## 2 Materials and Methods
### SIF Spectra Retrieval
#### 2.2.2 Reconstruction of Reflectance and SIF Spectra  
For full-spectrum SIF retrieval, the use of the SFM is necessary. According to the principle of the SFM, the key thing is to make an accurate estimation of the canopy reflectance and SIF spectra beforehand. However, for full-spectrum SIF retrieval, it is difficult to model the reflectance and SIF spectra accurately using simple mathematical functions, as was commonly done in earlier research. Inspired by earlier studies [31,32], PCA, a commonly used method for feature extraction, was employed in this study for the accurate estimation of both reflectance and SIF spectra. PCA is a statistical method that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called Principal Components (PCs), as described by Equation (5):  
\[\mathbf{\Phi}=\mathbf{X}\mathbf{W} \tag{5}\]  
where \(\mathbf{X}\) is the matrix representing the training dataset. In this case, the matrix has 2880 columns representing the 2880 samples of reflectance (or SIF) spectra in the simulated training dataset. The sample mean of each column is shifted to zero. \(\mathbf{\Phi}\) is the matrix representing the full set of PCs of the reflectance (or SIF); \(\mathbf{W}\) is a 2880-by-2880 matrix whose columns are the eigenvectors of \(\mathbf{X}^{\mathrm{T}}\mathbf{X}\). The reflectance and SIF can then be reconstructed using a linear combination of the first few PCs in \(\mathbf{\Phi}\), as described by Equations (6) and (7):  
\[\tilde{R}(\lambda)=\sum_{i=1}^{n_{r}}k_{i}\phi_{i}(\lambda) \tag{6}\]  
\[\tilde{SIF}(\lambda)=\sum_{i=1}^{n_{r}}j_{i}\varphi_{i}(\lambda) \tag{7}\]  
Where \(\tilde{R}(\lambda)\) and \(\tilde{SIF}(\lambda)\) are the reconstructed spectra of the reflectance and SIF respectively; \(\phi_{i}(\lambda)\) and \(\varphi_{i}(\lambda)\) are the \(i\)th PC of the reflectance and SIF, respectively; \(k_{i}\) and \(j_{i}\) are the weight coefficients of the \(i\)th PC; and \(n_{r}\) and \(n_{F}\) are the numbers of the reflectance and SIF PCs that were used, respectively. Based on the results of the study by [PERSON] and [PERSON] [32] and the results of the test with the simulated dataset, the first eight PCs of the reflectance and the first five PCs of the SIF, which account for about 99.999% of the spectral variance, were used in this study (Figure 2). The problem now turns to the estimation of \(k_{i}\) and \(j_{i}\). Figure 2: The first eight PCs of the reflectance (**a**) and first five PCs of the SIF (**b**) generated from the 2880 simulated spectra in the training dataset. The reflected radiation is the main component of the total upwelling radiance, especially in the atmospheric window region, where the contribution of the SIF is very small. The apparent reflectance \(\hat{R}\), (including the contribution of the SIF) outside the absorption bands can, therefore, be used to make an estimation of \(k_{{}_{i}}\), as described by Equation (8) [32]:  
\[\mathbf{k}=(\mathbf{\Phi}_{o}^{\mathrm{T}}\mathbf{\Phi}_{o})^{\mathrm{-1}}\mathbf{\Phi}_{o}^{ \mathrm{T}}\hat{\mathbf{R}}_{o} \tag{8}\]  
where \(\mathbf{k}=[k_{{}_{1}},k_{{}_{2}},...,k_{{}_{n}}]^{\mathrm{T}}\), \(\mathbf{\Phi}_{o}\) is the matrix consisting of the PCs (\(\phi_{i}(\lambda)\) ) without absorption bands, and (\(\mathbf{\Phi}_{o}^{\mathrm{T}}\mathbf{\Phi}_{o})^{\mathrm{-1}}\mathbf{\Phi}_{o}^{\mathrm{T}}\) is the pseudoinverse of the matrix \(\mathbf{\Phi}_{o}\). \(\hat{\mathbf{R}}_{o}\) is a vector consisting of the apparent reflectance (\(\hat{R}(\lambda)\) ) without absorption bands. In this way, the global shape of the reflectance spectra can be determined before the least-squares fitting for SIF retrieval is implemented, which should make the algorithm more robust.