GNSS time-series prediction plays an important role in the monitoring of crustal plate movement, and dam or bridge deformation, and the maintenance of global or regional coordinate frames. Deep learning is a state-of-the-art approach for extracting high-level abstract features from big data without any prior knowledge. Moreover, long short-term memory (LSTM) networks are a form of recurrent neural networks that have significant potential for processing time series. In this study, a novel prediction framework was proposed by combining a multi-scale sliding window (MSSW) with LSTM. Specifically, MSSW was applied for data preprocessing to effectively extract the feature relationship at different scales and simultaneously mine the deep characteristics of the dataset. Then, multiple LSTM neural networks were used to predict and obtain the final result by weighting. To verify the performance of MSSW-LSTM, 1000 daily solutions of the XJSS station in the Up component were selected for prediction experiments. Compared with the traditional LSTM method, our results of three groups of controlled experiments showed that the RMSE value was reduced by 2.1%, 23.7%, and 20.1%, and MAE was decreased by 1.6%, 21.1%, and 22.2%, respectively. Our results showed that the MSSW-LSTM algorithm can achieve higher prediction accuracy and smaller error, and can be applied to GNSS time-series prediction.

Removal of the common mode error (CME) is very important for the investigation of global navigation satellite systems’ (GNSS) error and the estimation of an accurate GNSS velocity field for geodynamic applications. The commonly used spatiotemporal filtering methods normally process the evenly spaced time series without missing data. In this article, we present the variational Bayesian principal component analysis (VBPCA) to estimate and extract CME from the incomplete GNSS position time series. The VBPCA method can naturally handle missing data in the Bayesian framework and utilizes the variational expectation-maximization iterative algorithm to search each principal subspace. Moreover, it could automatically select the optimal number of principal components for data reconstruction and avoid the overfitting problem. To evaluate the performance of the VBPCA algorithm for extracting CME, 44 continuous GNSS stations located in Southern California were selected. Compared to previous approaches, VBPCA could achieve better performance with lower CME relative errors when more missing data exists. Since the first principal component (PC) extracted by VBPCA is remarkably larger than the other components, and its corresponding spatial response presents nearly uniform distribution, we only use the first PC and its eigenvector to reconstruct the CME for each station. After filtering out CME, the interstation correlation coefficients are significantly reduced from 0.43, 0.46, and 0.38 to 0.11, 0.10, and 0.08, for the north, east, and up (NEU) components, respectively. The root mean square (RMS) values of the residual time series and the colored noise amplitudes for the NEU components are also greatly suppressed, with average reductions of 27.11%, 28.15%, and 23.28% for the former, and 49.90%, 54.56%, and 49.75% for the latter. Moreover, the velocity estimates are more reliable and precise after removing CME, with average uncertainty reductions of 51.95%, 57.31%, and 49.92% for the NEU components, respectively. All these results indicate that the VBPCA method is an alternative and efficient way to extract CME from regional GNSS position time series in the presence of missing data. Further work is still required to consider the effect of formal errors on the CME extraction during the VBPCA implementation.

Common Mode Error (CME) presents a kind of spatially correlated error that is widespread in regional Global Navigation Satellite System (GNSS) networks and should be eliminated during postprocessing of a GNSS position time series. Several spatiotemporal filtering methods have been developed to mitigate the effects of CME. However, such methodologies become inappropriate when missing and noisy data exists. In this research, we introduce a novel spatial filtering algorithm called Weighted Expectation Maximization Principal Component Analysis (WEMPCA) for detecting and removing CME from noisy GNSS position time series with missing values, among which formal errors of daily GNSS solutions are utilized to weight the input data. Compared with traditional PCA and the special case of EMPCA, simulation experiments demonstrate that the new WEMPCA algorithm always has outstanding performance over others. The WEMPCA algorithm was then successfully used to extract the CME from real noisy and missing GNSS position time series in Xinjiang province. Our results show that only the first principal component exhibits significant spatial response, with average values of 70.11%, 66.53%, and 52.45% for North, East, and Up (NEU) components, respectively, indicating that it represents the CME of this region. After removing CME, the canonical correlation coefficients and root mean square error of GNSS residual time series, as well as the amplitudes of power-law noises (PLN), are obviously decreased in all three directions. However, the white noise (WN) amplitudes are found to diminish exclusively in the North and East component, not in the Up components. Moreover, the average velocity differences before and after filtering CME are 0.19 mm/year, 0.03 mm/year, and −0.56 mm/year for the NEU components, respectively, indicating that CME has an influence on the GNSS station velocity estimation. The velocity uncertainty is also reduced by 43.51%, 38.64%, and 40.39% on average for the NEU components, respectively, implying that the velocity estimates are more reliable and accurate after removing CME. Therefore, we conclude that the new WEMPCA approach provides an efficient solution to detect and mitigate CME from the noisy and missing GNSS position time series.

Kinematic airborne platforms are becoming increasingly vital for Earth observation. They highlight the critical need for accurate tropospheric delay corrections across varying altitudes, especially as most existing models are limited to Earth's surface. Although analytical functions have been used to model vertical reductions in tropospheric delays, they struggle to capture the intricate vertical variations of atmospheric state. In response, we introduce a novel approach that utilizes deep neural networks (DNN) to reconstruct global three‐dimensional zenith hydrostatic delay (ZHD) and zenith wet delays (ZWD) derived from numerical weather models (NWM). Our method reconstructs NWM‐derived ZHD and ZWD globally up to 14 km above the Earth's surface, with average precision levels of 0.4 and 0.8 mm, respectively. Compared to the analytical third‐order exponential model, the DNN approach demonstrates substantial improvement with global average root‐mean‐square reductions of 63% for ZHD and 36% for ZWD.

High-speed railway suspension bridges (HSRSBs) have been constructed with the new advancements in technology. The deformation prediction for HSRSBs is essential to their safety and maintenance. The conventional prediction methods are developed for bridges without high-speed railway. Different factors, including temperature (TEMP), time delay compensation (TDC), train live load (TLL), are considered in these methods. However, the train side (TS) and train instantaneous position (TIP) have a significant impact on deformation for HSRSBs, and they are not used in the prediction. More importantly, the coupling issue among different factors is so significant that it cannot be neglected. In this study, we propose a deformation prediction model based on a backpropagation (BP) neural network. This model uses different factors as model input, including TEMP, TDC, TLL, TS, and TIP. The coupling issue is addressed by using the new model. The new model was evaluated using a dataset of 10-day field measurements. It achieves a mean absolute error (MAE) of 8.81 mm, a mean relative error (MRE) of 9.82%, and coefficient of determination (R²) of 0.94. The new model will provide high-precision prediction for deformation and will be used in the development of an early warning system.