# Evaluation of the Consistency of Three GRACE Gap-Filling Data
## 2 Materials and Methods
### Fusion Method Based on Triple-Collocation  
SSA, IGG, and QF all have the ability to fill in the missing data of GRACE, but we cannot treat any of them as a genuine observation of the \"truth\" due to the uncertainties in their data processing methods. In the absence of significant coarse errors in observations, the weighted average result can be considered the optimal estimate of the missing GRACE data. Here we introduced a triple configuration (TC) approach to fuse these three gap-filling data. The TC method was firstly proposed by [PERSON] to correct for scattermeter-driven ocean wind speed and estimate its error [36]. It has been widely used for estimating random error variances and uncertainties in remote sensing data, such as soil moisture [37; 38], leaf area index [39], sea surface salinity [40], and ocean winds [36; 41]. The key to the fusion of SSA, IGG, and QF data is the determination of the weights, which are closely related to the variance of the observations. The TC method can be a promising candidate for estimating the error variance of SHCs of SSA, IGG, and QF. Assuming that the errors of these data are independent of each other, a linear regression model is constructed:  
\[X_{i}=\alpha_{i}+\beta_{i}t+\varepsilon_{i} \tag{5}\]  
where \(X_{i}\) (\(i=1\), \(2\), \(3\)) represents the SHCs series of the SSA, QF, and IGG, respectively. \(t\) represents the true underlying value. \(\varepsilon_{i}\) represents the error of SHCs of SSA, QF, and IGG, respectively. \(\alpha_{i}\) and \(\beta_{i}\) are the intercept and slope, respectively. Assuming that the error expectations for these three data sources are zero and uncorrelated with each other and with \(t\), we can obtain the TC estimation equation for error variances:  
\[\left[ \begin{array}{c}\sigma_{\varepsilon_{1}}^{2}\\ \sigma_{\varepsilon_{2}}^{2}\\ \sigma_{\varepsilon_{3}}^{2}\end{array} \right]=\left[ \begin{array}{c}Q_{11} -\frac{Q_{12}Q_{13}}{Q_{23}}\\ Q_{22}-\frac{Q_{12}Q_{23}}{Q_{13}}\\ Q_{33}-\frac{Q_{13}Q_{23}}{Q_{12}}\end{array} \right] \tag{6}\]  
where \(Q_{11}\), \(Q_{12}\), \(Q_{13}\), \(Q_{22}\), \(Q_{23}\), and \(Q_{33}\) are the covariances. \(\sigma_{\varepsilon_{1}}^{2}\), \(\sigma_{\varepsilon_{2}}^{2}\), and \(\sigma_{\varepsilon_{3}}^{2}\) are the error variances of SSA, QF, and IGG, respectively. A detailed derivation process could refer to [PERSON] et al. (2014) [42]. Then we can use Equation (7) to incorporate these three data. \[Y=\omega_{1}X_{1}+\omega_{2}X_{2}+\omega_{3}X_{3} \tag{7}\]  
where \(Y\) is the result after fusion, and \(\omega_{i}\) represents weight calculated from  
\[\left[ \begin{array}{c}\omega_{1}\\ \omega_{2}\\ \omega_{3}\end{array} \right]=\left[\begin{array}{c}\frac{\delta_{ \varepsilon_{  
The Swarm solutions ([[http://icgem.gtz-potsdam.de/series/02_COST-G/Swarm](http://icgem.gtz-potsdam.de/series/02_COST-G/Swarm)]([http://icgem.gtz-potsdam.de/series/02_COST-G/Swarm](http://icgem.gtz-potsdam.de/series/02_COST-G/Swarm)), Potsdam, German, April 2022) for the same period are used as a reference for the calculation of rRMS values Equation (3). The average gravity field measured from GRACE for the period January 2003 to December 2010 is subtracted from the Swarm result. Considering the limited spatial resolution of the QF, only the low-degree terms (degree \(\leq\)11,140 coefficients in total) were compared. Figure 1 plots the rRMS values between the three datasets and Swarm solutions on the degree-order plane. The colored squares represent rRMS values and the white ones indicate the rRMS values are higher than 1.0. The numbers of coefficients with rRMS values between 0.5 and 1.0 and within the range <0.5 are enumerated separately and listed in the upper left corner of each subplot. According to Figure 1, the statistics of rRMS values for QF and SSA are largely comparable across the three periods. There are approximately 82% of rRMS values smaller than 1 and approximately 25% smaller than 0.5.