Accurate prediction of Global Navigation Satellite System-derived precipitable water vapor (GNSS-PWV), which is a crucial indicator for climate change monitoring, holds significant scientific value for climate disaster prevention and mitigation. In the study of GNSS-PWV prediction, the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) algorithm within a decomposition–integration framework effectively addresses the non-stationarity and complexity of PWV sequences, enhancing prediction accuracy. However, residual noise and pseudo-modes from decomposition can distort signals, reducing the predictor system’s reliability. Additionally, independent modeling of all decomposed components decreases computational efficiency. To address these challenges, this paper proposes a hybrid model combining the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN), autoregressive integrated moving average (ARIMA), and long short-term memory (LSTM) networks. Enhanced by local mean optimization and adaptive noise regulation, the ICEEMDAN algorithm effectively suppresses pseudo-modes and minimizes residual noise, enabling its decomposed intrinsic mode functions (IMFs) to more accurately capture the multi-scale features of GNSS-PWV. Sample entropy (SE) is used to quantify the complexity of IMFs, and components with similar entropy values are reconstructed into the following three sub-sequences: high-frequency, low-frequency, and trend. This process significantly reduces modeling complexity and improves computational efficiency. We propose different modeling strategies tailored to the dynamics of various subsequences. For the nonlinear and non-stationary high-frequency components, the LSTM network is used to effectively capture their complex patterns. The LSTM’s gating mechanism and memory cell design proficiently address the long-term dependency issue. For the stationary and weakly nonlinear low-frequency and trend components, linear patterns are extracted using ARIMA. Differencing eliminates trends and moving average operations capture random fluctuations, effectively addressing periodicity and trends in the time series. Finally, the prediction results of the three components are linearly combined to obtain the final prediction value. To validate the model performance, experiments were conducted using measured GNSS-PWV data from several stations in Hong Kong. The results demonstrate that the proposed model reduces the root mean square error by 56.81%, 37.91%, and 13.58% at the 1 h scale compared to the LSTM, EMD-LSTM, and ICEEMDAN-SE-LSTM benchmark models, respectively. Furthermore, it exhibits strong robustness in cross-month forecasts (accounting for seasonal influences) and multi-step predictions over the 1–6 h period. By improving the accuracy and efficiency of PWV predictions, this model provides reliable technical support for the real-time monitoring and early warning of extreme weather events in Hong Kong while offering a universal methodological reference for multi-scale modeling of geophysical parameters.

Zenith tropospheric delay (ZTD) is a significant atmospheric error that impacts the Global Navigation Satellite System (GNSS). Developing a high-precision, long-term forecasting model for ZTD can provide valuable insights into the overall trends of predicted ZTD, which is essential for improving GNSS positioning and analyzing changes in regional climate and water vapor. To address the challenges of incomplete information extraction and gradient explosion in a single neural network when forecasting ZTD long-term, this study introduces an Informer–LSTM Hybrid Prediction Model. This model employs a parallel ensemble learning strategy that combines the strengths of both the Informer and LSTM networks to extract features from ZTD data. The Informer model is effective at capturing the periodicity and long-term trends within the ZTD data, while the LSTM model excels at understanding short-term dependencies and dynamic changes. By merging the features extracted by both models, the prediction capabilities of each can complement one another, allowing for a more comprehensive analysis of the characteristics present in ZTD data. In our research, we utilized ERA5-derived ZTD data from 11 International GNSS Service (IGS) stations in Europe to interpolate the missing portions of GNSS-derived ZTD. We then employed this interpolated data from 2016 to 2020, along with an Informer–LSTM Hybrid Prediction Model, to develop a long-term prediction model for ZTD with a prediction duration of one year. Our numerical results demonstrate that the proposed model outperforms several comparative models, including the LSTM–Informer based on a serial ensemble learning model, as well as the Informer, Transformer, LSTM, and GPT3 empirical ZTD models. The performance metrics indicate a root mean square error (RMSE) of 1.91 cm, a mean absolute error (MAE) of 1.45 cm, a mean absolute percentage error (MAPE) of 0.60, and a correlation coefficient (R) of 0.916. Spatial distribution analysis of the accuracy metrics showed that predictive accuracy was higher in high-latitude regions compared to low-latitude areas, with inland regions demonstrating better performance than those near the ocean. This study introduced a novel methodology for high-precision ZTD modeling, which is significant for improving accurate GNSS positioning and detecting water vapor content.

The Zenith Tropospheric Delay (ZTD) is a significant atmospheric error affecting the accuracy of the Global Navigation Satellite System (GNSS). Accurate estimation of the ZTD is essential for enhancing GNSS positioning precision and plays a critical role in meteorological and climate-related applications. To address the limitations of current deep learning models in capturing long-term dependencies in ZTD sequences and overcoming computational inefficiencies, this study proposes SSMB-ZTD—an efficient deep learning model based on an improved selective State Space Model (SSM) architecture. To address the challenge of modeling long-term dependencies, we introduce a joint time and position embedding mechanism, which enhances the model’s ability to learn complex temporal patterns in ZTD data. For improving efficiency, we adopt a lightweight selective SSM structure that enables linear-time modeling and fast inference for long input sequences. To assess the effectiveness of the proposed SSMB-ZTD model, this study employs high-precision Zenith Tropospheric Delay (ZTD) products obtained from 27 IGS stations as reference data. Each model is provided with 72 h of historical ZTD inputs to forecast ZTD values at lead times of 3, 6, 12, 24, 36, and 48 h. The predictive performance of the SSMB-ZTD model is evaluated against several baseline models, including RNN, LSTM, GPT-3, Transformer, and Informer. The results show that SSMB-ZTD consistently outperforms RNN, LSTM, and GPT-3 in all prediction scenarios, with average improvements in RMSE reaching 31.2%, 37.6%, and 48.9%, respectively. In addition, compared with the Transformer and Informer models based on the attention mechanism, the SSMB-ZTD model saves 47.6% and 21.2% of the training time and 38.6% and 30.0% of the prediction time on average. At the same time, the accuracy is better than the two. The experimental results demonstrate that the proposed model achieves high prediction accuracy while maintaining computational efficiency in long-term ZTD forecasting tasks. This work provides a novel and effective solution for high-precision ZTD prediction, contributing significantly to the advancement of GNSS high-precision positioning and the utilization of GNSS-based meteorological information.