## 2 Materials and Methods
### Singular Vector Decomposition (SVD)  
The so-called Singular Vector Decomposition (SVD) retrieval method is a form of spectral fitting method for fluorescence retrieval [10; 14; 45]. It also represents the at-sensor radiance as the sum of the radiance reflected by the surface and the fluorescence contribution. The reflected radiance is constructed as the product of a spectrally smooth surface reflectance (modeled as a polynomial in wavelength) and the atmospheric absorption. However, instead of using explicit radiative transfer modeling to calculate atmospheric absorption along the spectral fitting window, this is modeled as a linear combination of orthogonal spectral functions derived from the data through Singular Vector Decomposition (similar to Principal Component Analysis). Figure 7: Flow chart of the fourth cluster of the HyPlant processing chain, presenting the workflow involved in processing HyPlant FLUO data from non-deconvolved/deconvolved at-sensor radiance to the final SIF products. The forward model M used by the SVD method to fit the at-sensor radiance spectrum can then be expressed as:  
\[M\Big{(}a_{i},\alpha_{j},F\Big{)}=\frac{\mu_{s}I_{o}}{\pi}\left[\sum_{i}a_{i} \lambda^{i}\sum_{j}\alpha_{j}u^{j}\right]+Fh_{f}\tau_{eff} \tag{1}\]  
where \(M\) describes the spectral radiance received by the sensor over a fluorescent target; \(\mu_{s}\) is the cosine of the sun zenith angle; \(I_{o}\) is the extraterrestrial solar irradiance; \(\lambda\) is the array of wavelengths within the fitting window; \(v\) is the set of singular vectors representing atmospheric absorption; a and \(\alpha\) are arrays of weights for the \(\lambda\) and \(\alpha\) vectors; \(F\) is the fluorescence intensity; \(h_{f}\) is the spectral shape of fluorescence; and \(\tau\) is an effective upward atmospheric transmittance calculated for the flight altitude and convolved to HyPlant's spectral response function. Improved Fraunhofer Line Discrimination (iFLD)  
The iFLD, as part of the HyPlant processing chain, is based on the iFLD method initially proposed by [39], but was adapted to allow SIF retrievals from HyPlant, in particular, the FLUO module [46]. As implemented, both telluric oxygen absorption bands centered around 687 nm (O\({}_{2}\)-B) and 760 nm (O\({}_{2}\)-A) are exploited, yielding SIF retrievals at these two wavelengths. The main retrieval challenge involves disentangling the emitted SIF radiation from the reflected radiance. Therefore, two measurements inside (\(i\)) and outside (\(o\)) of an absorption band are required. The radiative transfer, describing the radiance signal measured at the airborne sensor \(L_{\lambda}^{AIS}\) at these two wavelengths (\(\lambda\)), can be described as a function of the extraterrestrial irradiance (\(E_{j}^{0}\)), several atmospheric transfer functions, the surface reflectance (\(R\)), and the SIF emission (\(F\)), and can be expressed as:  
\[L_{\lambda}^{AIS}=\ \frac{\langle E_{\lambda}^{0}\cos\theta_{il}\rangle}{\pi} \Bigg{[}\langle\rho_{\infty}^{\lambda}\rangle+\frac{\Big{(}\langle\tau_{ss}^{ \lambda}\tau_{\omega}^{\lambda}\rangle+\langle\tau_{sd}^{\lambda}\tau_{oo}^{ \lambda}\rangle+\langle\tau_{ss}^{\lambda}\tau_{do}^{\lambda}\rangle+\langle \tau_{sd}^{\lambda}\tau_{do}^{\lambda}\rangle\Big{)}R_{\lambda}}{1-R_{\lambda} \langle\rho_{dd}^{\lambda}\rangle}\Bigg{]}+\frac{F_{\lambda}\Big{(}\langle\tau _{\omega}^{\lambda}\rangle+\langle\tau_{do}^{\lambda}\rangle\Big{)}}{1-R_{ \lambda}\langle\rho_{dd}^{\lambda}\rangle}\Bigg{]} \tag{2}\]  
where \(\theta_{il}\) is the illumination zenith angle; \(\rho_{so}\) is the path scattered radiance; \(\tau_{ss}\) is the direct transmittance for sunlight; \(\tau_{oo}\) is the direct transmittance in the view direction; \(\tau_{sd}\) is the diffuse transmittance of the atmosphere for sunlight; \(\tau_{do}\) is the hemispherical-directional transmittance in the view direction, and \(\rho_{dd}\) is the spherical albedo.