Long-sequence deep learning frameworks for multivariate forecasting of tropospheric parameters

This contribution presents the first multivariate deep learning frameworks for jointly forecasting four tropospheric parameters using nine long-sequence architectures representing Transformer, Convolutional Neural Network (CNN), Recurrent Neural Network (RNN), and Multi-Layer Perceptron (MLP) model families. Models were trained on seven years of hourly observations from 501 globally distributed stations with a 96-hour input and a 24-hour forecast horizon. The results demonstrate that the multivariate formulation consistently outperforms univariate forecasting, reducing 24-hour Root Mean Square Error (RMSE) by approximately 7% for both Zenith Tropospheric Delay (ZTD) and Zenith Wet Delay (ZWD) and providing typical full-horizon errors of 19.3 to 20.9 mm for ZTD and ZWD, 4.1 to 5.6 mm for Zenith Hydrostatic Delay (ZHD), and 3.1 to 3.3 mm for Precipitable Water Vapor (PWV). Moreover, the outcomes highlight that ZTD and ZWD can be forecasted at the sub-cm level in the 1–3 h time range. A central contribution of the study is the evaluation of physical consistency, which shows that the forecasted parameters preserve core atmospheric relationships, including the PWV/ZWD ratio and the short-term coupling between ZTD and ZWD, with violation rates below 0.01%. Although a few Transformer-based models show minor inconsistencies in the ZTD, ZHD, and ZWD closure, some architectures sustain high forecast accuracy with closure deviations constrained to 0.2 mm. These findings demonstrate the substantial benefit of multivariate deep learning for forecasting tropospheric parameters and highlight the need for future approaches that integrate explicit physical constraints to further enhance numerical stability and physical realism.