Mass redistribution of the atmosphere, oceans, and terrestrial water storage generates crustal displacements which can be predicted by environmental loading models and observed by the Global Positioning System (GPS). In this paper, daily height time series of 235 GPS stations derived from a homogeneously reprocessed Crustal Movement Observation Network of China (CMONOC) and corresponding loading displacements predicted by the Deutsche GeoForschungsZentrum (GFZ) are compared to assess the effects of loading corrections on the nonlinear variations of GPS time series. Results show that the average root mean square (RMS) of vertical displacements due to atmospheric, nontidal oceanic, hydrological, and their combined effects are 3.2, 0.6, 2.7, and 4.0 mm, respectively. Vertical annual signals of loading and GPS are consistent in amplitude but different in phase systematically. The average correlation coefficient between loading and GPS height time series is 0.6. RMS of the GPS height time series are reduced by 20% on average. Moreover, an investigation of 208 CMONOC stations with observing time spans of ~4.6 years shows that environmental loading corrections lead to an overestimation of the GPS velocity uncertainty by about 1.4 times on average. Nevertheless, by using a common mode component filter through principal component analysis, the dilution of velocity precision due to environmental loading corrections can be compensated.

Removal of the common mode error (CME) is very important for the investigation of global navigation satellite systems’ (GNSS) error and the estimation of an accurate GNSS velocity field for geodynamic applications. The commonly used spatiotemporal filtering methods normally process the evenly spaced time series without missing data. In this article, we present the variational Bayesian principal component analysis (VBPCA) to estimate and extract CME from the incomplete GNSS position time series. The VBPCA method can naturally handle missing data in the Bayesian framework and utilizes the variational expectation-maximization iterative algorithm to search each principal subspace. Moreover, it could automatically select the optimal number of principal components for data reconstruction and avoid the overfitting problem. To evaluate the performance of the VBPCA algorithm for extracting CME, 44 continuous GNSS stations located in Southern California were selected. Compared to previous approaches, VBPCA could achieve better performance with lower CME relative errors when more missing data exists. Since the first principal component (PC) extracted by VBPCA is remarkably larger than the other components, and its corresponding spatial response presents nearly uniform distribution, we only use the first PC and its eigenvector to reconstruct the CME for each station. After filtering out CME, the interstation correlation coefficients are significantly reduced from 0.43, 0.46, and 0.38 to 0.11, 0.10, and 0.08, for the north, east, and up (NEU) components, respectively. The root mean square (RMS) values of the residual time series and the colored noise amplitudes for the NEU components are also greatly suppressed, with average reductions of 27.11%, 28.15%, and 23.28% for the former, and 49.90%, 54.56%, and 49.75% for the latter. Moreover, the velocity estimates are more reliable and precise after removing CME, with average uncertainty reductions of 51.95%, 57.31%, and 49.92% for the NEU components, respectively. All these results indicate that the VBPCA method is an alternative and efficient way to extract CME from regional GNSS position time series in the presence of missing data. Further work is still required to consider the effect of formal errors on the CME extraction during the VBPCA implementation.