# Review of Satellite Remote Sensing of Carbon Dioxide Inversion and Assimilation
## 3 Carbon Satellite Remote Sensing Inversion Techniques
### Optimal Estimation Algorithm
#### 3.2.4 Empirical Algorithms  
Accurate retrieval of XCO\({}_{2}\) typically involves complex methods and significant computational resources, especially when processing data from orbiting satellites such as GOSAT-2, OCO-2, and OCO-3. Achieving high-precision XCO\({}_{2}\) measurements demands even greater computational power [89]. To enhance the spatial and temporal resolution of greenhouse gas (GHG) remote sensing satellite observations, there is a need for a substantial increase in the computational capacity for XCO\({}_{2}\) retrieval. In contrast, empirical algorithms offer a more efficient alternative. These algorithms leverage large datasets of observational samples to train models, bypassing the need for iterative optimization and forward radiative transfer calculations during the inversion process [90]. Unlike optimization algorithms, empirical methods do not provide an average kernel function or an error estimation matrix. Current empirical algorithms primarily include statistical regression and neural network approaches. Statistical Regression Inversion Algorithm  
The statistical inversion method estimates CO\({}_{2}\) concentrations by modeling the regression relationship between satellite spectral remote sensing data and atmospheric CO\({}_{2}\) levels. This approach involves fitting and analyzing observed data, thereby avoiding issues related to the adaptability of radiative transfer models and automatically correcting biases in the forward model. However, the accuracy of the inversion is influenced by the representativeness of the training samples and the chosen regression analysis method. In constructing the model, a large dataset of observations with known CO\({}_{2}\) concentrations is typically used as a training set. The weighting coefficients for individual observations are determined through regression analysis and optimization algorithms. Common regression methods include linear regression, polynomial regression, and support vector regression. For example, multivariate linear regression has been employed to model the linear relationships between GOSAT XCO\({}_{2}\) measurements and variables such as temperature, vegetation cover, and productivity [91]. During regression analysis, it is crucial to select an appropriate regression model and determine its parameters. However, the relationships between CO\({}_{2}\) and predictor variables can vary significantly over space and time and are often nonlinear. This complexity can make it challenging to accurately model these relationships using multiple linear regression alone. Therefore, data preprocessing steps--such as calibration, filtering, and interpolation--are often necessary to enhance the accuracy and stability of these algorithms. Neural Network Inversion Algorithm  
Sontag [92] theoretically demonstrated that any inversion problem can be addressed using a multilayer perceptron network with two hidden layers. This approach does not require linear assumptions regarding the radiative transfer equation or the assumption of normality in random variables, making it a powerful tool for solving nonlinear problems. With the advancement of artificial intelligence technology, although training neural networks is time-consuming and demands extensive a priori knowledge, they can perform efficient numerical iterative calculations during model application to meet the real-time data processing needs. Currently, this method is used to retrieve the mean atmospheric carbon dioxide concentration in the troposphere from meteorological satellites [93; 94]. The neural network inversion CO\({}_{2}\) algorithm estimates unknown CO\({}_{2}\) concentrations by creating a neural network model that relates observed remote sensing data to known CO\({}_{2}\) levels. This algorithm features adaptive learning and intelligent decision-making capabilities, with its core principle being the establishment of a nonlinear mapping between the observed data and CO\({}_{2}\) concentrations.