# XCO\({}_{2}\) Fusion Algorithm Based on Multi-Source Greenhouse Gas Satellites and CarbonTracker
## 3 Methods  
### Interpolation Algorithm  
OCO-2 and GOSAT have a large number of areas without XCO\({}_{2}\) observations, especially in high latitudes, oceans, and other areas. Therefore, based on the Kriging interpolation method [44], this paper realized the filling of these blank areas by combining the inherent change trend in time and space and other relevant characteristics of XCO\({}_{2}\) data. The Kriging interpolation method, also known as the spatial autocovariance optimal interpolation method, obtains the distance range affecting the interpolated points by analyzing the variation distribution of spatial attributes at the spatial position. According to the sampling points within this range, the attribute value of the interpolated points is estimated. This paper adopted the method of ordinary Kriging interpolation. The sample point to be measured is obtained from the measured value of sample points, and the estimated value of the sample point to be measured is:  
\[Z^{*}(x_{0})=\sum_{i=1}^{n}\lambda_{i}Z(x_{i}) \tag{1}\]  
where, \(\lambda\) (\(i\) = 1,2,3..., \(n\)) is the weight coefficient, which is a set of optimal coefficients that can meet the minimum difference between the estimated value and the real value at the point. The weight coefficient needs to meet the following two conditions: (1) the estimation is unbiased, that is, the mathematical expectation of the deviation is zero; (2) the estimation is optimal, that is, the variance between the estimated value and the actual value is the smallest. As shown in (2). \[\left\{ \begin{array}{l}E(\hat{z}_{o}-z_{o})=0\\ \min_{\lambda_{i}}\!\!Var(\hat{z}_{o}-z_{o})\end{array} \right. \tag{2}\]  
where: \(\hat{z}_{o}\) is the estimated value and \(z_{o}\) is the real value. Satisfy Kriging equations, as shown in (3). \[\left\{ \begin{array}{l}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\lambda_{j }\gamma\big{(}x_{i},x_{j}\big{)}+\mu=\gamma(x_{i},x_{0})\\ \sum\limits_{i=1}^{n}\lambda_{i}=1\end{array} \right. \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Site Area** & **Site** & **Site ID** & **Longitude/\({}^{\circ}\)** & **Latitude/\({}^{\circ}\)** \\ \hline \multirow{2}{*}{East Asia} & Saga, Japan [36] & JS & 130.29 & 33.24 \\  & Tsukuba, Ibaraki, Japan, & & 125 HR [37] & & 140.12 & 36.05 \\  & Reunion Island (Ile de La & & & \\ North America & Reunion), France [38] & PA & \(-90.27\) & & 45.94 \\  & Lamont, OK, USA [39] & OC & \(-97.49\) & & 36.6 \\ European & Izana, Tenerife, Spain [40] & IZ & 16.48 & & 28.3 \\  & Bremen, Germany [41] & BR & 8.85 & & 53.1 \\ Oceania & Wollongong, Australia [42] & WG & 150.88 & \(-34.41\) \\  & Darwin, Australia [43] & DB & 130.89 & \(-12.43\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Information of TCCON Sites Used in this Study. where: \(x_{i}\) and \(x_{0}\) represent the position of the \(i\)th sample and unknown point, and \(\upmu\) is the Lagrange constant. The weight coefficient can be obtained by solving the Kriging equations, and then the estimated value can be obtained through (1). Variogram plays an important role in Kriging interpolation, which mainly studies the relationship and spatial structure of regionalized variables [45]. In this paper, Kriging interpolation was carried out based on the screened data. By analyzing the spatial characteristics of the XCO\({}_{2}\) data and the accuracy results of different Kriging variogram, the exponential function model is selected. The variogram model formula used is as follows:  
\[\gamma(h)= \begin{cases}0&h=0\\ c_{0}+c\Big{(}1-e^{-\frac{h}{2}}\Big{)}&h>0\end{cases}  \tag{4}\]  
where, \(h\) is lag distance, \(c_{0},c\), and \(\alpha\) are the interpolation model parameters to be estimated. Due to the slow flow rate of CO\({}_{2}\) in the atmosphere, we believe that the distribution of CO\({}_{2}\) does not exhibit significant differences in its properties in various directions. Therefore, the isotropic parameters of the Kriging interpolation used in this study are true.