## 3 Results and Discussions
### Consistency Analysis between GNSS, GRACE and GGFC
#### 3.1.2 Quantitative Evaluation with Correlation Coefficient and WRMS Reduction  
Figures 4 and 5 display comparison results of the correlation coefficients of GNSS, GRACE and GGFC. Both results from GNSS vs. GRACE and GNSS vs. GGFC indicate a good correlation. Despite  
Figure 3: Harmonic annual VD derived from GNSS monthly time series after removing the GRACE (red) and GGFC (yellow) modeled VD. The length of the vector represents the seasonal amplitude, and the vector direction points directly to the direction of the initial phase angles. the number of stations with correlation larger than 0.5 between GNSS and GGFC being 2 more than that of between GNSS and GRACE, the number of stations with negative correlation for GNSS/GGFC is 5 more than that for GNSS/GRACE. The average of correlation coefficients is 0.45 and 0.49 for GNSS vs. GGFC and GNSS vs. GRACE, respectively, denoting that GRACE loading deformations are more consistent with GNSS series. Accordingly, to further determine the similarity between GNSS, GRACE and GGFC VD series, the WRMS reduction of GNSS signal is illustrated by subtracting the GRACE or GGFC VD from the GNSS VD series. The \(WRMS\) reduction is calculated as \(100\). (\(\frac{WRMS_{\mathcal{GPS}}-WRMS_{\mathcal{GPS}}-GACE_{\mathcal{GPS}}\cdot GGFC}{WRMS_{ \mathcal{GPS}}}\)), where \(WRMS_{\mathcal{GPS}}\) is the signal \(WRMS\) of the GNSS data and \(WRMS_{\mathcal{GPS}-GACE_{\mathcal{GPS}}\cdot GGFC}\). The signal \(WRMS\) of the GNSS corrected by GRACE solutions and GNSS corrected by GGFC solutions. Figure 6 and Table 2 present the reduction in the GNSS signal \(WRMS\) by subtracting GRACE and GGFC deformations. The mean value of the \(WRMS\) variations for all 186 stations is \(-12.3\%\) and \(-5.6\%\), respectively, for GNSS corrected by GRACE solutions and GNSS corrected by GGFC solutions. From Table 2, in terms of the station number, after correcting GNSS with GRACE and GGFC, there are \(80\%\) and \(74\%\) of stations displaying \(WRMS\) reduction. In terms of the \(WRMS\) reduction range, results for GGFC corrections have 24 more stations than that for GRACE for \(WRMS\) with \(20\%\) variations. For \(WRMS\) reduction between 0 and \(20\%\), results for GRACE corrections have 35 more stations than that for GGFC corrections. For the correction effect of \(WRMS\) reduction more than \(20\%\), the GNSS corrected by GGFC solutions are better than the GNSS with GRACE corrections in the number of stations (24 more stations). For the \(WRMS\) reduction interval of 0-20%, the GRACE correction effect is significantly better than the GGFC correction effect. The mean value of the \(WRMS\) reduction for the two types of correction results is close to each other, \(33\%\)/\(28\%\) for reduction more than \(20\%\) and \(10\%\)/\(11\%\) for reduction between 0 and \(20\%\), respectively, for GNSS corrected by GGFC solutions and GNSS corrected by GRACE solutions. Nevertheless, for the 49 stations with \(WRMS\) increase in GNSS corrected by GGFC solutions, the average \(WRMS\) increase is \(48\%\). Therefore, we conclude that the correction magnitude of GGFC loading is bigger than that of GRACE loading. However, the large magnitude is bidirectional, and the GGFC loading can bring better correction results for \(WRMS\) reduction more than \(20\%\); at the same time, the amount of the \(WRMS\) increase is also larger than that of GRACE corrections. The GRACE corrections are generally better for the global stations and can be a more reliable choice when correcting the non-tidal loading deformations in GNSS stations around the world. This is mainly because the deficiency of the GGFC model in islands. For GGFC, ATML is modeled using input from the same NCEP surface pressure grids, NTOL is modeled using the same ECCO data as input, and CWS is modeled using the same GLDAS data soil moisture and snow water equivalent. For all data sets, the GGFC provides a Fortran program that uses a bi-cubic interpolation scheme that allows the user to interpolate the grids to any site.