## 4 Conclusions  
To address the issue of improving the estimation of \(R_{nd}\) from \(R_{ni}\) or the instantaneous satellite data, several temporal expansion methods have been proposed. In particular, an empirical parameter \(C_{d}\), which was defined as the ratio between \(R_{nd}\) and \(R_{ni}\), under clear sky used to be popular because of its simple format and easy implementation. Although \(C_{d}\) could be calculated from the existing models easily, the robustness of these models around the globe has not been well verified. In this study, a \(C_{d}\) estimation model, namely New Model, has been proposed. Being different from the existing models, New Model could obtain \(C_{d}\) at any time during 9:30-14:30 hours under clear sky, only depending on \(LDt\). Against the comprehensive measurements collected from 105 global sites, New Model and two commonly used existing models (\(S\) model and \(R\) model) were validated and compared. Afterwards, the performance of New Model was further analyzed under various conditions. Finally, the estimated \(C_{d}\) from New Model were applied on the \(R_{ni}\) computed from the MODIS TOA data with [PERSON]'s model to obtain \(R_{nd}\), which were also compared with the ones computed from the sinusoidal model. According to all results, several main conclusions could be drawn as follows:  
(1) After comprehensive evaluation, the New Model proposed in this study exhibited superior performance compared to other two existing models, with the \(R_{nd\_Cd\_NM}\) obtained from the New Model \(C_{d}\) estimates yielding the best accuracy with an overall validated R\({}^{2}\) of 0.95, RMSE of 14.07 Wm\({}^{-2}\), and Bias of \(-\)0.21 Wm\({}^{-2}\), followed by the \(R_{nd}\) calculated from \(S\) model and \(R\) model with their validation R\({}^{2}\) values of 0.91 and 0.89, RMSEs of 18.26 Wm\({}^{-2}\) and 21.36 Wm\({}^{-2}\), and Biases of \(-\)1.03 Wm\({}^{-2}\) and 1.33 Wm\({}^{-2}\), respectively. (2) Generally speaking, New Model exhibited robust performance under various cases in terms of time scales, moments, spatial distribution, and land cover types. However, New Model performed relatively better over the vegetated surface (i.e., DBF, EBF, DNF, and MF) than that over the non- or less-vegetated surface (i.e., GRA, OSH, and WET). Furthermore, New Model is suggested to be applied near noontime at a time scale within 60 min. (3) The two existing models are strongly suggested to be applied after calibrating with the comprehensive samples. After calibration, the estimation accuracy represented by \(R_{nd}\) of the two models could be improved significantly by decreasing the RMSE values by 19.59 and 23.00 wm\({}^{-2}\) and the magnitude of Bias by 30.53 wm\({}^{-2}\) and 38.26 wm\({}^{-2}\), respectively. Regarding the vegetated and non-vegetated surfaces, similar results were also observed for \(S\) and \(R\) models as that for New Model, but they both tended to be overestimated for non-vegetated surface. (4) The \(C_{d}\) from New Model has been successfully applied to expand the \(R_{ni}\) estimated from the MODIS TOA data into \(R_{nd}\), which had a much better accuracy than the ones from the same \(R_{ni}\) but using the sinusoidal model, yielding overall R\({}^{2}\) values of 0.88 and 0.84, RMSEs of 19.60 and 27.70 Wm\({}^{-2}\), and Bias of \(-\)0.76 and 8.88, respectively. In addition, the more \(R_{ni}\) estimates in one day, the more accurate of the final obtained \(R_{nd\_Cd\_NM}\). In summary, the proposed New Model could estimate \(C_{d}\) more accurately and conveniently. Therefore, this model has a strong potential to be used for \(R_{nd}\) estimation from the satellite data, especially the ones with low temporal resolutions, such as the Landsat data. However, so far, the new \(C_{d}\) model could be only applied during 9:30-14:30 hours at flat surfaces under clear sky; thus, the application of the new \(C_{d}\) model is limited.