# Review of Satellite Remote Sensing of Carbon Dioxide Inversion and Assimilation
## 4 Carbon Flux Data Assimilation and Inversion Studies
### Framework and Techniques for Assimilation of Atmospheric CO\({}_{2}\)
#### 4.3.3 Satellite CO\({}_{2}\) Column Concentration Data-Assimilation Process  
Data-assimilation techniques for atmospheric CO\({}_{2}\) aim to determine the optimal CO\({}_{2}\) flux by minimizing the difference between simulated and observed concentrations. Different assimilation methods use distinct cost functions. The assimilation process is divided into two parts: the forecast process of CO\({}_{2}\) flux (state forecast) and the analysis process (state analysis). The state forecast step calculates the current state variable (CO\({}_{2}\) flux) and estimates the error covariance to create a prior estimate for the next time step. Typically, a simple forecast model, such as the identity matrix I, is used for this. The state analysis step combines the prior estimate with new measurement data to produce an improved posterior estimate. The CO\({}_{2}\) flux from the forecast process serves as input for simulating current CO\({}_{2}\) concentrations using atmospheric transport models (e.g., TMS, GEOS-Chem). The simulated CO\({}_{2}\) concentrations are then compared with observed concentrations to refine the CO\({}_{2}\) flux and optimize carbon sources and sinks.  
The satellite column concentration-assimilation-cycle integrates observations, atmospheric transport models, state forecasts, and state analysis. Unlike other data-assimilation processes, it incorporates state variables across multiple time steps within the same carbon-assimilation cycle (with a time step of one week). This approach accounts for the time lag between CO\({}_{2}\) flux and concentration, as the observed CO\({}_{2}\) concentration at any given time reflects the accumulation of CO\({}_{2}\) flux transport and diffusion from previous periods. This lag issue is described mathematically as a \"lag window\" in the assimilation system. The entire assimilation process includes several cycles, with each cycle's duration determined by the assimilation window. For clarity, the \"lag window\" is set to 5 weeks. Figure 24 illustrates the assimilation-cycle process, where the state variable (CO\({}_{2}\) flux) is denoted as \(x_{i}\)(0,1,2,3,4). The numbers in parentheses (0,1,2,3,4) indicate the number of times the state variable (CO\({}_{2}\) flux) has been assimilated in previous cycles, and the subscript \(i\) represents the set of state variables (CO\({}_{2}\) flux). The light blue box shows the background field (forecast field) of the state variable, with data representing the background state vectors. The blue area denotes the assimilation-cycle process, while the light green box represents the analysis field of the state variable, with data as the analysis state vectors.  
The complete assimilation inversion CO\({}_{2}\)-cycle process is described as follows:  
1. In the first analysis cycle, the background state vector is updated using observational data. At time \(t\), the driving data \(x_{i}\)(0, 1, \(\cdots\), 4) within the light blue box is used to drive the atmospheric transport model to simulate the CO\({}_{2}\) concentration values from CO\({}_{2i}\)  
the cycle to start a new assimilation process. Afterward, in each analysis cycle, the analysis state vector from the previous cycle is used as the background state vector for the current cycle.