# 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
For the \(k\) ensembles \(X\) at time step \(n-1\), the predicted state variables are as follows:  
\[x_{k,n}^{f}=Mx_{k,n-1}^{a}+\varepsilon_{k,n-1} \tag{24}\]  
where \(M\) is the prediction model; \(\varepsilon_{k,n-1}\) is the corresponding error term. By using these randomly simulated sets, the covariance of the sample error can be estimated. \[P_{n}^{f}=\frac{1}{K-1}\sum_{i=1}^{K}[x_{k,n}^{f}-\overline{x_{n}^{f}}][x_{k, n}^{f}-\overline{x_{n}^{f}}]^{T} \tag{25}\]  
\[\overline{x_{n}^{f}}=\frac{1}{k}\sum_{i=1}^{k}x_{k,n}^{f} \tag{26}\]  
where \(x_{k,n}^{f}\) is the forecast value of the \(k\) ensemble at the nth moment. Theoretically, as the number of ensembles \(N\) rises, the Monte Carlo sampling error decreases proportionally to \(1/\sqrt{N}\)[147], but the rise in the number of ensembles brings with it a greater amount of computation and memory. The state update equation can also be obtained for the set Kalman:  
\[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]0  
\[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]1  
\[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]2  
where \(M\) and \(H\) are the model operator and observation operator respectively; \(R\) is the observation error covariance, and \(y_{k,n-1}^{obs}\) is also the observed value of the \(k\) ensemble at the \(n\)th moment; \(x_{k,n}^{a}\) is the value obtained from the \(k\) ensemble corrected at the \(n\)th moment. The analyzed error covariance can be calculated as follows:  
\[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]3  
The size of the set ranges from dozens to hundreds depending on the situation. As the number of sets increases, the estimation accuracy is expected to be higher. However, the computational complexity also increases. If the number of sets is too small, the average value cannot truly represent the objective state. Therefore, further research is needed to determine the most appropriate number of sets to choose. The advent of remote sensing satellites, the increase in observational data, and the rising demand for accuracy necessitate more sophisticated variational methods and Kalman filter models to meet application needs and mitigate implementation challenges. However, these methods continue to grapple with issues related to model errors, parameter estimation, and error covariance specification [150]. Recently, machine learning (ML) has shown promise in approximating nonlinear systems and extracting meaningful insights from high-dimensional data [151; 152]. ML algorithms can complement or even replace traditional methods by addressing their assumptions and limitations, correcting model errors, and facilitating real-time improvements and adjustments during data assimilation and prediction. Typically, model correction methods involve using statistical models, such as neural networks, to adjust knowledge-based physical models. One straightforward approach is to apply a correction to each model. [PERSON] et al. [153] proposed an offline learning method that alternates between data assimilation (DA) and ML to learn system states and observations in sparse and noisy environments. This method, which minimizes a defined loss function through coordinate descent [154], is illustrated in \[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]4 Additionally, [PERSON] et al. [155; 156; 157] developed an online learning method that replaces offline learning, utilizing enhanced state DA techniques to estimate system states and statistical correction parameters. Due to the complexity of system models, online parameter estimation for high-dimensional dynamical systems has long been a challenge [158]. For handling high-dimensional big data, reduced-order modeling and low-dimensional surrogate models generated by machine learning algorithms have been increasingly applied to enhance algorithm efficiency. Traditionally, dimension reduction methods included Principal Component Analysis (PCA) and Empirical Orthogonal Function (EOF) techniques. [PERSON] et al.