Abstract
To improve the reliability of Global Positioning System (GPS) signal extraction, the traditional variational mode decomposition (VMD) method cannot determine the number of intrinsic modal functions or the value of the penalty factor in the process of noise reduction, which leads to inadequate or over-decomposition in time series analysis and will cause problems. Therefore, in this paper, a new approach using improved variational mode decomposition and wavelet packet transform (IVMD-WPT) was proposed, which takes the energy entropy mutual information as the objective function and uses the grasshopper optimisation algorithm to optimise the objective function to adaptively determine the number of modal decompositions and the value of the penalty factor to verify the validity of the IVMD-WPT algorithm. We performed a test experiment with two groups of simulation time series and three indicators: root mean square error (RMSE), correlation coefficient (CC) and signal-to-noise ratio (SNR). These indicators were used to evaluate the noise reduction effect. The simulation results showed that IVMD-WPT was better than the traditional empirical mode decomposition and improved variational mode decomposition (IVMD) methods and that the RMSE decreased by 0.084 and 0.0715 mm; CC and SNR increased by 0.0005 and 0.0004 dB, and 862.28 and 6.17 dB, respectively. The simulation experiments verify the effectiveness of the proposed algorithm. Finally, we performed an analysis with 100 real GPS height time series from the Crustal Movement Observation Network of China (CMONOC). The results showed that the RMSE decreased by 11.4648 and 6.7322 mm, and CC and SNR increased by 0.1458 and 0.0588 dB, and 32.6773 and 26.3918 dB, respectively. In summary, the IVMD-WPT algorithm can adaptively determine the number of decomposition modal functions of VMD and the optimal combination of penalty factors; it helps to further extract effective information for noise and can perfectly retain useful information in the original time series.
Highlights
IntroductionProposed a new signal multiscale time–frequency analysis and processing method, variational mode decomposition (VMD) [30]
The study and analysis of the Global Navigation Satellite Systems (GNSS) time series are conducive to obtaining accurate positions and velocities of stations, reasonably understanding plate tectonic movements, and establishing and maintaining dynamic earth reference frames, and they contribute to the study of relevant geodynamic processes
Step 1: The parameter range of the variational mode decomposition (VMD) algorithm was set and the parameters of the grasshopper optimisation algorithm (GOA) algorithm were initialised in [34,35] the modal component number K ∈ [2, 8], K ∈ [2, 10] and penalty factor α ∈ [1000, 10000]; in [40,41], K takes an integer in the interval of [2, 8], and this paper focused on VMD applying to the Global Positioning System (GPS); the authors considered that the range of K and α was [2, 8], [1000, 10000], and the population number of the GOA algorithm N = 30, and the maximum cycle number L = 10 [34,35]
Summary
Proposed a new signal multiscale time–frequency analysis and processing method, variational mode decomposition (VMD) [30]. This method is based on VMD to denoise mechanical signals, and the denoising effect is better than wavelet and EMD denoising methods to varying degrees. In view of the advantages of VMD in analysing complex nonlinear, multiscale and nonstationary data, its algorithm has good antinomies performance, but the number of modal functions and penalty factors in the VMD method needs to be set in advance, and the use of inappropriate parameter combinations will result in insufficient noise reduction, so it is not adaptive [31,32,33,34,35].
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