## 2 Materials and Methods
### Method  
The data-driven PCA algorithm regards the radiance signals (\(L_{TOA}\)) as the sum of non-fluorescence and fluorescence as follows:  
\[L_{TOA}=I_{sol}.\frac{\mu_{0}}{\pi}.\left[\rho_{0}+\frac{\rho_{s}\cdot T_{ \uparrow\downarrow}}{1-S\cdot\rho_{s}}\right]+\frac{F_{s}\cdot T_{\uparrow}}{ 1-S\cdot\rho_{s}} \tag{1}\]  
where \(I_{sol}\) is the sun irradiance from the top of the atmosphere (TOA), \(\mu_{0}\) is the cosine of the sun's zenith angle, \(\rho_{0}\) refers to the atmospheric backscattering, \(S\) is the reflection from the hemisphere, \(\rho_{s}\) is the surface reflectance, \(T_{\uparrow}\) is the atmospheric upward transmittance, and \(F_{s}\) is the fluorescence signals, which are our final demands. The first term on the right side in Equation (1) is the non-fluorescent spectra, and the second term is the fluorescent spectra. \begin{table}
\begin{tabular}{c c} \hline \hline  & **Hyperspectral Data** \\ \hline Study area & Danzhou, Hainan Province, China \\ Spatial resolution & \(1\) m \(\times\) 1 m \\ Spectral resolution & 0.11 nm \\ Spectral band number & 1004 \\ Spectral regions & 669.84 nm\(-\)780.32 nm \\ Acquiring time & 25 May 2020 \\ Data format & BIL \\ Data unit & Radiance (mW/cm\({}^{2}\cdot\)str\(\cdot\)um) \(\times\) 1000.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: An introduction of the study area and data. The non-fluorescence can be decomposed into high-frequency signals (the absorption of Earth's atmosphere through solar radiation's descent) and low-frequency signals (including \(\rho_{0}\), \(S\), and \(\rho_{s}\)). The low-frequency signals can be characterized as a low-order polynomial function of the wavelength (\(\lambda\)), while the high-frequency term can be regarded as a linear formula of the non-fluorescent characteristics. Therefore, the PCA retrieval method was used to extract features from the training samples, which means that the first term of Equation (1) can be expressed as  
\[I_{sol}.\frac{\mu_{0}}{\pi}\cdot\left[\rho_{0}+\frac{\rho_{s}\cdot T_{\uparrow \downarrow}}{1-S\cdot\rho_{s}}\right]\approx I_{sol}.\frac{\mu_{0}}{\pi}\sum_{ i=0}^{n_{p}}\left(\alpha_{i}\cdot\lambda^{i}\right)\cdot\sum_{j=1}^{n_{p}}\left( \beta_{j}\cdot PC_{j}\right) \tag{2}\]  
where \(PC_{j}\) is the principal component vectors, \(\beta_{j}\) is the coefficient of the principal component, \(n_{p}\) is the order of the polynomial, and \(n_{pc}\) is the number of PCs. On the other hand, the fluorescence spectra could be composed of the \(F_{s}\) fluorescence signals and the high-frequency signals in the atmospheric upstream. The latter can be further represented by PCs as well. Therefore, the second term in Equation (1) can be expressed as  
\[\frac{F_{s}\cdot T_{\uparrow}}{1-S\cdot\rho_{s}}\approx F_{s}\cdot exp\left[ \ln\left(\sum_{j=1}^{n_{pc}}\left(\beta_{j}\cdot PC_{j}\right)\right)\frac{ \sec\theta_{v}}{\sec\theta_{v}+\sec\theta_{0}}\right] \tag{3}\]  
where \(\theta_{v}\) and \(\theta_{0}\) are the view zenith angle and the solar zenith angle, respectively, and the upward atmospheric radiation is ignored in the calculations. At the same time, assuming that within the selected fitting window, fluorescence spectra obey the Gaussian distribution. Thus, Equation (3) can be simplified as  
\[\frac{F_{s}\cdot T_{\uparrow}}{1-S\cdot\rho_{s}}\approx F_{s}\cdot h_{f} \tag{4}\]  
where \(h_{f}\) is an SIF spectral function. Figure 1: Location of the study area. In summary, the final on-plane radiance on the fluorescent surface can be simplified as follows:  
\[L_{\text{TO}A}(\alpha,\beta,F_{\text{s}})=I_{\text{sol}}.\frac{\mu_{0}}{\pi} \cdot\sum_{i=0}^{n_{p}}\Bigl{(}\alpha_{i}\cdot\lambda^{i}\Bigr{)}\cdot\sum_{j=1} ^{n_{F}}\bigl{(}\beta_{j}\cdot\text{PC}_{j}\bigr{)}+F_{\text{s}}\cdot h_{f} \tag{5}\]  
where \(h_{f}\) is as follows:  
\[\text{h}_{\text{F}}=\exp\left(\frac{-(\lambda-\text{u}_{\text{h}})^{2}}{2 \cdot\sigma_{\text{h}}^{2}}\right) \tag{6}\]  
Therefore, this linear least squares problem paves the way for solving SIF signals.