A Selective State-Space-Model Based Model for Global Zenith Tropospheric Delay Prediction

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.