## 3 Results and Discussions
### Consistency Analysis between GNSS, GRACE and GGFC
#### 3.1.1 Similarity of Annual Harmonic Signals  
In order to get an overview of general properties of annually recurring signals for a larger number of stations, we follow the expression means of [25]. The amplitudes and phases of vertical annual signals estimated by GNSS, GRACE and GGFC observations are shown in Figure 2. The length of the vector represents the seasonal amplitude, and the vector direction points directly to the direction of the initial phase angles (counterclockwise from the beginning of the east).  
Figure 2: Harmonic annual vertical deformation (VD) derived from GNSS (blue), Gravity Recovery and Climate Experiment (GRACE) (red) and Global Geophysical Fluid Center (GGFC) (yellow) for monthly time series. The length of the vector represents the seasonal amplitude, and the vector direction points directly to the direction of the initial phase angles.  
Generally, at most stations, there are good consistency between GNSS, GGFC and GRACE. In terms of annual amplitudes, the range of GNSS amplitude is 0.3-10.5 mm. For the GRACE results, the annual amplitude range is 0.5-8.3 mm, and for the GGFC results, the annual amplitude range is 0.5-8.0 mm. The mean annual amplitude is 3.5 mm for GNSS, 2.7 mm for GRACE and 2.6 mm for GGFC. Therefore, GNSS annual amplitudes are larger than those of both GRACE and GGFC, on average by a factor of about 1.3. Although GRACE and GGFC derived non-tidal deformations can interpret most of seasonal annual signals observed by GNSS, there are still annual amplitude variations remaining in the GNSS time series in addition to the surface deformation caused by hydrological, non-tidal ocean and atmospheric factors. For annual phases, differences between GNSS, GRACE and GGFC are small. The mean value of the annual phase difference for the GNSS versus (vs.) GRACE and GNSS vs. GGFC is 8.8\({}^{\circ}\) and 19.6\({}^{\circ}\), indicating that there is no significant systematic phase difference between GNSS and GRACE data.  
Figure 3 shows the harmonic annual VD estimated from GNSS monthly time series after removing the GRACE and GGFC modeled VD. Generally, seasonal amplitudes reduced with different levels for different stations. The amplitude of the GRACE correction result is larger than the GGFC correction result in the northern hemisphere, while the southern hemisphere is the opposite in most cases. The annual phase of GNSS-GRACE and GNSS-GGFC solutions is overall consistent, and the inconsistency mainly occurs in the Greenland region of the Northern Hemisphere and Central Europe.  
Table 1 lists statistical results for annual signals from different techniques and combination solutions. On the whole, the annual amplitude of the GNSS time series is the most significant, followed by GRACE, and GGFC is the smallest, suggesting that in addition to the surface deformation caused by hydrological, non-tidal ocean and atmospheric factors, there are still annual amplitude variations remaining in the GNSS time series. After correcting the GNSS results with GRACE and GGFC loading deformations, the amplitude of the GNSS corrected by GGFC solutions is more optimal than the amplitude of the GNSS corrected by GRACE solutions, which may be owing to the filtering of the GRACE gravity field SHCs preprocessing. To some extent, this is to be expected, as the Gaussian averaging used to smooth the GRACE data also has the effect of reducing the amplitude of the true signal.