The Consideration of Formal Errors in Spatiotemporal Filtering Using Principal Component Analysis for Regional GNSS Position Time Series  
[PERSON]  
1 College of Geomatics, Shandong University of Science and Technology, Qingdao 266590, China; [EMAIL_ADDRESS]  
[PERSON]  
22 College of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, China  
be too optimistic, with a factor of 5-11 [10]. Therefore, temporal noise should be estimated correctly. Generally, they are estimated with maximum likelihood estimation [11], least squares variance component estimation [7; 12] and minimum norm quadratic unbiased estimation [13]. Temporal noise is reduced after the removal of CME [11; 14], thus it directly affects velocity estimation, especially accuracy estimates. Our previous work has shown that the velocity accuracy increased by about 28% after CME was removed [15]. [PERSON] et al. [16] also confirmed that velocity accuracy was improved after CME was removed in the analysis of GNSS time series in Central Europe; therefore, they highly recommended the CME be filtered out before velocity estimation. Removing CME from a regional GNSS network can enhance the signal to noise ratio and is helpful for the detection of weak transient signals. The process of removing CME is called spatiotemporal filtering [17]. For a small regional GNSS network, both Stacking [17] and weighted Stacking [18], which assumes that CME is spatially homogenous, are widely used to compute CME, of which the latter takes formal errors into account. However, the spatial limit of the homogenous response of GNSS networks is unclear [1]. Thankfully, a feature of Principal Component Analysis (PCA) is exactly right to extract spatially inhomogeneous CME and was initially employed by [PERSON] et al. [1] in the spatiotemporal filtering of the Southern California Integrated GPS Network. For both Stacking and PCA, the maximum area for which CME could be acquired remained unspecified [19]. However, the correlation coefficients between the GNSS position time series showed that the stations were considered uncorrelated for a network larger than 6000 km [20]. They adopted a distance-dependent weighting factor that approached zero for sites further than 2000 km [20]. According to the correlation among stations and the environmental loadings, [PERSON] et al. [21] divided the GNSS station network into blocks and performed regional filtering block by block, which enabled higher precision geodetic application. Still, PCA has its own limitations, namely that it requires GNSS time series with equivalent intervals. However, due to various reasons, such as equipment failure and replacement and physical disturbance, missing data occurs in almost all GNSS stations, but at different missing rates. Iterative filling with principal components [1], Regularized Expectation Maximization algorithm [3; 22], and other interpolations such as Kriged Kalman Filter [23] can be adopted to fill the missing data beforehand. Since CME is generally dealt with in the residual time series [1; 8], which has irregular high frequency variation, it is relatively more difficult to fill in. Considering that a time series can be reproduced with its principal components, [PERSON] et al. [24] modified the PCA approach, with no need to fill or interpolate the missing data, and successfully applied this to analyze the 27 permanent stations of the Crustal Movement Observation Network of China (CMONOC). In traditional PCA, the GNSS position time series data of each station at all epochs is the unit weight [25], which is not reasonable because the precision of each station at different epochs must be different. The KLE (Karhunen-Loeve expansion) approach, in which the position time series of each station is normalized by the station related variance [1], can be employed to suppress strong local effects.