## 2 Materials and Methods
### SIF Spectra Retrieval
#### 2.2.3 Spectral Fitting  
Using \(\tilde{SIF}(\lambda)\) to replace \(\tilde{SIF}(\lambda)\) in Equation (4), the total upwelling radiance at the top of canopy near the three absorption bands can be written as:  
\[L(\lambda)=\begin{cases}\dfrac{E(\lambda)}{\pi}(\alpha_{Ha}\tilde{R}(\lambda) +\beta_{Ha})+\sum_{i=1}^{n}j_{i}\varphi_{i}(\lambda),&\lambda\in[653,662]\\ \dfrac{E(\lambda)}{\pi}(\alpha_{B}\tilde{R}(\lambda)+\beta_{B})+\sum_{i=1}^{ n}j_{i}\varphi_{i}(\lambda),&\lambda\in[683,692]\\ \dfrac{E(\lambda)}{\pi}(\alpha_{A}\tilde{R}(\lambda)+\beta_{A})+\sum_{i=1}^{n} j_{i}\varphi_{i}(\lambda),&\lambda\in[757,771]\end{cases} \tag{10}\]  
The underdetermined coefficients \(\alpha\), \(\beta\) and \(j\) can be estimated with the least-squares fitting method. Although correction coefficients (\(\alpha\) and \(\beta\)) are employed, the accuracy of the estimated reflectance \(\tilde{R}(\lambda)\) still has an influence on the accuracy of the SIF retrieval. The difference between \(R(\lambda)\) and \(\tilde{R}(\lambda)\) is caused by the small contribution due to the SIF outside the absorption bands, the value of which is unknown beforehand. Inspired by the work of [PERSON] _et al._[29], here we introduce an iterative process to minimize the SIF contribution, as described below,  
1. Calculate the apparent reflectance (\(\hat{R}(\lambda)\) ) using the total upwelling radiance and downwelling irradiance.
2. Estimate the weights of the reflectance PCs using the apparent reflectance (\(\hat{R}(\lambda)\) ) without absorption bands according to Equation (8) and then reconstruct the reflectance \(\tilde{R}(\lambda)\) using the weights of the different reflectance PCs according to Equation (6).
3. Estimate the weights of the SIF PCs according to Equation (10) using the least-squares fitting method for the parts of the spectrum within the absorption bands and reconstruct the SIF spectrum \(\tilde{SIF}(\lambda)\) according to Equation (7).
4. Remove the estimated SIF spectrum \(\tilde{SIF}(\lambda)\) from the total upwelling radiance and then calculate a new apparent reflectance \(\hat{R}(\lambda)\) ; go back to steps (2) and (3) to reconstruct a new SIF spectrum \(\tilde{SIF}(\lambda)\).
5. Iterate until the reconstructed SIF spectrum \(\tilde{SIF}(\lambda)\) is stable and then let \(\tilde{SIF}(\lambda)\) equal \(\tilde{SIF}(\lambda)\).  
The efficiency of the iterative process was also investigated. Figure 3 shows the correlation between the true reflectance (\(R(\lambda)\)) and the reconstructed reflectance (\(\tilde{R}(\lambda)\) ) for the first three iterations. As a result of the iterative process, the accuracy of the estimated reflectance can be significantly improved. The Root Mean Square Error (RMSE) (integrated over the spectral range of each absorption band--the ranges are as indicated in Equaiton (10)) for the second run is only about 3\(-\)10 percent of that for the first run. For the second and third iterations, the difference between the true and reconstructed reflectance is very tiny--an RMSE of less than 0.06% for all three bands.  
Figure 4 shows the how the average Relative Root Mean Square Error (RRMSE) (integrated over the full spectral range of 645-805 nm) in the value of the SIF retrieved from the 2304 test datasets (with an SR of 0.3 nm and without noise) varies with the number of iterations. The results show that the RRMSE for the second run is significantly smaller than that for the first run. However, when the number of iterations is greater than 3, the RRMSE in the SIF becomes quite steady. Therefore, the iterative process is effective at improving the accuracy of full-spectrum SIF retrieval. In this study, we used three iterations because a higher number of iterations does not help to improve the accuracy and makes the algorithm less efficient.