The effective extraction and elimination of the Global Positioning System's (GPS) common mode error (CME) is of great significance to improving the signal-to-noise ratio of the coordinate time series and accurately estimating the characteristics of the crustal deformation. In this study, the principal component analysis (PCA), Karhunen-Loeve expansion (KLE) and stacking filtering methods are used to extract the CMEs of Greenland GPS station coordinates in the East-North-Up (ENU) directions. Their influence on GPS coordinate residuals and the relationship amongst the observations are also analyzed. The results show that the PCA, KLE and stacking filtering methods can effectively eliminate the common mode noise in the GPS station coordinate time series and reduce their uncertainty. The PCA method is better than KLE and stacking methods, and the filtering effect in the up (U) direction is better than those in the east (E) and north (N) directions. Then, as the CME can be detected periodically in the U direction, its physical mechanism is studied. On the one hand, the period of CME maximum power is 1 year but is not constant. On the other hand, the correlation between the GPS CME and GRACE ice loading deformation is 0.5. When the ice loading deformation is deduced by the GRACE method, the CME of the GPS network can be eliminated by 25.6%, which indicates that the ice loading deformation is part of the source of GPS CME in Greenland. Finally, the influence of the U direction CME on the station noise characteristics is analyzed using maximum likelihood estimation (MLE). The results show that the CME noise type is mainly WN + FN. Moreover, the estimated coordinate velocities and most of the annual amplitudes of the stations have an increasing trend after CME elimination, whereas the uncertainty has a decreasing trend. These findings are consistent with the trend of noise component change. The above results indicate that the CME elimination of the GPS stations cannot be ignored, as they can help to improve the estimation of station velocity and annual amplitude and reduce the uncertainty.