## 3 SIF Retrieval Methods  
### FLD and SFM Methods  
The family of FLD-methods, typically applied to the O\({}_{2}\)-A band or to Fraunhofer lines to retrieve SIF, are based on the exploitation of two or more channels inside and outside the selected absorption band. Assuming a Lambertian behaviour of surface reflectance, radiance measured from a vegetated target at surface level \(\overline{L}\) can be expressed as two complementing contributions: (1) the reflected solar irradiance, and (2) the SIF emission. Mathematically, this can be expressed at a given wavelength (\(\lambda\)):  
\[\overline{L}(\lambda)=\rho(\lambda)\cdot\frac{\overline{E}(\lambda)}{\pi}+F(\lambda) \tag{5}\]  
where \(\overline{E}\) is the total solar (direct and diffuse) irradiance reaching the surface, \(\rho\) is the Lambertian surface reflectance, and \(F\) is the fluorescence emission. Commonly to all FLD-family techniques, the radiances inside (\(\overline{L}_{in}\)) and outside (\(\overline{L}_{out}\)) the absorption band are related by assuming a simplified behaviour of the \(\rho\) and \(F\). For the FLD, a constant wavelength response is assumed [19], whereas a spectral dependency is introduced for the iFLD method [22]. In view of these assumptions, the FLD and 3 FLD methods can be considered as particular cases of the iFLD formulation [22]:  
\[F=\frac{\alpha_{R}\cdot\overline{E}_{out}\cdot\overline{L}_{in}-\overline{E}_ {in}\cdot\overline{L}_{out}}{\alpha_{R}\cdot\overline{E}_{out}-\alpha_{F} \cdot\overline{E}_{in}} \tag{6}\]  
where \(\alpha_{R}=\frac{\rho_{out}}{\rho_{in}}\) and \(\alpha_{F}=\frac{F_{out}}{\overline{I}_{in}}\) are the coefficients that introduce the spectral variation of \(\rho\) and \(F\) in the absorption band region. Note that the FLD method is the particular case where \(\alpha_{R}=1\) and \(\alpha_{F}=1\)[22], and the 3 FLD formulation is equivalent to the original FLD formulation when \(\overline{L}_{out}\) and \(\overline{E}_{out}\) are obtained by linear interpolation of two bands at each side of the absorption panel [21].  
Unlike the simpler FLD methods, the SFM approach exploits all the spectral information around the selected absorption band [23; 24], or even the complete spectral SIF range from 650 nm to 800 nm [25]. According to the selected spectral interval, reflectance and fluorescence spectra can bedescribed by appropriate mathematical functions. Therefore, a spectral difference, \(\epsilon(\lambda)\), exists between the modelled and observed radiance at TOC:  
\[arg_{\overrightarrow{x_{p}},\overrightarrow{x_{k}}}min\left(\overline{L}( \lambda)-\underbrace{\left(\rho_{MOD}(\lambda)\cdot\frac{\overline{E}(\lambda) }{\pi}+F_{MOD}(\lambda)\right)}_{\overline{L}_{MOD}}\right) \tag{7}\]  
where \(\overline{L}(\lambda)\) is the observed radiance at TOC from Equation (5), and \(\overrightarrow{x_{p}},\overrightarrow{x_{F}}\) are vectors containing the parameters used to reproduce \(\rho_{MOD}\) and \(F_{MOD}\) spectral functions. For narrow spectral intervals covering the O\({}_{2}\)-A region, the SFM typically models \(\rho_{MOD}\) and \(F_{MOD}\) by polynomial (quadratic or cubic) functions [44]. Thus, the inversion process disentangles \(F_{MOD}\) from \(\rho_{MOD}\) by finding the coefficients of the parametric functions for \(\rho_{MOD}\) and \(F_{MOD}\) that minimizes \(\epsilon(\lambda)\)[24].