## 2 Materials and Methods
### Singular Vector Decomposition (SVD)
All required atmospheric transfer functions are obtained from MODTRAN5 [32] simulations, in combination with the MODTRAN5 interrogation technique [47; 48; 49; 50]. Four unknowns remain in Equation (2), namely \(R\) and \(F\), inside and outside the O\({}_{2}\)-absorption band, respectively. The iFLD-based retrieval relates \(R\) and \(F\) inside and outside both absorption bands using non-linear functions. This step reduces the number of unknowns to two and, given the fact that two \(L_{\lambda}^{AIS}\) measurements are considered, yields a solvable set of equations to eventually retrieve \(F\) (cf. [24] for details). For the airborne data, an empirical constraint based on non-vegetated reference surfaces is additionally implemented to account for uncertainties in the characterization of the atmosphere and remaining sensor artifacts (i.e., spectral shifts and detector mis-calibration). Details of this approach can be found in [51]. The empirical correction yields precise SIF retrievals at 760 and 687 nm but limits the applicability of the method to flight lines with sufficient non-vegetated reference pixels across track along the whole flight line. Spectral Fitting Method (SFM)  
The SFM retrieval algorithm developed for HyPlant is inspired by the original version, proposed by [52; 53], and successive improvements by [54; 55]. The SFM is a quasi-physical approach, which consists of two main components: (i) atmospheric radiation transfer modeling; and (ii) decoupling reflectance and fluorescence based on the _Spectral Fitting_ technique. The atmospheric spectra are computed by means of MODTRAN5, in which the model input parameters are derived from sun photometer measurements (i.e., aerosol optical thickness (AOT) at 550 nm, water vapor column (WVC) and surface pressure) and geometry parameters derived from the navigation data of a HyPlant image cube. The _Spectral Fitting_ uses general mathematical functions to model the canopy fluorescence and reflectance at different wavelengths within spectral windows centered on the oxygen absorption features (750-780 nm for the O\({}_{2}\)-A and 684-697 nm for the O\({}_{2}\)-B feature, respectively). Specifically, the retrieval algorithm employs the Lorentzian function to model the fluorescence emission peak and polynomial or piece-wise cubic splines to fit the reflectance. These two canopy contributions are combined and propagated through the atmosphere according to Equation (3) in order to simulate the at-sensor radiance observed by HyPlant. A comprehensive description of Equation (3) can be found in [54; 56], but some details are reported here for clarity. \[L\ =t_{1}\left[t_{2}+t_{8}r_{so}+\ \frac{t_{9}+t_{14}\overline{r_{sd}}}{1- \overline{r_{dd}}\ t_{3}}r_{do}+\frac{t_{10}\overline{r_{sd}}+t_{11}\overline{ r_{dd}}}{1-\overline{r_{dd}}\ t_{3}}\right]+t_{6}F_{so}+\ \left[\frac{\overline{F_{hem}}(t_{7}+t_{13}r_{do})}{1- \overline{r_{dd}}\ t_{3}}\right] \tag{3}\]  
The terms \(t_{i}\) represent the atmospheric transfer functions, obtained by combining different atmospheric functions in high resolution and convolving them to the instrument spectral response. The reflectance is modeled by four BRDF terms: \(r_{so}\) is the target bi-directional reflectance factor; \(r_{do}\) is the target directional reflectance for diffuse incidence; \(\overline{r_{sd}}\) is the average surroundings diffuse reflectance for solar irradiance; \(\overline{r_{dd}}\) is the average surroundings diffuse reflectance for diffuse incidence. \(F_{so}\) is the sun-induced fluorescence radiance of the target in the observer's direction, and \(\overline{F_{hem}}\) is the hemispherical fluorescence flux of the surroundings. Currently, Equation (3) is employed under the Lambertian assumption for both reflectance and fluorescence, and therefore the different reflectance (\(r_{so},\ r_{do},\ r_{sd}\) and \(r_{dd}\)) and fluorescence (\(F_{so}\) and \(F_{hem}\)) terms are considered equal.