Zenith tropospheric delay model in mainland China based on geographically convolutional neural network weighted regression and tensor spline function

In recent years, the global navigation satellite system (GNSS)-derived zenith tropospheric delay (ZTD) has emerged as a vital component in positioning and meteorological studies. However, limitations arise from the scarcity of GNSS stations and missing data issues. To address these challenges, we introduce a novel model: the geographically convolutional neural network weighted regression-tensor spline function (GCNNWR-TSF). This model accurately estimates ZTD at target locations using solely station location details and meteorological parameters. Experimental evidence supports its effectiveness: (1) There is a strong linear and spatial correlation observed between elevation, air pressure, temperature and ZTD. (2) The GCNNWR-TSF model significantly outperforms spatial interpolation models [the inverse distance weighted interpolation (IDW), Kriging] and the global pressure and temperature (GPT) family models (HGPT2, GPT3) in estimation accuracy. Compared to these models, GCNNWR-TSF demonstrates improvement ratios of MAE/RMSE: IDW (75.3%/75.4%), Kriging (72.7%/73.5%), HGPT2 (34.4%/30.2%), GPT3 (27.6%/25.0%), and GCNNWR (8.7%/6.3%). (3) The MAE and RMSE of the GCNNWR-TSF model compared to the GCNNWR model show superior accuracy in different scenarios, including 9.1% and 6.5% improvement at 00:00 UTC, 10% and 10.3% in winter, 9.5% and 7.1% for sites with altitude over 1000 m and 6.3% and 9.1% in the North China Plain, respectively. (4) The ZTD calculated by the GCNNWR-TSF reduces the convergence time of precise point positioning (PPP) in the U-direction by an average of 6.68%.