## 2 Materials and Methods
### The Data-Driven SIF Retrieval Algorithm  
The first two terms in Equation (1) are non-fluorescent contributions, while the last term is the fluorescent contribution. According to the basic idea of the data-driven algorithm, the non-fluorescence contribution of \(\text{L}_{\text{TOA}}\) is a combination of high-frequency information (including Fraunhofer lines and atmospheric absorption lines) and low-frequency information. The high-frequency part (caused by the absorption of the solar atmosphere and Earth's atmosphere) has a specific pattern, which can be extracted from the training dataset composed of non-vegetated targets and reconstructed with a small number of feature spectra. The low-frequency part (caused by atmospheric radiation and surface reflection) can be expressed by a polynomial of wavelength (\(\lambda\)) [11; 19]. Moreover, SIF spectra usually have a specific shape and can be expressed by a mathematical function. Thus, Equation (1) can be rewritten as:  
\[\text{L}_{\text{TOA}}=\left(\sum_{\text{i}=0}^{\text{n}_{\text{p}}}\text{a}_{ \text{i}}\lambda^{\text{i}}\right)\cdot\left(\sum_{\text{j}=1}^{\text{n}_{ \text{SV}}}\text{a}_{\text{j}}\cdot\text{v}_{\text{j}}\right)+\text{F}_{\text {S}_{0}}\cdot\text{h}_{\text{F}}\cdot\text{T}_{\uparrow} \tag{4}\]  
where \(\text{v}_{\text{j}}\) is the feature vector decomposed by the singular vector decomposition (SVD) method, \(\text{a}_{\text{i}}\) and \(\text{a}_{\text{j}}\) are the coefficients of the polynomial and the singular vectors, respectively, \(\text{n}_{\text{p}}\) is the order of polynomial, \(\text{n}_{\text{SV}}\) is the number of selected singular vectors, \(\text{F}_{\text{s}0}\) is the SIF signal at the specified wavelength, and \(\text{T}_{\uparrow}\) is the upward atmospheric transmittance. \(\text{h}_{\text{F}}\) is expressed as a Gaussian function with an average of \(\text{\mu}_{\text{h}}\) and a standard deviation of \(\text{\sigma}_{\text{h}}\):  
\[\text{h}_{\text{F}}=\exp\left[\frac{-(\lambda-\text{\mu}_{\text{h}})^{2}}{2 \text{\sigma}_{\text{h}}{}^{2}}\right] \tag{5}\]  
According to [PERSON] et al. [13], the upward atmospheric transmittance \(\text{T}_{\uparrow}\) can be estimated before retrieval, so when \(\text{n}_{\text{p}}\), \(\text{n}_{\text{SV}}\), \(\text{\mu}_{\text{h}}\) and \(\text{\sigma}_{\text{h}}\) were determined, only \(\text{a}_{\text{j}},\text{ }\text{ }\text{a}_{\text{j}}\), and \(\text{F}_{\text{S}_{0}}\) were left as unknown parameters; then, the retrieval could be turned into a linear least-squares problem. The parameters mentioned above are mostly set by experience. For example, \(\text{\mu}_{\text{h}}\) is generally 740 nm for far-red fluorescence, and 692 nm for red fluorescence, \(\text{n}_{\text{p}}\) is generally 2-4. However, the influence of different parameters on the results remains to be analyzed. In addition to parameters in the forward model, the fitting window, i.e., the spectral range in Equation (5), is also a key parameter, which determines the number of absorption lines used. Meanwhile, because the random noise of the instrument also fills the absorption line, it interferes with the retrieval so different SNR levels were also considered in this study. For red SIF retrieval, the spectral range covered by O\({}_{2}\)-B absorption lines (682-697 nm) covered nearly all the fitting windows selected in previous studies [18; 20; 31]. For far-red SIF retrieval, the water vapor absorption band (range from 712 nm to 745 nm), the O\({}_{2}\)-A band (range from 755 nm to 775 nm), and the atmospheric window between them were commonly used. Thus, we completed a series of experiments to preliminarily select the range of the far-red fitting window (Table 3). Table 3 shows the retrieval results with the minimum RMSE in the fitting windows after trial and error. Fitting windowswithin the O\({}_{2}\)-A band were used in Exp. 1 and 2, Exp. 3-6 used the fitting window containing water vapor absorption lines, while the fitting window of Exp. 5 and 6 also contained the atmospheric window range from 747 nm to 758 nm.