## 2 Materials and Methods
### Retrieval Methodology  
#### 2.2.1 The Forward Model  
The top-of-atmosphere (TOA) radiance spectrum, denoted as L\({}_{\rm TOA}\), acquired by the satellite from nadir over a Lambertian reflecting surface, neglecting atmospheric bidirectional reflectance effects [17], can be expressed as  
\[{\rm L_{TOA}}={\rm E}_{0}\cdot\cos\theta_{\rm SZA}(\rho_{0}+{\rm T}_{\downarrow \uparrow}\frac{\rho_{\rm s}}{\pi})+{\rm F}_{\rm S}\cdot{\rm T}_{\uparrow} \tag{1}\]  
where E\({}_{0}\) represents extraterrestrial solar irradiance, \(\theta_{\rm SZA}\) is the solar zenith angle, \(\rho_{0}\) is the atmospheric path reflectance, \(\rho_{\rm s}\) is the surface reflectance, T denotes atmospheric transmittance with arrows indicating the direction of light propagation, and F\({}_{\rm S}\) represents the canopy solar-induced fluorescence (SIF) emission. The L\({}_{\rm TOA}\) can be considered as a superposition of non-SIF contributions and SIF emissions. In the forward model, the non-SIF contribution typically includes an n-order polynomial to characterize slow-varying components such as land surface reflectance spectra [5; 18; 19]. However, due to the smoothness of land surface reflectance spectra within the retrieval window proposed in this study and the inability to eliminate the slope features of reflectance spectra in the singular vectors through slope normalization of non-SIF spectral sets, we directly perform Singular Value Decomposition (SVD) on the non-SIF spectral set to obtain singular vectors that contain information about land surface reflectance and weak atmospheric absorption. As the proposed retrieval window introduces only a narrow portion of atmospheric absorption lines and F\({}_{\rm S}\) is only influenced by the upward atmospheric transmittance T\({}_{\uparrow}\), and accurate estimation of unidirectional atmospheric transmittance faces certain challenges [20], we neglect the atmospheric impact on the SIF spectral shape. The forward model is represented as the direct summation of non-SIF contributions and the TOA SIF spectrum. \begin{table}
\begin{tabular}{c c} \hline
**Quality_Flag = 0 (Best)** & **Quality_Flag = 1 (Good)** \\ \hline
28 \(\leq\) continuum radiance at 757 nm \(\leq\) 195 (W\(\cdot\)m\({}^{-2}\cdot\)sr\({}^{-1}\cdot\)um\({}^{-1}\)) & 28 \(\leq\) continuum radiance at 757 nm \(\leq\) 195 (W\(\cdot\)m\({}^{-2}\cdot\)sr\({}^{-1}\cdot\)um\({}^{-1}\)) \\ \(\chi^{2}\) at 755 nm \(\leq\) 2.0 & 2.0 \(<\chi^{2}\) at 755 nm \(\leq\) 3.0 \\ \(\chi^{2}\) at 770 nm \(\leq\) 2.0 & 2.0 \(<\chi^{2}\) at 770 nm \(\leq\) 3.0 \\ 0.85 \(\leq\) O\({}_{2}\) ratio \(\leq\) 1.5 & 0.85 \(\leq\) O\({}_{2}\) ratio \(\leq\) 1.5 \\ 0.5 \(\leq\) CO\({}_{2}\) ratio \(\leq\) 4 & 0.5 \(\leq\) CO\({}_{2}\) ratio \(\leq\) 4 \\ \(\theta_{\rm SZA}\)\(\leq\) 80\({}^{\circ}\) & 98\({}_{\rm SZA}\)\(\leq\) 80\({}^{\circ}\) \\ Land fraction = 100\% & 80\% \(\leq\) Land fraction \(<\) 100\% \\ \hline \end{tabular}
\end{table} 
Table 1: Criteria of quality flags, with best and good for the GOSAT SIF product. Soundings that do not meet either set of criteria are flagged as having failed (2). \[\mathrm{F}(\omega,\mathrm{F_{s}})=\sum_{\mathrm{i=1}}^{\mathrm{m_{v}}}\omega_{ \mathrm{i}}\mathrm{Sv_{i}}+\mathrm{F_{s}^{TOA}V_{Fs}} \tag{2}\]  
where \(\omega_{\mathrm{i}}\) and \(\mathrm{F_{s}^{TOA}}\) represent the weights of the singular vectors \(\mathrm{Sv_{i}}\) in the reconstructed spectrum and the intensity information of the TOA SIF spectrum, respectively. Considering that within the inversion window, the shape of the vegetation canopy's SIF spectrum does not change when transmitted to the top of the atmosphere, we used the state vector \(\mathrm{V_{Fs}}\) of the canopy steady-state fluorescence \(\mathrm{F_{s}}\) to characterize the shape of the TOA SIF spectrum.Its slight shape error compared to the actual SIF spectrum has little impact on the retrieval results [17; 21; 22].