# Estimation of Chlorophyll Fluorescence at Different Scales: A Review
## 4 SIF Retrieval Methods in Space Scale
### The Physical Methods
#### 4.2.4 Simplified Radiative Transfer Method  
Equation (9) gives the expression of the radiance at sensor. In some special window, such as the potassium (K) I absorption line, the CaII line near 866 nm, and some narrow spectrum including the Fraunhofer lines, the atmospheric scattering and absorption may be neglected (\(\rho_{\text{so}}\) = 0, \(\rho\) = 0, \(\tau_{\text{ss}}\) + \(\tau_{\text{sd}}\) = 1,\(\tau_{\text{do}}\) + \(\tau_{\text{oo}}\) = 1); the radiance received by the sensor can be simplified as follows [33]:  
\[(L_{TOA})^{*}=(\frac{\rho\cdot E_{0}\cdot\cos\theta}{\pi}+S\overset{\rightarrow }{IF})^{*}=K\cdot E^{*}+S\overset{\rightarrow}{IF} \tag{19}\]  
in which, \(E\) is the high-resolution solar irradiance spectrum from Kurucz; the asterisk * shows that it is convoluted with respect to the instrumental line shape.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Meaning** \\ \hline \(\overset{\rightarrow}{I}_{0}\) & High-resolution solar transmission spectrum \\ \(F_{s}^{rel}\) & Relative fluorescence signal \\ \(<\) \(<\) & Convolution symbol (with the instrumental line shape) \\ \(\overset{\rightarrow}{\underset{i=0}{\overset{\rightarrow}{\pi}}}a_{i}\lambda^{i}\) & Polynomial item (the continuum radiance), in which \(a_{i}\) is the coefficient of the polynomial, \(\lambda^{i}\) is the wavelength. \\ \(\overset{\rightarrow}{y}\) & Logarithm of the measurement vector \\ \(S_{\varepsilon}\) & Diagonal measurement error covariance matrix \\ \hline \hline \end{tabular}
\end{table}
Table 5: Parameters used in Equations (17) and (18).  
Similarly, in the range of interest, atmospheric scattering and absorption are considered constant values, and the received radiation is rewritten in the following form [33; 44]:  
\[\left(L_{TOA}\right)^{*}=K^{\prime}\cdot E^{*}+\varepsilon\cdot SIF \tag{20}\]  
In Equations (19) and (20), \(L_{TOA}\) and \(E^{*}\) are known parameters, and other unknown parameters can be obtained through a standard, weighted least squares fitting method. These two equations show that neglecting the atmospheric effects only result in a slight scale factor \(\varepsilon\). Some researchers proved that this scale factor was about 0.3, which can result in a 0.6% error [33]. In previous literature, K I lines from the GOSAT TANSO-FTS data, which have a super-fine spectral resolution, were mostly used as the retrieved window [33]. To improve the SNR, the retrieved window will be widened (i.e., 769.9-770.25 nm including the K I line, 758.45-758.85 nm near the oxygen absorption band, and 863.5-868.8 nm including the CaII line) [44]. With respect to GOSAT data, the other simplified radiative transfer method GARLiC was proposed by [PERSON] [38]. The retrieval window was 755-759 nm, and the upward transmittance on a clear day was 1, and \(\rho\) was much smaller than 1; therefore, Equation (9) can be simplified as follows [38]:  
\[L_{TOA}=\frac{E_{0}\cdot\cos\theta}{\pi}\cdot\left(\rho_{so}+\rho\cdot\tau_{ \downarrow}\right)+SIF \tag{21}\]  
In Equation (21), under the assumption that the atmospheric scattering and surface reflectance are functions of the wavelength, this expression can be further simplified as follows [38]:  
\[L_{TOA}=E^{*}\cdot\left(\alpha_{0}+\alpha_{1}\cdot\lambda\right)+SIF \tag{22}\]  
in which, \(\alpha_{0}\) and \(\alpha_{1}\) are the parameters describing the effects of atmospheric scattering and surface reflectance, respectively. Lastly, the least squares fitting method is applied to obtain \(\alpha_{0}\), \(\alpha_{1}\), and \(SIF\).  
The physical methods are commonly used in the fluorescence retrieved. All these methods have a clear physical meaning; but they were developed from different strategies. In these methods, the atmospheric effects can be neglected. However, in future research, the atmospheric effects should be paid more attention.