Keywords: 
• genetic algorithm /
• variational mode decomposition /
• multi-scale permutation entropy /
• global navigation satellite system /
• coordinate time series /
• denoising

Abstract: Objectives: Global navigation satellite system (GNSS) coordinate time series provide important data support for the study of crustal movement and deformation, and plate tectonics. Due to the noise caused by various external factors, the GNSS coordinate time series cannot reflect the real motion information of the station well. To effectively reduce the noise in the GNSS time series, we adopted a noise-reduction method, GA-VMD, combining genetic algorithm (GA) and variational mode decomposition (VMD). Methods: Firstly, the genetic algorithm was used to optimize VMD parameters, and the envelope entropy of the input signal was used as the fitness function of the genetic algorithm to find the optimal VMD parameter combination suitable for the signal. According to the optimized parameters, the signal was decomposed by VMD to obtain A series of modal components. Then we calculated the multi-scale permutation entropy (MPE) of each component and then regarded the MPE as the criterion of the noise component. Finally, according to the MPE, the noise components were identified and removed, and the remaining components were reconstructed to obtain the noise-reduced signal. In this paper, the noise reduction effect of GA-VMD was analyzed through the example of noise reduction of analog signal and observation data, and compared with wavelet denoising (WD) and empirical mode decomposition (EMD) methods. Results: The results show that:(1) the noise reduction results from the analog signals show that WD and EMD have the incomplete and excessive troubles on the noise reduction, respectively. However, GA-VMD can effectively eliminate noise and retain effective signals. From the evaluation index, compared with WD and EMD, the signal-to-noise ratio were increased by 5.18dB and 2.91dB, the correlation coefficient by 0.05 and 0.02, respectively, when using GA-VMD. (2) For the complex observation, we used the noise and velocity uncertainty as accuracy indicators to evaluate the noise reduction effects of the three methods. The results show that WD can only extract a part of the white noise, while EMD and GA-VMD can completely remove the white noise. GA-VMD can reduce the flicker noise to the range of 0 to 6 mm·a-0.25. For the velocity uncertainty, the average gain rates of GA-VMD relative to the WD and EMD is 69% and 15.33%, respectively. GA-VMD has an average correction rate of 79.89% and 84.46% for the velocity uncertainty and flicker noise of GNSS coordinate time series. Conclusion: Therefore, GA-VMD is the most effective one among the three noise reduction methods, which can effectively reduce the noise in the GNSS time series and improve its accuracy. However, in this paper, we only discussed the effect of GA on VMD parameter optimization without comparing it with other method. Hence, it will be the key for studying the advantages and shortcomings of those optimization algorithms in the selection of VMD, and improving the accuracy on the GNSS time series in the future.