# 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  
This part introduces 3D-VAR, 4D-VAR, Kalman Filter (KF), EnKF algorithm, and Neural Network algorithm. 3D-Var & 4D-Var  
Three-dimensional variational analysis (3D-Var) compares the observed fields at different time steps with the analysis fields obtained by forward calculation from the initial fields. By defining a function for the initial fields and using a certain descent algorithm, the gradient function of the corresponding initial fields is solved to minimize the function and obtain the optimal initial fields. Compared to optimal interpolation, three-dimensional variational analysis has a larger amount of data analysis and can perform global analysis, avoiding local discontinuity issues. Three-dimensional variational analysis is based on the theoretical methods of Bayesian filtering and maximum likelihood estimation. The objective function of 3D-Var [142] is as follows:  
\[J(x_{0})=\frac{1}{2}(x_{0}-x_{b})^{T}B^{-1}(x_{0}-x_{b})+\frac{1}{2}[H(x_{0}) -y^{obs}]^{T}R^{-1}[H(x_{0})-y^{obs}] \tag{15}\]where \(x_{0}\) represents the variable to be solved for. The formula can also be interpreted as assuming that the background field and error covariance are known, and ultimately the goal is to minimize the total analysis error covariance. The purpose of solving the cost function is to find an optimal initial field \(x_{0}\) that minimizes the cost function. The solution of the cost function is mainly achieved through the gradient function, which is iteratively solved using the following equation:  
\[\
abla J=\
abla J_{b}+\
abla J_{0}=B^{-1}(x_{0}-x_{b})+H^{T}R^{-1}[H(x_{0})-y^ {obs}] \tag{16}\]  
where \(H\) represents the tangent linear mode of the observation operator; \(H^{T}\) is the concomitant operator of this tangent linear mode. The number of iterations can be set artificially or stopped by setting \(\
abla J\) a decreasing limit. Three-dimensional variational assimilation (3D-Var) utilizes observations from specific time points, whereas four-dimensional variational assimilation (4D-Var) extends this by incorporating a forecast operator, thus integrating observations over time. [PERSON] and [PERSON] [143] significantly advanced 4D-Var by introducing the adjoint model, simplifying its solution and fostering its development. Unlike 3D-Var, the cost function in 4D-Var assimilation includes both the observation operator and the forecast model, thus extending the analysis across the temporal dimension. However, this temporal extension increases the computational cost and memory usage of 4D-Var and complicates the solution of the tangent linear adjoint model, which remains a challenging task [144]. The objective function of 4D-Var is:  
\[ \begin{split} J(x_{0})=\frac{1}{2}(x_{0}-x_{0}^{b})^{T}B^{-1}(x_ {0}-x_{0}^{b})+\frac{1}{2}\sum_{n=0}^{N}\{H_{n}\{M_{n}[\cdots M_{0}(x_{0})]\}- \\ y_{n}^{obs}\}^{T}R_{n}^{-1}\Big{\{}H_{n}\{M_{n}[\cdots M_{0}(x_{0} )]\}-y_{n}^{obs}\Big{\}}\end{split}  \tag{17}\]  
where \(x_{0}\) is the state at the initial moment; \(x_{0}^{b}\) is the initial background field; \(y_{n}^{obs}\) is the observation at the nth moment; \(M(t_{0},t_{i})(x_{0})\) is the state quantity predicted by the nonlinear model at the \(t_{i}\) moment; \(H_{i}\) is the observation operator that maps the state variables to the observation space at the \(t_{i}\) moment; \(B\) is the background error covariance; \(R\) is the observation error covariance at the \(t_{i}\) moment [145]. 4D-Var variational assimilation is a nonlinear optimization problem under constraint conditions. The solution to the four-dimensional variational assimilation problem adopts a similar algorithm to the three-dimensional variational assimilation, using the adjoint method and the optimal control theory of partial differential equations to solve it. The gradient of the objective function in 4D-Var is obtained as follows:  
\[