Abstract
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.
Highlights
In the last three decades, global navigation satellite system (GNSS) technology has provided abundant, high-accuracy position information for the Earth, which allows researchers to investigate many types of geophysical phenomena, such as geocenter motion [1,2], crustal deformation [3,4], seismic monitoring [5,6], and glacial isostatic adjustment [7,8]
The probabilistic principal component analysis (PPCA) method was first introduced by Gruszczynski et al [29] to filter out common mode error (CME) from time series with few common observational epochs, yet the PPCA method is sometimes sensitive to the initial parameters and prone to face the over-fitting problem due to the usage of maximum likelihood criterion to update model parameters [30]
CME has a certain influence on the velocity estimation of the GNSS stations
Summary
In the last three decades, global navigation satellite system (GNSS) technology has provided abundant, high-accuracy position information for the Earth, which allows researchers to investigate many types of geophysical phenomena, such as geocenter motion [1,2], crustal deformation [3,4], seismic monitoring [5,6], and glacial isostatic adjustment [7,8]. For regional GNSS networks, strong CME, coming from the reference frame error, mis-modeling of satellite orbits and clocks, large-scale environmental effects, etc., presents in the position time series [11]. This makes it difficult to discern weak and transient tectonic signals, such as slow slip events and aseismic episodic tremor, from GNSS data. The aforementioned methods are unable to identify stations with strong local effects that could affect the CME detection, and the suitable weights for all stations at each epoch still need to be investigated Another way to remove the CME is to transform the solutions into a regional reference frame [17,18]. The probabilistic principal component analysis (PPCA) method was first introduced by Gruszczynski et al [29] to filter out CME from time series with few common observational epochs, yet the PPCA method is sometimes sensitive to the initial parameters and prone to face the over-fitting problem due to the usage of maximum likelihood criterion to update model parameters [30]
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