# Review of Satellite Remote Sensing of Carbon Dioxide Inversion and Assimilation
## 4 Carbon Flux Data Assimilation and Inversion Studies
### Data-Assimilation Algorithms
#### 4.2.2 Data-Assimilation Algorithms
\begin{split}\frac{\partial J[x(t_{0})]}{\partial x}=B^{-1}[x(t_ {0})-x^{b}(t_{0})]+H_{0}^{T}d_{0}+M_{0}^{T}[H_{1}^{T}d_{1}+\\ M_{1}^{T}(H_{2}^{T}d_{2}+\cdots+M_{n}^{T}(H_{n}^{T}d_{n})\cdots] \end{split}  \tag{18}\]  
In the equation, \(d_{n}=R_{n}^{-1}[H_{n}(x_{n})-y_{n}^{obs}]\). During the minimization process, the adjoint matrix is used to search for the gradient information corresponding to the initial state in order to reduce the value of the objective function. This involves forward integration of the forecast model once and backward integration of the adjoint equation to the initial time to adjust the gradient of the cost function. The adjusted initial field is then used in the forward model. This process is repeated in a loop until the convergence condition is met. Kalman Filter Algorithm (KF)  
The classic Kalman filter assimilation algorithm is an optimal recursive data processing method. It estimates the state variables by utilizing the statistical characteristics of the observations, as well as the errors between the model and the observations. The basic idea is as follows: first, make a forecast of the state variables, then introduce the observation data, reanalyze the state variables based on the observation data, and then make another forecast. This completes the cycle of forecast-analysis-forecast [146]. The Kalman filter consists of two parts: prediction and analysis, with a total of 5 equations, expressed as follows [147]. Prediction part:  
\[x_{n}^{f}=Mx_{n-1}^{a} \tag{19}\]  
\[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]  
The first equation is the prediction of the state variable, \(x_{n}^{f}\) denotes the prediction operator \(M\) forward integrating to obtain the predicted value of the state quantity at the n moment; the second equation is the prediction of the prediction error covariance \(P_{n}^{f}\), which can be considered to be composed of two parts, one of which is the part of the initial value as well as the part of the error caused by the evolution of its model, and the other part of which is the model's prediction error at the nth moment \(Q_{n-1}\), which is caused by the model itself, such as the driving data and the error of boundary conditions, the error of model parameters, etc. [148], \(A_{n-1}\) is the analyzed error of the previous moment. Analysis part:  
\[K_{n}=P_{n}H^{T}(HP_{n}H^{T}+R_{n})^{-1} \tag{21}\]  
\[x_{n}^{a}=x_{n}^{f}+K_{n}(y^{obs}-Hx_{n}^{f}) \tag{22}\]  
\[A_{n}=(1-K_{n}H)P_{n} \tag{23}\]  
where \(K_{n}\) denotes the Kalman gain at the n moment, \(H\) is the observation operator used to project the state variables into the observation space, and \(R_{n}\) is the observation error covariance; a posteriori adjustments to the state quantities can be made using the second equation, which is the same as the OI and the update formula in the variational assimilation and is obtained by multiplying the predicted value (or the background value) plus an increase in the observed value by a certain weight, \(K\); the third equation concludes with an analytical error covariance that can be used to predict the prediction error covariance at the next moment. The whole process of Kalman filtering is shown in Figure 17.  
Ensemble Kalman Filter Algorithm (EnKF)  
The ensemble Kalman filter algorithm is a method that uses a limited ensemble and is based on the Monte Carlo algorithm to estimate the error covariance [136; 149]. It generates a certain number of ensembles by adding perturbations to the background field and observation values and obtains the corresponding analysis values. Then, based on the differences in these values, it statistically calculates the error covariance matrix. The biggest advantage of the ensemble Kalman filter algorithm is that it does not require the development of tangent linear and adjoint models, and it can explicitly provide initial  
Figure 17: Assimilation-cycle process. perturbations for ensemble forecasts.