# Estimation of Chlorophyll Fluorescence at Different Scales: A Review
## 4 SIF Retrieval Methods in Space Scale
### The Physical Methods  
#### 4.2.1 FLD-Like Methods  
The FLD algorithm is successfully used near the ground without considering the atmospheric effects. FLD-like methods are often combined with MODTRAN to extract the SIF from airborne or space-borne data [99; 100; 86; 101; 102].  
Following Equation (9), the expression can be written in another form in the interested window:  
\[\begin{array}{l}L=L^{P}+\frac{E^{S}\cdot\rho/\pi+SIF}{1-S\cdot\rho}\cdot \tau_{\uparrow}\\ L^{P}=\frac{E_{0}\cos\theta}{\pi}\rho_{so}\\ E^{S}=E_{0}\cos\theta\cdot\tau_{\downarrow}\cdot S\end{array} \tag{10}\]  
in which, \(L^{P}\) is the atmospheric path radiance, \(E^{S}\) is the irradiance including the direct and diffuse fluxes arriving at the surface. Other parameters are described in Table 3. The oxygen absorption band is chosen for the retrieved window. Two bands, one inside oxygen absorption band (i: 760 nm), and the other outside (o: 753 nm), are expressed as follows [86]:  
\[\begin{array}{l}L_{i}=L^{P}_{i}+\frac{E^{S}_{i}p_{i}/\pi+SIF_{i}}{1-S\cdot p _{i}}\cdot\tau_{\uparrow i}\\ L_{o}=L^{P}_{o}+\frac{E^{S}_{o}\rho_{so}+SIF_{o}}{1-S\cdot p_{o}}\cdot\tau_{ \uparrow o}\end{array} \tag{11}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Meaning** \\ \hline \(L_{\text{TOA}}\) & Radiance at the top of atmosphere \\ \(\rho_{so}\) & Hemispherical reflectance \\ E\({}_{0}\) & extraterrestrial solar irradiance on a plane perpendicular to the sun’s rays \\ \(\theta\) & Solar zenith angle \\ \(\rho\) & Surface reflectance \\ SIF & Fluorescence radiance at the top-of-canopy (TOC) \\ S & Spherical reflectance of the atmosphere back to the surface \\ \(\tau_{\uparrow}\) & Upward transmittance \\ \(\tau_{\downarrow}\) & Downward transmittance \\ \hline \hline \end{tabular}
\end{table}
Table 3: The definitions of parameters used in Equation (9).  
Combining the two previous equations, SIF can be calculated as follows [86]:  
\[SIF_{i}=B\left[\frac{X_{i}(E^{g}_{o}+\pi X_{o}S_{o})-AX_{o}(E\textit{S}_{i}+\pi X _{i}S_{i})}{B(E^{g}_{o}+\pi X_{o}S_{o})-A(E^{g}_{i}+\pi X_{i}S_{i})}\right] \tag{12}\]  
\[X_{j}=\frac{L_{j}-L_{j}^{p}}{\tau_{j}},j=i,o \tag{13}\] \[\rho_{i}=A\cdot\rho_{o}\] \[SIF_{i}=B\cdot\textit{SIF}_{o}\]  
A is the factor relating \(\rho_{i}\) to \(\rho_{o}\), \(B\) is the factor linking \(SIF_{i}\) with \(SIF_{o}\). Assuming the reflectance and fluorescence have a linear relationship in the oxygen absorption bands without considering the variation of spectral shape, A can be computed through the linear interpolation of surface reflectance through two channels (758 nm and 771 nm) outside the oxygen absorption band (760 nm) (Equation (7)) [86]; B is fixed at 0.8 [80; 103] using field and simulated analysis.  
\[A=\frac{\rho_{758}\cdot\omega_{1}+\rho_{771}\cdot\omega_{2}}{\rho_{758}},\ \omega_{1}=\frac{771-760}{771-758},\ \omega_{2}=\frac{760-758}{771-758} \tag{14}\]  
in which, \(\omega_{1}\) shows the proportion of the right shoulder to the total width of the absorption band; \(\omega_{2}\) shows the proportion of the left shoulder.  
This method develops from the measurement near the ground and has a clear physical meaning; however, it needs to know the accurate atmospheric parameters. As we know, the atmospheric parameters are difficult to obtain, and are mainly obtained through simulation using the radiation transfer model. Therefore, these methods have been applied in the limited range and are not fit for measuring the SIF with conventional methods.