## 4 Result
### The Impacts of the Residual Zenith Tropospheric Delay on the SPP Solution
Near the equator, the dZTD RMSs can exceed 20 cm and accordingly cause over 100 cm biases on the SPP solutions in the vertical direction. In contrast, more significant asymmetry between southern and northern hemispheres is observed by using the UNB3 model. The performance of UNB3 model is significantly better in the northern hemisphere, especially around middle and high latitudes (40\({}^{\circ}\)N\(-\)90\({}^{\circ}\)N), where the dZTD RMSs are below 5 cm and RMSs of SPP vertical biases are below 20 cm. Equation (6) indicates that the residual ZTD will be projected to the SPP solution with a mapping coefficient vector \(\mathbf{m}=\mathbf{D}_{\hat{x}\hat{x}}\mathbf{A}^{\intercal}\mathbf{D}_{\alpha}^{\intercal}\mathbf{a}\). To further analyze the relationship between dZTD and the SPP biases on the vertical direction, the means of the mapping coefficient  
Figure 6: RMSs of the SPP solutions on the vertical direction produced by _dZTD_ of the 9 models (saas+norm/GPT/GPT2/GPT2w/GPT3, EGNOS, UNB3, UNB3m, VMF3) in different seasons. Figure 7: _dZTD_ RMSs (upper panels) and RMSs of the SPP biases on the vertical direction (lower panels) generated by _dZTD_ from saas+GPT (left panels) and UNB3 (right panels) models at globally distributed IGS stations (unit: cm). between dZTD and the SPP biases on the vertical direction at globally distributed IGS stations are shown in Figure 8. It can be seen that the mapping coefficient is negative and approximately symmetrical along the equator. The mean value reaches the minimum (around \(-4.6\)) at mid-latitude zones, and then increases to around \(-5.4\) at equatorial, and around \(-6\) at polar zones. This implies that the dZTD values would be adversely magnified about five times on the SPP vertical solutions globally, and the impacts are slightly smaller at mid-latitude zones. In Figure 7 we can see that the SPP biases show significant discrepancies along different latitudes. Therefore, the globally distributed stations are divided into groups with a latitude interval of \(15^{\circ}\), and RMS of the vertical SPP biases caused by dZTDs from the nine models were further calculated for each latitude region. The station numbers at different latitude regions are shown in Figure 9 and RMSs of the vertical SPP biases by using the nine models at different latitude regions are shown in Figure 10.  
Figure 8: Means of the mapping coefficient between dZTD and the SPP biases on the vertical direction at globally distributed IGS stations. Figure 9: Number of stations at different latitude ranges. Figure 9 shows that nearly 50% of these stations are located on 30 degN-60 degN, and stations beyond 60 degS-60 degN are rare. Figure 10 shows that RMS of the vertical SPP biases caused by dZTDs from the nine models are significantly different at different latitudes, and tendencies among models are inconsistent. Around equatorial regions (30 degS-30 degN), the SPP biases for saas+norm/GPT models reach the largest (around 100 cm) which are 2-3 times, as much as those for UNB3/UNB3m/EGNOS and saas+GPT2/GPT2w/GPT3. While the model discrepancies decrease relatively at middle latitude regions (30 Sdeg-60 degS and 30 Ndeg-75 degN), and RMSs of SPP biases by using these models are all reduced to below 40 cm. In the regions of 60 Sdeg-90 degS, RMSs of SPP biases by using UNB3/UNB3m/EGNOS and saas+norm/GPT models gradually increase to 50 cm, and those by using saas+GPT2/GPT2w/GPT3 further reduced to around 10 cm. In the regions of 45 Ndeg-90 degN, RMSs of SPP biases by using UNB3/UNB3m/EGNOS and saas+GPT2/GPT2w/GPT3 models are nearly stable, all less than 20 cm, while those for saas+norm/GPT models increase slightly to around 30 cm and 50 cm respectively. The VMF3 model performs well globally, with a relatively larger uncertainty around the equator, where the RMSs of SPP vertical biases are still less than 10 cm. Figure 10: RMSs of the vertical SPP biases caused by dZTDs from the 9 models (saas+norm/GPT/GPT2/GPT2w/GPT3, EGNOS, UNB3, UNB3m, VMF3) at different latitude regions (unit: cm).