## 2 Materials and Methods
### Frequency Shift Correction Algorithm  
In order to correct for frequency shifts in satellite spectra, a reference spectrum devoid of frequency shifts relative to the Earth's surface is employed. This reference spectrum  
Figure 1: Singular vectors in the forward model and the state vector of Fs within two retrieval windows: (**a**) FLs band, (**b**) joint retrieval for FLs-O\({}_{2}\) absorption bands.  
is defined as the product of the solar irradiance spectrum and the standard atmospheric transmittance spectrum:  
\[\mathrm{I^{ref}(v)=E(v)exp\left[-\left(\frac{1}{\cos\theta_{SZA}}+\frac{1}{\cos \theta_{ZA}}\right)\tau(v)\right] \tag{3}\]  
where \(\mathrm{I^{ref}(v)}\) represents the radiance spectrum at the top of the atmosphere (TOA), Eis the solar radiance spectrum, \(\tau\) is the atmospheric optical thickness, and \(\theta_{SZA}\) and \(\theta_{ZA}\) represent the solar zenith angle and satellite zenith angle, respectively. Convolution of \(\mathrm{I^{ref}}\) with the GOSAT satellite's instrumental line shape (ILS) yields the reference spectrum:  
\[\mathrm{S^{ref}_{P/S}(i)=\sum_{j=-N_{W}}^{N_{W}}ILS_{P/S}(-j\delta v)\mathrm{ I^{ref}(v_{i}+j\delta v)\delta v}} \tag{4}\]  
where P and S represent two polarization directions, \(\mathrm{S^{ref}_{P/S}(i)}\) denotes the intensity at the i-th sampling point of the reference spectrum, \(\delta v\) is the convolution step size (usually set to 0.01 cm\({}^{-1}\)), and \(\mathrm{N_{W}}\)\(\delta v\) represents the convolution half-width, set to 20 cm in this study.  
To determine the actual wavenumber axis of the satellite spectrum, a lookup table consisting of n wavenumber axes with certain frequency shifts is constructed. The wavenumber axis corresponding to the reference spectrum is interpolated in the lookup table to build the corresponding reference spectrum matrix \(\mathrm{M^{ref}_{P/S}}\). The correlation between the satellite spectrum and each reference spectrum in \(\mathrm{M^{ref}_{P/S}}\) is then calculated using Equation (5), where \(\mathrm{V_{correlation}}\) is a row vector of length n representing the magnitudes of the correlations between the satellite and reference spectra, and Sounding represents the satellite spectrum, a column vector of length 1755.  
\[\mathrm{V_{correlation}=Sounding_{P}}^{T}\times\mathrm{M^{ref}_{P}+Sounding_{S} }^{T}\times\mathrm{M^{ref}_{S}} \tag{5}\]  
According to the actual wavenumber axis of the satellite spectrum obtained, the satellite spectrum is interpolated in the SIF retrieval window, effectively eliminating frequency shifts between satellite spectra. The effectiveness of the wavenumber correction is illustrated in Figure 2.  
Figure 2: The identical set of spectra before and after wavenumber correction. For the sake of clarity, only a limited portion of the FTS-Band1 spectrum is displayed.  
The solar irradiance spectrum database is provided by Toon [27]. This database offers a merged solar pseudo-transmittance spectrum, ranging from 600 to 33,300 cm\({}^{-1}\), by consolidating measurements derived from various methods. The pseudo-transmittance spectrum has a spectral resolution of 0.01 cm\({}^{-1}\), which can be directly used for convolution with ILS. For the accuracy of the reference spectrum, the solar radiation continuum of TSIS SIM was utilized in this study to convert the solar pseudo-transmittance spectrum into the solar irradiance spectrum.