# 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
[159] proposed a deep learning (DL) model that integrates Kalman filter-based reduced-order modeling, neural networks, and data assimilation, as shown in \[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]5 This model combines the efficiency of reduced-order modeling (ROM), the accuracy of data assimilation (DA), and the speed of deep learning. It effectively addresses issues such as dimensionality constraints, noisy data, and low-quality data, with PCA enabling DA to be conducted in a reduced-dimensional space. [PERSON] et al. [160] introduced Generalized Latent Assimilation (GLA), a system that integrates a reduced-order surrogate model using Autoencoder (AE) and Long Short-Term Memory (LSTM) techniques with advanced data-assimilation methods. This system adjusts model parameters in real-time by incorporating observational data. Additionally, it employs a local smoothing surrogate function to connect the spatial representation of encoded system variables with current observations, facilitating variational data assimilation while reducing computational costs. The flowchart for Generalized Latent Assimilation is depicted in \[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]6  
\[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]7 Integration diagram of DA, PCA and NNs. \[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]8 Flowchart of LA with heterogeneous potential space. \[P_{n}^{f}=MA_{n-1}M^{T}+Q_{n-1} \tag{20}\]9 Schematic representation of the system correction. The DA step is used to estimate the system state from sparse and noisy observations, while the ML step is used to estimate the parameters of statistical calibration based on the estimated state. Error covariance matrix modeling is a crucial component in data-assimilation algorithms, as it significantly impacts prediction accuracy. Estimating these covariances typically relies on empirical assumptions and physical constraints, which can be imprecise and computationally demanding, particularly for high-dimensional systems. [PERSON] et al. [161] developed and tested an innovative data-driven approach using Long Short-Term Memory Recurrent Neural Networks (LSTM) to enhance the accuracy and efficiency of covariance specification for dynamic data-assimilation problems. During training, synthetically generated observation sequences are used as inputs to the LSTM. A key advantage of this data-driven approach is its ability to handle both non-parametric and parametric covariance models. [PERSON] et al. [162] introduced a new data-driven method called HDA-MLP to address data-assimilation challenges, as illustrated in \[K_{n}=P_{n}H^{T}(HP_{n}H^{T}+R_{n})^{-1} \tag{21}\]0 This method involved formal optimization of individual data-assimilation techniques (3D-Var and EnKF) using Multi-Layer Perceptrons (MLP), while another neural network model implicitly updated the background error covariance matrix. By leveraging short-term predictions to correct current assimilation results, this approach reduced dependence on traditional data-assimilation methods. The results demonstrated improved robustness and a better capability to capture variations in state variables, significantly enhancing both the quality of the analysis and computational efficiency. Recent research has increasingly employed machine learning (ML) techniques to develop end-to-end learning solutions for entire data-assimilation (DA) systems, mapping observed data directly to target state sequences or model parameters. [PERSON] et al. [163] introduced a framework that integrates neural networks (NN) with physical models via a data-assimilation algorithm to enhance model error correction and prediction accuracy, as depicted in \[K_{n}=P_{n}H^{T}(HP_{n}H^{T}+R_{n})^{-1} \tag{21}\]1 The effectiveness of this framework was demonstrated through various case studies and sensitivity analyses. The figure illustrates the overall process of the Deep Data-Assimilation (DDA) model.