# Review of Satellite Remote Sensing of Carbon Dioxide Inversion and Assimilation
## 3 Carbon Satellite Remote Sensing Inversion Techniques
### Orthogonal Modelling and Inversion Principles  
#### 3.1.1 Basic Principles of Remote Sensing Inversion  
The ultimate goal of satellite remote sensing observations is to analyze and calculate the physical and chemical properties of the atmosphere or target from spectra or images observed by satellite instruments, and to use these input parameters as information about the actually observed atmosphere. To achieve this, appropriate inversion algorithms are required. The main components of remote sensing inversion are the forward model, the inversion method, and the error analysis. The forward model is the basis of the inversion algorithm. The forward process of satellite remote sensing describes the physical process  
Figure 10: Contribution of papers from different countries. between atmospheric covariates to observations, using the physical process of radiative transfer to simulate the radiant brightness of the sensor and returning a spectrum of simulated radiances and the partial derivatives of these radiances concerning some of the atmospheric and surface state attributes, which also referred to as Jacobi determinants. The Jacobi matrix is used in the inversion process to estimate the state covariates required to minimize the difference between the observed and simulated spectra. The assumed atmospheric state is modified to improve the fit to the measured spectra by controlling the margin of error between the two sides by solving for residuals, and the process is repeated until convergence conditions are met. Once the atmospheric state that produces the best fit to the observed spectrum is found, the algorithm determines the errors from different sources (e.g., vertical smoothing, measurement noise, etc.) in XCO\({}_{2}\), XCO2, and the XCO\({}_{2}\) column-averaged kernel function (CAK), and obtains the corresponding atmospheric state results. The relationship between the atmospheric composition measurements, the forward model, and the inversion algorithm is shown in Figure 11.  
The closer the description of each part of the forward model is to the actual situation, the more accurate the simulated spectra will be, and the higher the accuracy of the inversion based on this fitting will be. The inversion algorithm is based on the mapping relationship provided by the forward model to find the optimal set of conditions from all possible states, so that the simulation results of the forward model are the closest to the actual results of the instrument, and the inversion value is obtained at this time. Combined with the satellite observation model, considering the solar radiation and the satellite instrument modeling function, the forward process can be expressed as follows:  
\[y=F(x,b)+\varepsilon \tag{1}\]  
where \(y\) is the measurement vector, \(x\) is the inverse state vector, \(b\) is the non-inverse state vector, and \(\varepsilon\) is the error vector. \(F(x,b)\) is a function of the state vector as the forward model, which describes the entire physical process of the measurement. The input parameters to the a priori state vector consist of atmospheric parameters and the observed spectra, and  
Figure 11: Forward model and inversion process. the cost function represents the cost of different minimum deviations by optimizing the difference between the measured and simulated spectra as well as the difference between the state vector and the a priori information. The solution equation can be used to minimize the bias cost function by means of an optimization algorithm [43]. \[J(x)=[F(x)-y]^{T}S_{\epsilon}^{-1}[F(x)-y]+(x-x_{a})^{T}S_{a}^{-1}(x-x_{a}) \tag{2}\]  
where \(S_{a}\) is the a priori error variance matrix, \(x_{a}\) is the a priori state vector, \(S_{\epsilon}\) is the error covariance matrix, and \(T\) denotes the matrix transpose matrix. XCO\({}_{2}\) can be computed directly from the \(x\) of the cost function formulation, where the forward model is rerun at each iteration in the minimization of the cost function solver, but needs to use the a priori information gained in the previous iteration.