COA–VMPE–WD: A Novel Dual-Denoising Method for GPS Time Series Based on Permutation Entropy Constraint
Abstract
1. Introduction
2. Data and Methods
2.1. Data
2.2. Methods
2.2.1. Crayfish Optimization Algorithm
2.2.2. COA Optimization VMD
2.2.3. Multi-Scale Permutation Entropy Judgment of COA-VMD Reconstructed Signal
2.2.4. COA–VMPE–WD Algorithm
2.2.5. Evaluation Index
3. Results
3.1. Simulation Signal Decomposition
3.2. Evaluation of Noise Reduction Effect of Simulated Signals
3.3. Application of COA–VMPE–WD for GPS Time Series Denoising Analysis
3.4. Comparative Analysis of COA–VMPE–WD
4. Discussion
4.1. Analysis of Optimal Noise Models Using Different Methods
4.2. Analysis of GPS Station Velocity and Annual Term Using Different Methods
5. Conclusions
- (1)
- Simulation signal experiments show that compared to traditional EMD and WD methods, VMD method can effectively alleviate modal aliasing and endpoint effects and has better signal feature extraction ability. In terms of noise reduction, the COA–VMPE–WD method outperforms the WD, EMD, EEMD, and CEEMDAN methods in terms of RMSE, R, and SNR evaluation metrics, effectively removing noise from the original data.
- (2)
- Experimental data show that WD, EMD, EEMD, and CEEMDAN methods can remove white noise from the original coordinate time series and significantly reduce the amplitude of colored noise. Compared with WD, EMD, EEMD, and CEEMDAN methods, COA–VMPE–WD has a more significant noise reduction effect and better preserves the characteristics of the original signal.
- (3)
- The WD, EMD, EEMD, CEEMDAN COA–VMPE–WD methods have a significant impact on the U component in terms of the velocity and uncertainty of the reference station. The COA–VMPE–WD method reduced station velocity by an average of 50.00%, 59.09%, 18.18%, and 64.00% compared to the WD, EMD, EEMD, and CEEMDAN methods. The noise reduction effect is higher than the other four methods. The above results verify the effectiveness and reliability of the COA–VMPE–WD denoising method.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
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Modal Component | IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 | IMF7 | IMF8 |
---|---|---|---|---|---|---|---|---|
MPE | 0.4059 | 0.6033 | 0.7113 | 0.7668 | 0.7955 | 0.8085 | 0.8113 | 0.8229 |
Noise Reduction Methods | Evaluation | |||
---|---|---|---|---|
RMSE/mm | SNR/dB | MAE/mm | R | |
WD | 1.1971 | 12.8760 | 0.9734 | 0.8777 |
EMD | 1.3119 | 25.1553 | 0.2329 | 0.9722 |
EEMD | 1.2384 | 12.8041 | 0.9913 | 0.9300 |
CEEMDAN | 1.2136 | 13.1049 | 1.0139 | 0.9145 |
COA-VMPE-WD | 0.2291 | 27.8397 | 0.1780 | 0.9757 |
Station | N | E | U | Station | N | E | U |
---|---|---|---|---|---|---|---|
CQCS | PLWN | FNRWWN | FNWN | SXLF | FNWN | PLWN | FNWN |
FJPT | PLWN | FNRWWN | FNRWWN | SXLQ | FNWN | PLWN | FNWN |
FJWY | PLWN | FNRWWN | FNRWWN | XJBL | PLWN | PLWN | PLWN |
FJXP | PLWN | FNWN | FNRWWN | XJFY | PLWN | FNWN | FNWN |
GSMX | FNRWWN | FNRWWN | FNRWWN | XJHT | PLWN | FNWN | PLWN |
GZSC | PLWN | FNWN | FNRWWN | XJJJ | PLWN | FNWN | FNWN |
HAHB | FNWN | PLWN | FNRWWN | XJKC | PLWN | FNWN | FNRWWN |
HAJY | FNWN | FNWN | FNWN | XJKE | PLWN | FNWN | FNWN |
HBES | PLWN | PLWN | FNRWWN | XJML | PLWN | FNWN | FNWN |
HECC | PLWN | PLWN | FNWN | XJSH | PLWN | FNWN | FNWN |
HECD | PLWN | PLWN | FNWN | XJWQ | PLWN | FNWN | FNWN |
HECX | PLWN | PLWN | FNRWWN | XJWU | PLWN | PLWN | PLWN |
HELQ | PLWN | PLWN | FNWN | XJZS | PLWN | FNWN | FNWN |
HEYY | PLWN | FNRWWN | FNWN | YNCX | PLWN | FNWN | FNRWWN |
LNYK | FNWN | PLWN | FNWN | YNJD | PLWN | FNWN | FNRWWN |
NMAG | PLWN | PLWN | FNWN | YNJP | PLWN | PLWN | FNRWWN |
NMER | PLWN | PLWN | FNWN | YNLA | PLWN | PLWN | FNRWWN |
NMWJ | PLWN | PLWN | FNWN | YNMJ | PLWN | FNWN | PLWN |
NMZL | FNWN | PLWN | FNWN | YNML | PLWN | PLWN | FNRWWN |
SCJL | FNRWWN | FNRWWN | PLWN | YNMZ | FNRWWN | FNWN | FNRWWN |
SCSP | PLWN | FNRWWN | FNWN | YNRL | PLWN | PLWN | FNRWWN |
SCTQ | FNRWWN | PLWN | FNWN | YNSM | FNRWWN | PLWN | PLWN |
SCXC | PLWN | PLWN | PLWN | YNTC | PLWN | PLWN | FNRWWN |
SDCY | PLWN | FNWN | FNWN | YNTH | FNWN | FNWN | PLWN |
SDLY | PLWN | PLWN | FNWN | YNYL | PLWN | FNWN | FNRWWN |
SDRC | PLWN | FNRWWN | FNWN | YNYS | PLWN | FNWN | FNWN |
Component | Origin | WD | EMD | EEMD | CEEMDAN | COA–VMPE–WD |
---|---|---|---|---|---|---|
N | PLWN | PLWN | PLWN | PLWN | PLWN FNRW | PLWN FNRW |
E | PLWN | PLWN | PLWN | PLWN | PLWN | PLWN |
FNRWWN | PLWN | PLWN | PLWN | FNRWWN | FNRW | |
U | FNRWWN | PLWN | PLWN | PLWN | PLWN | PLWN |
FNWN | PLWN | PLWN | PLWN | PLWN | FNRW |
Component | Origin | WD | EMD | EEMD | CEEMDAN | COA-VMPE-WD |
---|---|---|---|---|---|---|
N | 0.018 | 0.018 | 0.020 | 0.021 | 0.018 | 0.010 |
E | 0.023 | 0.021 | 0.022 | 0.023 | 0.023 | 0.018 |
U | 0.022 | 0.018 | 0.022 | 0.011 | 0.025 | 0.009 |
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Wang, Z.; He, X. COA–VMPE–WD: A Novel Dual-Denoising Method for GPS Time Series Based on Permutation Entropy Constraint. Appl. Sci. 2025, 15, 10418. https://doi.org/10.3390/app151910418
Wang Z, He X. COA–VMPE–WD: A Novel Dual-Denoising Method for GPS Time Series Based on Permutation Entropy Constraint. Applied Sciences. 2025; 15(19):10418. https://doi.org/10.3390/app151910418
Chicago/Turabian StyleWang, Ziyu, and Xiaoxing He. 2025. "COA–VMPE–WD: A Novel Dual-Denoising Method for GPS Time Series Based on Permutation Entropy Constraint" Applied Sciences 15, no. 19: 10418. https://doi.org/10.3390/app151910418
APA StyleWang, Z., & He, X. (2025). COA–VMPE–WD: A Novel Dual-Denoising Method for GPS Time Series Based on Permutation Entropy Constraint. Applied Sciences, 15(19), 10418. https://doi.org/10.3390/app151910418