# XCO\({}_{2}\) Fusion Algorithm Based on Multi-Source Greenhouse Gas Satellites and CarbonTracker
## 3 Methods
### TC Fusion Algorithm  
The Triple Collocation (TC) algorithm can evaluate the random error of three independent datasets [46]. Assuming that the error is irrelevant, each observation data in the dataset can be expressed as:  
\[\theta_{i}^{N}=\theta^{N}+\varepsilon_{i}^{N} \tag{5}\]  
where, \(\theta_{i}^{N}\) an \(\varepsilon_{i}^{N}\) respectively represent the \(N\)th observation of the \(i\)th dataset and its corresponding random error, and \(i\in\{1,2,3\}\). \(\theta^{N}\) is the true value corresponding to the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Season** & **Months** & **Latitude Zone** & **Length/\({}^{\circ}\)** & **Width/\({}^{\circ}\)** \\ \hline \multirow{8}{*}{Winter} & & 65\({}^{\circ}\) N–90\({}^{\circ}\) N & 4.95 & 5.41 \\  & & 30\({}^{\circ}\) N–65\({}^{\circ}\) N & 11.26 & 8.11 \\  & & 10\({}^{\circ}\) S–30\({}^{\circ}  
\(N\)th observation. There may be different degrees of systematic deviation between the observed dataset and the true value. Assuming that there is a linear relationship between the observation and the true value, (5) can be written as:  
\[\theta_{i}^{N}=a_{i}\theta^{N}+b_{i}+\varepsilon_{i}^{N} \tag{6}\]  
where, \(a_{i}\) and \(b_{i}\) are the regression coefficient between the observation and the true value. The TC algorithm obtains the error term \(\varepsilon_{i}\) in (6) by calculating the covariance between the datasets:  
\[Cov(\theta_{i},\theta_{j})=a_{i}a_{j}\sigma_{\theta}^{2}+a_{i}Cov(\theta, \varepsilon_{j})+a_{j}Cov(\theta,\varepsilon_{j})+Cov(\varepsilon_{i}, \varepsilon_{j}) \tag{7}\]  
where, \(\sigma_{\theta}^{2}\) is the variance of the true dataset \(\theta\). Since the TC algorithm needs to meet the following assumptions: 1\(E(\varepsilon_{i})=0\); 2\(Cov(\theta,\varepsilon_{i})=0\); 3\(Cov(\varepsilon_{i},\varepsilon_{j})=0\) (\(i\
eq j\)), (7) can be rewritten as:  
\[C_{ij}=Cov(\theta_{i},\theta_{j})=\left\{\begin{array}{l}a_{i}a_{j}^{2} \sigma_{\theta}^{2},i\
eq j\\ a_{i}^{2}a_{\theta}^{2}+\varepsilon_{i}^{2},i=j\end{array}\right. \tag{8}\]  
According to (8), the estimation expression of random error term \(\varepsilon_{i}\) is:  
\[\varepsilon_{i}=\left\{\begin{array}{l}\sqrt{C_{11}-\frac{C_{12}C_{13}}{C_{ 23}}},i=1\\ \sqrt{C_{22}-\frac{C_{12}C_{23}}{C_{13}}},i=2\\ \sqrt{C_{33}-\frac{C_{13}C_{23}}{C_{12}}},i=3\end{array}\right. \tag{9}\]  
GOSAT, OCO-2, and CT XCO\({}_{2}\) have different prior distributions and mean kerns, which describe the sensitivity of the true state of the atmosphere. Recent studies have shown that the influence of column averaging on satellite XCO\({}_{2}\) is relatively small compared to the measurement accuracy, so the influence of different mean cores and the difference in the distribution of prior CO\({}_{2}\) are not considered in this paper. The three XCO\({}_{2}\) datasets used in this study have their own characteristics and uncertainties due to different observation frequencies or inversion algorithms. On this basis, the uncertainties were quantitatively analyzed using the TC algorithm, and the three datasets are further fused by linearly combining the three datasets of GOSAT, OCO-2, and CT:  
\[\theta_{new}=w_{1}\theta_{1}+w_{2}\theta_{2}+w_{3}\theta_{3} \tag{10}\]  
where, \(\theta_{new}\) is the fused dataset; \(w_{i}\) are weight coefficients, \(\theta_{1},\theta_{2},\theta_{3}\) are the OCO-2, GOSAT, and CT dataset, respectively. In order to ensure that the fusion XCO\({}_{2}\) value is unbiased, it is needed to meet the condition: \(w_{1}+w_{2}+w_{3}=1\), where \(w_{i}\) can be determined by the random error of TC estimation:  
\[w_{i}=\left(I^{T}C^{-1}I\right)^{-1}I^{T}C^{-1}(i=1,2,3) \tag{11}\]