Objectives  The global navigation satellite system(GNSS) coseismic displacements are always subject to temporal and spatial colored errors, which may lead to the incorrect interpretation of some geophysical phenomenon. Therefore, considering the nonlinear and nonstationary of GNSS seismic signal, a denoising method of multiscale multiway principal components analysis(MSMPCA) based on complete ensemble empirical mode decomposition(CEEMD) (C-MSMPCA) is used to denoising the high-frequency GNSS coseismic displacements for obtaining high precise instantaneous coseismic deformation.

  Methods  The C-MSMPCA method integrates the merits of CEEMD and MSMPCA to denoise the high-frequency GNSS coseismic displacements. Firstly, CEEMD replaces wavelet transform (WT) to decompose GNSS coseismic signal into various intrinsic mode function(IMF). Then, instead of directly deleting the previous high-requency IMF, all IMFs are grouped based on their frequency bands (FB) and each group of modes with the same FB is denoised using MSMPCA. Finally, the processed IMF is reconstructed and handled using multiway principal component analysis(MPCA) to obtain the denoised coseismic signal.

  Results  High-frequency(5 Hz) GNSS observations of 14 stations of the California real-time GNSS network (CRTN) in southern California during the 2010 EI Mayor-Cucapah earthquake(Mw 7.2) are denoised using C-MSMPCA. The performance of C-MSMPCA is compared with that of MSMPCA. The results show that: (1) Take coseismic displacement time series before and after filtering at P496 station as example, as the amplitude of the seismic signal decreases, in terms of the peak displacements and the long-term stability, the coseismic displacements obtained from C-MSMPCA are better than those obtained from MSMPCA. C-MSMPCA not only eliminates the noise in the GNSS seismic displacements, but also effectively retains the seismic signal. (2) Take GNSS residual and error curve time series before and after filtering at P496 station, as observed in the 600 s before the earthquake as example, the raw residual and error curve time series have centimeter-level low-frequency colored noise and high-frequency random noise. Compared with the error curves time series obtained from MSMPCA, the fluctuations of those obtained from C-MSMPCA are more obvious, indicating that the performance of C-MSMPCA is better than MSMPCA. (3) Each root mean square error(RMSE) and average RMSE of the original and denoised residual signals of the 14 GNSS stations, as observed in the 600 s before the earthquake show that RMSE of C-MSMPCA is less than those of MSMPCA, regardless of the coordinate components. The average RMSE values of the original GNSS solutions are determined to be 1.82, 2.55, and 7.16 mm for the east, north, and vertical components, respectively, while those of the denoising results obtained by C-MSMPCA are 0.89, 0.96, and 4.27 mm, respectively, representing RMSE reductions of 51.10%, 62.35%, and 40.36%, respectively. (4) The results of power spectral analysis show that high-frequency GNSS displacements contain high-frequency white noise and low-frequency colored noise. MSMPCA and C-MSMPCA both can significantly eliminate high-frequency white noise at 1×10¹—1×10² s periods. C-MSMPCA has the better performance than MSMPCA at 1×10²—1×10³ s and 1×10³—1×10⁴ s periods. Over all, the denoising results of MSMPCA are smoother than those of C-MSMPCA, but the latter can effectively retains higher-frequency seismic signal.

  Conclusions  Denoising is an essential step in earthquake-induced coseismic displacements. Considering the nonlinear and nonstationary of GNSS seismic signal and how to effectively preserve seismic signal, a denoising method of C-MSMPCA is used to denoising the high-frequency GNSS coseismic displacements. Over all, the performance of C-MSMPCA is better than that of MSMPCA. The high performance of C-MSMPCA is that the method is not directly remove the first high-frequency IMF but group IMF based on their frequency bands for adaptively MSMPCA processing. The method avoids the loss of high-frequency effective information and effectively preserves high-frequency seismic waveform of high-frequency GNSS displacements, which is particularly critical for the research and application of earthquake-induced coseismic displacements.