# Review of Satellite Remote Sensing of Carbon Dioxide Inversion and Assimilation
## 3 Carbon Satellite Remote Sensing Inversion Techniques
### Advances in Remote Sensing Algorithms for Satellite Atmospheric CO\({}_{2}\)
#### 3.2.1 Physical Inversion Algorithms  
The conventional Differential Optical Absorption Spectroscopy (DOAS) algorithm was first proposed by [PERSON] et al. [57], which mainly focused on the UV-visible band (200\(\sim\)760 nm) for the inversion of trace gases in ground-based observations, but the algorithm introduces large errors in the inversion of infrared bands. [PERSON] et al. [58; 59; 60] developed the WFM-DOAS algorithm based on the DOAS algorithm, which was applied to the inversion of SCIAMACHY near-infrared spectra for atmospheric CO\({}_{2}\) column concentrations. WFM-DOAS is an unconstrained linear least squares method based on the scaling (or shifting) of preselected vertical profiles, and it obtains information on atmospheric CO\({}_{2}\) by matching the logarithms of irradiance of the observed spectrum to the simulated spectrum. The effect of the slow variations in the spectra is removed by the differential absorption. The WFM-DOAS algorithm, with accurate calibration of the observed spectra, performs a first-order Taylor expansion of the natural logarithm of the solar normalized radiation intensity at the wavelength \(\lambda_{i}\), expressed as:  
\[ \begin{split}\ln I_{i}^{means}(V^{t},b^{t})\approx\ln I_{i}^{mod} (\overline{V},\overline{b})+\frac{\partial\ln I_{i}^{mod}}{\partial V_{j}}\mid \overline{\
u}\times(\hat{V}-\overline{V})+\\ \sum_{j=1}^{F}\frac{\partial\ln I_{i}^{mod}}{\partial b_{j}}\mid \overline{b_{j}}\times(b_{j}-\overline{b}_{j})+P_{i}\end{split}  \tag{5}\]  
where \(I_{i}^{means}\) and \(I_{i}^{mod}\) are the observed and simulated values of normalized radiation intensity at \(\lambda_{i}\), and \(V^{t}\), \(\bar{V}\) and \(\hat{V}\) are the true, simulated, and inverted values of the total column of atmospheric components respectively. \(b_{j}\) represents atmospheric temperature, pressure, aerosol, cloud parameters and surface albedo, etc., \(P_{i}\) is a low-order polynomial. Then the simulated spectral constants and column weight functions normalized by the sun can be obtained by the atmospheric radiative transfer simulation software SCIATRAN (V4.6) calculations, and the vertical column concentrations of the inverse-performing gases are based on the least-squares fitting of the spectra to the observed spectra:  
\[\left\|\ln I_{i}^{means}(V^{t},b^{t})-\ln I_{i}^{mod}(\overline{V},\overline{ b})\right\|^{2}\equiv\left\|RES_{i}\right\|^{2}\longrightarrow min \tag{6}\]  
In overview, the WFM-DOAS implementation process [61] is shown in Figure 13.  
The initial WFM-DOAS inversion faced challenges due to its inability to address aerosol scattering effects and its low accuracy [62]. To correct for temperature and pressure dependencies of gas absorption cross-sections in the infrared spectrum, various international research groups have developed improved inversion algorithms based on conventional DOAS principles. For example, [PERSON] et al. [63] from the University of Heidelberg developed the IMAP-DOAS algorithm, an enhanced iterative maximum a posteriori probability DOAS method for SCIAMACHY. [PERSON] et al. [64] further refined this approach with the Full Spectral Initiation (FSI)-WFM-DOAS algorithm, which incorporates an a priori profile with actual atmospheric parameters and accounts for three aerosol modes, increasing computational complexity. [PERSON] et al. [65] and [PERSON] et al. [66] improved the WFM-DOAS algorithm by including SCIAMACHY's M-factor, a multiplicative correction related to absolute radiative calibration, which reduced errorsfrom aerosol and cloud scattering. This enhanced algorithm improved the accuracy of XCO\({}_{2}\) concentration retrieval over long time series and expanded SCIAMACHY's applicability to non-clear-sky conditions. Additionally, WFM-DOAS has been further enhanced with constant aerosol vertical profiles and cloud-detection algorithms in radiative transfer simulations [67]. [PERSON] et al.