Extracting Common Mode Errors of Regional GNSS Position Time Series in the Presence of Missing Data by Variational Bayesian Principal Component Analysis

Missing Data Imputation in GNSS Monitoring Time Series Using Temporal and Spatial Hankel Matrix Factorization

GNSS time series for static reference stations record the deformation of monitored targets. However, missing data are very common in GNSS monitoring time series because of receiver crashes, power failures, etc. In this paper, we propose a Temporal and Spatial Hankel Matrix Factorization (TSHMF) method that can simultaneously consider the temporal correlation of a single time series and the spatial correlation among different stations. Moreover, the method is verified using real-world regional 10-year period monitoring GNSS coordinate time series. The Mean Absolute Error (MAE) and Root-Mean-Square Error (RMSE) are calculated to compare the performance of TSHMF with benchmark methods, which include the time-mean, station-mean, K-nearest neighbor, and singular value decomposition methods. The results show that the TSHMF method can reduce the MAE range from 32.03% to 12.98% and the RMSE range from 21.58% to 10.36%, proving the effectiveness of the proposed method.

A New Spatial Filtering Algorithm for Noisy and Missing GNSS Position Time Series Using Weighted Expectation Maximization Principal Component Analysis: A Case Study for Regional GNSS Network in Xinjiang Province

Common Mode Error (CME) presents a kind of spatially correlated error that is widespread in regional Global Navigation Satellite System (GNSS) networks and should be eliminated during postprocessing of a GNSS position time series. Several spatiotemporal filtering methods have been developed to mitigate the effects of CME. However, such methodologies become inappropriate when missing and noisy data exists. In this research, we introduce a novel spatial filtering algorithm called Weighted Expectation Maximization Principal Component Analysis (WEMPCA) for detecting and removing CME from noisy GNSS position time series with missing values, among which formal errors of daily GNSS solutions are utilized to weight the input data. Compared with traditional PCA and the special case of EMPCA, simulation experiments demonstrate that the new WEMPCA algorithm always has outstanding performance over others. The WEMPCA algorithm was then successfully used to extract the CME from real noisy and missing GNSS position time series in Xinjiang province. Our results show that only the first principal component exhibits significant spatial response, with average values of 70.11%, 66.53%, and 52.45% for North, East, and Up (NEU) components, respectively, indicating that it represents the CME of this region. After removing CME, the canonical correlation coefficients and root mean square error of GNSS residual time series, as well as the amplitudes of power-law noises (PLN), are obviously decreased in all three directions. However, the white noise (WN) amplitudes are found to diminish exclusively in the North and East component, not in the Up components. Moreover, the average velocity differences before and after filtering CME are 0.19 mm/year, 0.03 mm/year, and −0.56 mm/year for the NEU components, respectively, indicating that CME has an influence on the GNSS station velocity estimation. The velocity uncertainty is also reduced by 43.51%, 38.64%, and 40.39% on average for the NEU components, respectively, implying that the velocity estimates are more reliable and accurate after removing CME. Therefore, we conclude that the new WEMPCA approach provides an efficient solution to detect and mitigate CME from the noisy and missing GNSS position time series.

Independent Component Extraction from the Incomplete Coordinate Time Series of Regional GNSS Networks

Independent component analysis (ICA) is one of the most effective approaches in extracting independent signals from a global navigation satellite system (GNSS) regional station network. However, ICA requires the involved time series to be complete, thereby the missing data of incomplete time series should be interpolated beforehand. In this contribution, a modified ICA is proposed, by which the missing data are first recovered based on the reversible property between the original time series and decomposed principal components, then the complete time series are further processed with FastICA. To evaluate the performance of the modified ICA for extracting independent components, 24 regional GNSS network stations located in North China from 2011 to 2019 were selected. After the trend, annual and semiannual terms were removed from the GNSS time series, the first two independent components captured 17.42, 18.44 and 17.38% of the total energy for the North, East and Up coordinate components, more than those derived by the iterative ICA that accounted for 16.21%, 17.72% and 16.93%, respectively. Therefore, modified ICA can extract more independent signals than iterative ICA. Subsequently, selecting the 7 stations with less missing data from the network, we repeatedly process the time series after randomly deleting parts of the data and compute the root mean square error (RMSE) from the differences of reconstructed signals before and after deleting data. All RMSEs of modified ICA are smaller than those of iterative ICA, indicating that modified ICA can extract more exact signals than iterative ICA.

Modeling Seasonal Variations in Vertical GPS Coordinate Time Series Using Independent Component Analysis and Varying Coefficient Regression

Common seasonal variations in Global Positioning System (GPS) coordinate time series always exist, and the modeling and correction of the seasonal signals are helpful for many geodetic studies using GPS observations. A spatiotemporal model was proposed to model the common seasonal variations in vertical GPS coordinate time series, based on independent component analysis and varying coefficient regression method. In the model, independent component analysis (ICA) is used to separate the common seasonal signals in the vertical GPS coordinate time series. Considering that the periodic signals in GPS coordinate time series change with time, a varying coefficient regression method is used to fit the separated independent components. The spatiotemporal model was then used to fit the vertical GPS coordinate time series of 262 global International GPS Service for Geodynamics (IGS) GPS sites. The results show that compared with least squares regression, the varying coefficient method can achieve a more reliable fitting result for the seasonal variation of the separated independent components. The proposed method can accurately model the common seasonal variations in the vertical GPS coordinate time series, with an average root mean square (RMS) reduction of 41.6% after the model correction.

Impacts of Local Effects and Surface Loads on the Common Mode Error Filtering in Continuous GPS Measurements in the Northwest of Yunnan Province, China

While seasonal hydrological mass loading, derived from Gravity Recovery and Climate Experiment (GRACE) measurements, shows coherent spatial patterns and is an important source for the common mode error (CME) in continuous global positioning system (cGPS) measurements in Yunnan, it is a challenge to quantify local effects and detailed changes in daily GPS measurements by using GRACE data due to its low time and spatial resolutions. In this study, we computed and compared two groups of CMEs for nine cGPS sites in the northwest Yunnan province; rCMEs were computed with the residual cGPS time series having high inter-station correlations, while oCMEs were computed with all the GPS time series. The rCMEs-filtered time series had smaller variances and larger root mean square (RMS) reductions than those that were oCMEs-filtered, and when the stations local effects were not removed, spurious transient-like signals occurred. Compared with hydrological mass loading (HYDL), its combination with non-tidal atmosphere pressure and ocean mass reached a better agreement with the CME in the vertical component, with the Nash–Sutcliffe efficiency (NSE) increasing from 0.28 to 0.55 and the RMS reduction increasing from 15.19% to 33.4%, respectively. Our results suggest that it is necessary to evaluate the inter-station correlation and remove the possible noisy stations before conducting CME filtering, and that one should carefully choose surface loading models to correct the raw cGPS time series if CME filtering is not conducted.