The global navigation satellite system (GNSS) position time series provides essential data for geodynamic and geophysical studies. Interpolation of the GNSS position time series is necessary because missing data will produce inaccurate conclusions made from the studies. The spatio-temporal correlations between GNSS reference stations cannot be considered when using traditional interpolation methods. This paper examines the use of machine learning models to reflect the spatio-temporal correlation among GNSS reference stations. To form the machine learning problem, the time series to be interpolated are treated as output values, and the time series from the remaining GNSS reference stations are used as input data. Specifically, three machine learning algorithms (i.e., the gradient boosting decision tree (GBDT), eXtreme gradient boosting (XGBoost), and random forest (RF)) are utilized to perform interpolation with the time series data from five GNSS reference stations in North China. The results of the interpolation of discrete points indicate that the three machine learning models achieve similar interpolation precision in the Up component, which is 45% better than the traditional cubic spline interpolation precision. The results of the interpolation of continuous missing data indicate that seasonal oscillations caused by thermal expansion effects in summer significantly affect the interpolation precision. Meanwhile, we improved the interpolation precision of the three models by adding data from five stations which have high correlation with the initial five GNSS reference stations. The interpolated time series for the North, East, and Up (NEU) are examined by principal component analysis (PCA), and the results show that the GBDT and RF models perform interpolation better than the XGBoost model.

The global navigation satellite system (GNSS) has seen tremendous advances in measurement precision and accuracy, and it allows researchers to perform geodynamics and geophysics studies through the analysis of GNSS time series. Moreover, GNSS time series not only contain geophysical signals, but also unmodeled errors and other nuisance parameters, which affect the performance in the estimation of site coordinates and related parameters. As the number of globally distributed GNSS reference stations increases, GNSS time series analysis software should be developed with more flexible format support, better human–machine interaction, and with powerful noise reduction analysis. To meet this requirement, a new software named GNSS time series noise reduction software (GNSS-TS-NRS) was written in MATLAB and was developed. GNSS-TS-NRS allows users to perform noise reduction analysis and spatial filtering on common mode errors and to visualize GNSS position time series. The functions’ related theoretical background of GNSS-TS-NRS were introduced. Firstly, we showed the theoretical background algorithms of the noise reduction analysis (empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD)). We also developed three improved algorithms based on EMD for noise reduction, and the results of the test example showed our proposed methods with better effect. Secondly, the spatial filtering model supported five algorithms on a separate common model error: The stacking filter method, weighted stacking filter method, correlation weighted superposition filtering method, distance weighted filtering method, and principal component analysis, as well as with batch processing. Finally, the developed software also enabled other functions, including outlier detection, correlation coefficient calculation, spectrum analysis, and distribution estimation. The main goal of the manuscript is to share the software with the scientific community to introduce new users to the GNSS time series noise reduction and application.

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.

Accurate noise model identification for GNSS time series is crucial for obtaining a reliable GNSS velocity field and its uncertainty for various studies in geodynamics and geodesy. Here, by comprehensively considering time span and missing data effect on the noise model of GNSS time series, we used four combined noise models to analyze the duration of the time series (ranging from 2 to 24 years) and the data gap (between 2% and 30%) effects on noise model selection and velocity estimation at 72 GNSS stations spanning from 1992 to 2022 in global region together with simulated data. Our results show that the selected noise model have better convergence when GNSS time series is getting longer. With longer time series, the GNSS velocity uncertainty estimation with different data gaps is more homogenous to a certain order of magnitude. When the GNSS time series length is less than 8 years, it shows that the flicker noise and random walk noise and white noise (FNRWWN), flicker noise and white noise (FNWN), and power law noise and white noise (PLWN) models are wrongly estimated as a Gauss–Markov and white noise (GGMWN) model, which can affect the accuracy of GNSS velocity estimated from GNSS time series. When the GNSS time series length is more than 12 years, the RW noise components are most likely to be detected. As the duration increases, the impact of RW on velocity uncertainty decreases. Finally, we show that the selection of the stochastic noise model and velocity estimation are reliable for a time series with a minimum duration of 12 years.

To solve the problems of difficult to model parameter selections, useful signal extraction and improper-signal decomposition in nonlinear, non-stationary dam displacement time series prediction methods, we propose a new predictive model for grey wolf optimization and variational mode decomposition and long short-term memory (GVLSTM). Firstly, we used the grey wolf optimization (GWO) algorithm to optimize the parameters of variable mode decomposition (VMD), obtaining the optimal parameter combination. Secondly, we used multiscale permutation entropy (MPE) as a standard to select signal screening, determining and recon-structing the effective modal components. Finally, the long short-term memory neural network (LSTM) was used to learn the dam deformation characteristics. The result shows that the GVLSTM model can effectively reduce the estimation deviation of the prediction model. Compared with VMDGRU and VMDANN, the average RMSE and MAE value of each station is increased by 19.11%~28.58% and 27.66%~29.63%, respectively. We used determination (R²) coefficient to judge the performance of the prediction model, and the value of R² was 0.95~0.97, indicating that our method has good performance in predicting dam deformation. The proposed method has outstanding advantages of high accuracy, reliability, and stability for dam deformation prediction.

Noise model selection criteria has a significant impact on identifying the stochastic noise proper-ties of any GNSS daily coordinate time series. The low-frequency random walk noise existing in these time series could lead to overestimation of the tectonic rate, so it is of great significance to accurately detect the random walk component. This study focuses on noise model estimation cri-terion (BIC_tp) derived from the AIC and the BIC by introducing 2π factors. It is more sensitive to abnormal steps (random jumps). Using observation data from 72 GNSS stations from 1992 to 2022 and simulated data, four combined noise models are used to explore the impacts of time se-ries lengths (ranging from2 to 24 years) and data loss (between 2% and 30%) on noise models and velocity estimation. The results show that as the time length increased, the selected optimal noise model, and the estimated uncertainty of the tectonic trend with different data gap, gradually con-verge. When the time length is short (less than 8 years), it could lead to the FNRWWN, FNWN, and PLWN models being mistakenly estimated as GGMWN models, thereby affecting the accura-cy of determining the station velocity parameters. When the time length is 12 years, the RW noise component is more probably detected, As the time length increases, the impact of RW on velocity uncertainty is weakened. Finally, we conclude that for a time series with a minimum time length of 12 years, both the selection of the optimal stochastic noise model and the estimation of the ve-locity parameters are reliable.