## 3 SIF Retrieval Methods
### O\({}_{2}\) Transmittance Compensation on FLD and SFM  
Following the initial FLD assumptions, i.e., \(F(\lambda_{out})\sim F(\lambda_{in})\) and \(\rho(\lambda_{out})\sim\rho(\lambda_{in})\); and introducing Equations (1) and (3) into Equation (6), SIF can be formulated as:  
\[F=\frac{\frac{L_{in}}{t_{in}^{\dagger}}\cdot\alpha_{R}\cdot E_{out}\cdot t_{ out}^{\downarrow}-\frac{L_{out}}{t_{out}^{\downarrow}}\cdot E_{in}\cdot t_{in}^{\downarrow}}{ E_{out}\cdot\alpha_{R}\cdot t_{out}^{\downarrow}-E_{in}\cdot\alpha_{F}\cdot t_{in}^{ \downarrow}} \tag{8}\]  
where Equation (8) represents the improved formulation of the iFLD method to compensate for the O\({}_{2}\) transmittance. Note that Equation (8) is also valid for the FLD and 3 FLD methods when the spectral fluorescence and reflectance correction factors are \(\alpha_{F}=\)1 and \(\alpha_{R}=\)1, respectively. Outside the O\({}_{2}\)-A absorption band, upward and downward transmittances (\(t^{\uparrow}\) and \(t^{\downarrow}\)) can be set to unity, which simplifies Equation (8). In the case where solar irradiance is measured at TOC level, then \(E\cdot t^{\downarrow}\) is replaced by \(\overline{E}\); and thus, no O\({}_{2}\) correction needs to be applied to the solar irradiance term. Inside the O\({}_{2}\)-A absorption band, upward and downward oxygen transmittance can be simulated using an atmospheric radiative transfer code, such as the MODerate resolution atmospheric TRANsmission (MODTRAN, [42]), or using the HIgh-resolution TRANsmission molecular absorption database (HITRAN, [45]). Alternatively, empirical O\({}_{2}\) transmittance approximations [37] as shown in Equations (9) and (10) can also be used for moderate spectral resolution (\(\sim\)0.22 nm) measurements:  
\[t(\lambda)=exp[-(10^{C^{\prime}(\lambda)}\Big{(}\frac{p}{p_{0}}\Big{)}^{n} \Big{(}\frac{T_{0}}{T}\Big{)}^{m}U\Big{)}^{a}] \tag{9}\]  
\[U=0.7732\cdot 10^{-4}M\rho_{a}Z \tag{10}\]  
where \(p\)(atm), \(T\)(K), \(M\)(ppmv) and \(\rho_{a}\)(g\(\cdot\)m\({}^{-3}\)) are the pressure, temperature, absorber concentration, air density conditions, respectively, for a particular path length height (e.g., the distance between the TOC and the sensor); and \(U\)(atm \(\cdot\) cm) is the total absorber amount in the path length, \(Z\)(km). Finally, the subscript \(T_{0}\), \(p_{0}\) indicates the standard conditions, i.e., 273.16 K and 1 atm, and the values of the parameters \(a\), \(m\) and \(n\) are set to 0.5641, 0.9353 and 0.1936, respectively. See also [PERSON] et al., 1986 [37] for more details and default values typically assumed for the C\({}^{\prime}\) spectral coefficients. When using the SFM for proximal-sensing retrievals, then the observed radiance at sensor level (\(L\) from Equation (7)), is affected by atmospheric transmittance between the TOC target and the sensor. Therefore, assuming that the solar irradiance (\(E\)) and the up-welling radiance (\(L\)), are both measured away from the surface and using the relationship from Equation (3), the modified formulation of the SFM for a first order O\({}_{2}\) compensation would be:\[arg_{\frac{S^{2}_{xy},y^{2}}{2}}min\left(\frac{L}{t^{\uparrow}_{vza}}-\left(\frac{E \cdot t^{\downarrow}_{vza}}{\pi}\cdot\rho_{MOD}+F_{MOD}\right)\right)\Longleftrightarrow arg _{\frac{S^{2}_{xy},y^{2}}{2}}min\left(\frac{L-\left(\frac{E\cdot t^{\downarrow}_ {vza}}{\pi}\cdot\rho_{MOD}+F_{MOD}\right)t^{\uparrow}_{vza}}{S_{sensor}}\right) \tag{11}\]  
where the \(t^{\downarrow}_{vza}\) and \(t^{\uparrow}_{vza}\) are the (wavelength-dependent) downward and upward atmospheric transmittances. Sufixes \(sza\) and \(vza\) indicate the main dependence on the solar illumination and the sensor acquisition geometry, respectively. In Equation (11), the spectral dependency of each of the functions involved is omitted for brevity.