## 3 Results and Discussion
### Accuracy Assessment for the KPPP-Estimated OTL  
To validate the accuracy of the GPS KPPP estimates of the eight tidal constituents, according to Equations (5), (6) and (13), we calculated the RMS of the total phasor differences, the systematic phasor differences and the corresponding residual phasor differences between the GPS estimates (using the observed data for 2008-2017) and the nine OTL models, for each of the eight tidal constituents at 12 stations. The RMS values of the eight tidal constituents are shown in Figure 5.  
Figure 5 shows that the systematic RMS of the N2 and Q1 tidal constituents between the GPS estimates and the different OTL models varies from 0.02 to 0.40 mm and from 0.12 to 0.40 mm, respectively. The systematic RMS varies significantly with the different OTL models, which shows that the systematic of the N2 and Q1 tidal constituents was mainly caused by the uncertainties of the OTL models. The uncertainties of the OTL models result in the systematic differences. After removing the systematic differences, the RMS of the residual phasors of the N2 and Q1 tidal constituents are all around 0.2 mm. The residual RMS does not change with the different OTL models, so the residual RMS reflects the uncertainty of the GPS estimates. Figure 5: RMS values of the eight tidal constituents between the GPS estimates and the nine OTL models. As shown in Figure 5, the systematic RMS change of the M2 and O1 tidal constituents with the various OTL models is from 0.01 to 1.02 mm and 0.20 to 1.01 mm, respectively. These results indicate that the systematic RMS of the M2 and O1 constituents was mainly caused by the uncertainties of the OTL models. After removing the systematic differences, the residual RMS of the O1 tidal constituent was around 0.2 mm, and no longer changed with the various OTL models. The residual RMS values of the M2 tidal constituent between the GPS estimates and the six OTL models of GOT4.10c + STW, FES2014b + STW, FES2014b + PREM, TPXO.8 + PREM, NAO99b + GB, and OSU12 + GB were all 0.2 mm. However, the residual RMS of the M2 tidal constituent between the GPS estimates and the other three OTL models (DTU10 + GB, EOT11a + GB, HAMTIDE + GB) were around 0.35 mm. This indicates that the residual RMS of the M2 tidal constituent changes slightly with the different OTL models. Therefore, the residual RMS should absorb the errors of the three OTL models, which may not exist entirely as systematic differences, because the M2 tidal resolution of the DTU10, EOT11a, and HAMTIDE ocean tide models is 1/8\({}^{\circ}\)\(\times\) 1/8\({}^{\circ}\) smaller than the 1/2\({}^{\circ}\)\(\times\) 1/2\({}^{\circ}\) size of Hong Kong. Overall the uncertainty of the GPS-estimated M2 tidal constituent is at the 0.2 mm level. Figure 5 also shows that there was obvious systematic RMS of the K1, P1, and K2 tidal constituents between the GPS estimates and the nine OTL models, of around 6 mm, 2 mm, and 9 mm, respectively. This indicates that the K1, K2, and P1 tidal constituents are affected by the significant common-mode errors for all the stations. These errors may include: (1) the unmodeled body tide; (2) the OTL model uncertainty caused by the ocean tide model and mantle inelasticity; and (3) GPS-related error, such as satellite orbit error, which has almost the same effect for all stations. Firstly, for the three tidal constituents of K1, K2, and P1, the unmodeled body tide error should be at the sub-millimeter level [12]. Secondly, the OTL displacement differences for the K1, P1, and K2 tidal constituents among the predictions by the nine different OTL models are at the sub-millimeter level, which indicates that that the OTL models are unlikely to have caused such misfits. Therefore, the common-mode errors are likely caused by the GPS-related errors. After removing the systematic differences of the K1, P1, and K2 tidal constituents, the residual RMS was around 0.8 mm, 0.5 mm, and 1 mm, respectively.