## 3 Results
### New Model
#### 3.1.2 Models Inter-Comparison  
In order to evaluate objectively, the three \(C_{d}\) models were inter-compared against the common validation samples (No. = 517), and the results were further divided into vegetated and non-vegetated surfaces as shown in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Condition** & **c1** & **c2** & **c3** & **d1** & **d2** & **d3** \\ \hline NDVI \(\geq\) 0.1 & 0.9204 & \(-\)0.0052 & 0.0280 & \(-\)0.0039 & 0.1146 & \(-\)0.9468 \\ NDVI \(<\) 0.1 & 0.9041 & \(-\)0.0070 & 0.0519 & \(-\)0.0036 & 0.0939 & \(-\)0.7710 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Coefficients of the New Model for vegetated (NDVI \(\geq\) 0.1) and non-vegetated (NDVI \(<\) 0.1) surfaces, respectively.  
Figure 6: The validation accuracy of the estimated \(R_{nl-Cd-NM}\) by using the \(C_{d}\) from the New Model against (**a**) all validation samples, and the validation samples over (**b**) vegetated and (**c**) non-vegetated surfaces.  
Figure 7: The validation accuracy of (**a**) the \(R_{nl-Cd-SM}\) from the calibrated \(S\) Model, (**b**) the \(R_{nl-Cd-RM}\) from the calibrated \(R\) Model, and (**c**) the \(R_{nd-Cd-NM}\) from the New Model against the common validation samples. The red and blue points represent the estimates over the vegetated and non-vegetated surfaces, respectively.  
From Figure 7, the overall validation accuracy of the \(R_{nd\text{-}Cd\text{-}NM}\) from the New Model (Figure 7c) was the best with a RMSE of 14.20 Wm\({}^{-2}\) and Bias of 0.21 Wm\({}^{-2}\), followed by the \(R_{nd\text{-}Cd\text{-}SM}\) from \(S\) model (Figure 7a). The performance of the \(R_{nd\text{-}Cd\text{-}RM}\) from \(R\) model (Figure 7b) was the worst with its validated RMSE and Bias being larger than that of the \(R_{nd\text{-}Cd\text{-}NM}\) by 9.65 and 2.33 Wm\({}^{-2}\), respectively. In terms of the performance of the three models on the vegetated and non-vegetated surfaces, the three models worked generally better over the vegetated surface than that over the non-vegetated surface, yielding rRMSEs of 13.96% and 19.51% for \(S\) model, 17.23% and 26.80% for \(R\) model, and 10.44% and 15.50% for New Model, respectively. However, the tendency of underestimation for the vegetated surface was the most remarkable for \(S\) model (Bias = \(-\)7.07 Wm\({}^{-2}\)), and tendency of overestimation for the non-vegetated surface for \(S\) and \(R\) models with the corresponding Biases of 6.78 and 7.77 Wm\({}^{-2}\), respectively. Only the New Model has no tendency, yielding Biases of 0.27 and 0.11 Wm\({}^{-2}\).  
Furthermore, by considering New Model could work at any time, the \(R_{nd\text{-}Cd\text{-}NM}\) was estimated and validated by using the same validation samples of \(S\) and \(R\) model (Table 3) for further comparison. The results are provided in Figure 8.  
Figure 8a presents the validation results of the estimated \(R_{nd\text{-}Cd\text{-}NM}\) using the validation samples corresponding to \(S\) model. Compared with Figure 4, the overall validation RMSE (rRMSE) of New Model witnessed a reduction by 3.44 Wm\({}^{-2}\) (rRMSE = 3.27%), accompanied by a decrease in Bias amplitude by 1.02 Wm\({}^{-2}\). At the three specific moments, the Bias ranged from \(-\)0.54 to 0.27 Wm\({}^{-2}\) and rRMSE remained below \(\sim\)14.45%, which were both superior to that of \(S\) model. Regarding Figure 8b, the performance of New Model with the same validation samples of \(R\) model (Figure 5) was much better, yielding an R\({}^{2}\) value of 0.95, a Bias of 2.50 Wm\({}^{-2}\), and an RMSE (rRMSE) of 13.54 Wm\({}^{-2}\) (11.2%).  
To sum up, the New Model proposed in this study explored more about \(C_{d}\) and thus outperformed the existing models.