) \cdot R_{nN}\right]\\ &=\frac{1}{24}\left[LDt\cdot\frac{2R_{nmax}}{\pi}+(24-  
However, after analysis, it was found that the value of the actual \(C_{d}\) was a little bit different from that of \(C_{d\_theoretical}\), and their difference was closely related to \(t\). Besides that, the ratio of \(R_{nN}\) and \(R_{n\_max}\) was related to \(LDt\) in the form of a quadratic function (as shown in Figure 3), so \(\frac{R_{nN}}{R_{n\_max}}\) could be calculated according to Equation (17a).  
Hence, after multiple experiments, the final \(C_{d}\) was expressed as follows:  
\[C_{d}(t)=c_{1}\bigg{[}\frac{LDt}{12\pi}+\bigg{(}1-\frac{LDt}{24}\bigg{)}\cdot \frac{R_{nN}}{R_{n\_max}}\bigg{]}\cdot\frac{1}{sin\Big{(}\Big{(}\frac{1}{2}+ \frac{t-t_{max}}{LDt}\Big{)}\pi\Big{)}}+c_{2}\cdot t+c_{3} \tag{17}\]  
\[\frac{R_{nN}}{R_{n\_max}}=d_{1}\cdot LDt^{2}+d_{2}\cdot LDt+d_{3} \tag{17a}\]  
where \(c_{1}\sim c_{3}\) and \(d_{1}\sim d_{3}\) are the empirical coefficients, and \(t_{max}\) is the time of \(R_{n\_max}\) defined as 12:30 hour by referring to [PERSON] et al. (2014). From Equation (17), the \(C_{d}\) for any specific location at any time could be obtained theoretically as long as the \(LDt\) (Equation (10)) was known. However, by considering the extremely unstable values of \(C_{d}\) around sunrise and sunset due to the very small \(R_{nii}\), as well as the transit times of most polar-orbiting satellites, the range of \(t\) in Equation (17) was defined as 9:30-14:30 hours at local time. Additionally, according to previous study ([PERSON] et al., 2017), the NDVI that is usually used to characterize the surface vegetation condition is closely related to \(R_{n}\), and the variations in \(R_{n}\) over the vegetated surface were different from that over the non-vegetated surface. Hence, it is suggested to apply the new \(C_{d}\) model for vegetated and non-vegetated surfaces separately. For assessment, the New Model, \(S\) model, and \(R\) model were trained or calibrated by first using the corresponding training samples (Table 3), respectively. Afterwards, the accuracy of their \(R_{nd}\) estimates, as well as the ones from the two existing models but with their original coefficients, were validated against their corresponding validation samples (Table 3). For better comparison, the three \(C_{d}\) models were inter-compared using the common validation samples (No. of samples = 517). If more than one \(R_{nd}\) estimate was obtained in one day, then the final \(R_{nd}\) was the average of all estimates. Table 3 lists the training and validation samples used for each case. Note that the samples were different for the three \(C_{d}\) models because of their different use requirements. Figure 3: (**a**) The scatter plots showing relationships between the ratio of \(R_{nN}\) and \(R_{n\_max}\) and \(LDt\), as well as (**b**) the corresponding boxplot. After that, the \(C_{d}\) estimated from New Model was applied to expand the \(R_{ni}\), which was estimated from the MODIS TOA data (Table 2) by implementing [PERSON]'s model into \(R_{nd}\) (namely \(R_{nd\text{-}cd\text{-}NM}\)). Meanwhile, the \(R_{ni}\) was also used with the sinusoidal method to obtain \(R_{nD}\), which was then used to calculate \(R_{nd}\) (namely \(R_{nd\text{-}SIN}\)) according to the empirical relation between \(R_{nD}\) and \(R_{nd}\). More details could be referred to [PERSON] et al. [11]. Afterwards, the two results were validated and compared.