A tropospheric delay grid prediction model based on non-isotropy using NARX neural network

Tropospheric delay is a key error source in high-precision Global Navigation Satellite System (GNSS) positioning. In GNSS data processing, slant path delay (SPD) is usually calculated by multiplying zenith tropospheric delay by the mapping functions. However, mapping functions ignore horizontal atmospheric variations, leading to isotropic SPD. To reduce errors from such neglect, horizontal gradient corrections are often introduced, yet these assume anisotropy SPD. Previous studies show neither isotropic nor anisotropic assumptions accurately describe SPD's horizontal features. SPD is non-isotropic, with similar values in some azimuth directions and larger differences in others. Given the difficulty of obtaining real-time meteorological data and the low accuracy of existing SPD models that ignore non-isotropy, this paper proposes a non-isotropy-based tropospheric delay grid prediction model using NARX neural networks combined with non-isotropic classification. To verify the model's accuracy, 20 global IGS stations were selected, with 2019-2020 data as training sets to predict 2021 SPD. Experimental results show that the prediction accuracy of grid-point parameters output by the NARX network reaches the millimeter level. Adopting the non-isotropic classification method reduces NARX training data demand and shortens training time. A higher spatial-resolution grid model delivers higher SPD prediction accuracy and better reflects SPD non-isotropy. The 0.25°×0.25° grid model achieves an average RMS of 37.26 mm across all stations and elevation angles. This grid model shows high prediction accuracy and better fits real atmospheric characteristics. Introducing the non-isotropic model improves SPD prediction accuracy, supporting high-precision SPD prediction for GNSS and GNSS meteorology research.