Different mathematical methods for ZTD spatial prediction and their performance in BDS PPP augmentation using GNSS network of China

The mathematical method of ZTD (zenith tropospheric delay) spatial prediction is important for precise ZTD derivation and real-time precise point positioning (PPP) augmentation. This paper analyses the performance of the popular optimal function coefficient (OFC), sphere cap harmonic analysis (SCHA), kriging and inverse distance weighting (IDW) interpolation in ZTD spatial prediction and Beidou satellite navigation system (BDS)-PPP augmentation over China. For ZTD spatial prediction, the average time consumption of the OFC, kriging, and IDW methods is less than 0.1 s, which is significantly better than that of the SCHA method (63.157 s). The overall ZTD precision of the OFC is 3.44 cm, which outperforms those of the SCHA (9.65 cm), Kriging (10.6 cm), and IDW (11.8 cm) methods. We confirmed that the low performance of kriging and IDW is caused by their weakness in modelling ZTD variation in the vertical direction. To mitigate such deficiencies, an elevation normalization factor (ENF) is introduced into the kriging and IDW models (kriging-ENF and IDW-ENF). The overall ZTD spatial prediction accuracies of IDW-ENF and kriging-ENF are 2.80 cm and 2.01 cm, respectively, which are both superior to those of the OFC and the widely used empirical model GPT3 (4.92 cm). For BDS-PPP enhancement, the ZTD provided by the kriging-ENF, IDW-ENF and OFC as prior constraints can effectively reduce the convergence time. Compared with unconstrained BDS-PPP, our proposed kriging-ENF outperforms IDW-ENF and OFC by reducing the horizontal and vertical convergence times by approximately 13.2% and 5.8% in Ningxia and 30.4% and 7.84% in Guangdong, respectively. These results indicate that kriging-ENF is a promising method for ZTD spatial prediction and BDS-PPP enhancement over China.