The rapid convergence of precise point positioning real-time kinematics (PPP-RTK) with centimeter-level accuracy is of utmost importance for many applications. One way of accelerating this convergence is to explore the use of ionospheric models and multiple global navigation satellite system (GNSS) observations, e.g., Global Positioning System (GPS) and Galileo Satellite Navigation System (Galileo) observations. Because the temporal and spatial variations of the ionosphere are significant, convergence analysis of PPP-RTK should be investigated in networks with different scales, especially networks with large differences in their scales. This study describes the convergence performance of PPP-RTK using GPS/Galileo observations derived from networks with different scales under medium ionospheric conditions. Slant ionospheric corrections were first estimated from the reference network and then imported as virtual observations to enhance the convergence performance of PPP-RTK at the user interface. The results show that for the 165-km reference site spacing, the portions of single-differenced (SD) ionospheric residuals within 0.3 total electron content units (TECU) were 85.2% and 81.7% for the GPS and Galileo observations, respectively. Considering the 90th percentile of horizontal position errors, the PPP-RTK convergence time within the network with 165-km spacing was shortened from 2.5 min for GPS-only observations to 2.0 min for integrated GPS + Galileo observations. For the network of about 50 km, the proportions of the SD ionospheric residuals of the GPS and Galileo constellation within 0.3 TECU were 95.9% and 82.8%, respectively. The PPP-RTK convergence time of the 90th percentile horizontal positioning errors based on GPS-only observations was 2.0 min but 1.5 min based on integrated GPS + Galileo observations. Using GPS and Galileo observations, the convergence time could be reduced by 25% for the network with 50-km spacing. Our results suggest that the convergence time of PPP-RTK depends on the scale of the reference network and becomes shorter as the scale of the network decreases. Compared with the GPS-only PPP-RTK, the GPS/Galileo PPP-RTK could shorten the convergence time further. Graphical Abstract

The correct ambiguity resolution of real-time kinematic precise point positioning (PPP-RTK) plays an essential role in achieving fast, reliable, and high-precision positioning. However, the ambiguity of incorrect fixing will cause poor PPP-RTK positioning performance. Hence, it is essential to optimize the selected strategy of the ambiguity subset to obtain a more reliable ambiguity resolution performance for PPP-RTK. For this reason, a partial ambiguity resolution (PAR) method combining quality control and Schmidt orthogonalization (Gram–Schmidt) is proposed in this study. To investigate the performance of global positioning system (GPS) dual- and three-frequency PPP-RTK comprehensively, the PAR method based on the Gram–Schmidt method was analyzed and compared with the highest elevation angle method, which considered the satellite with the highest elevation angle as the reference satellite. The performance of ambiguity fixing, atmospheric corrections, and positioning were evaluated using five stations in Belgium and its surrounding area. The results showed average epoch fixing rates of 81.01%, 95.92%, 82.05%, and 97.93% in the dual-frequency highest elevation angle (F2-MAX), dual-frequency Gram–Schmidt (F2-ALT), three-frequency highest elevation angle (F3-MAX), and three–frequency Gram–Schmidt (F3-ALT), respectively. In terms of the time to first fix (TTFF), 89.02%, 94.25%, 90.24%, and 95.69% of the single-differenced (SD) narrow lane (NL) ambiguity fell within 3 min in F2-MAX, F2-ALT, F3-MAX, and F3-ALT, respectively. As far as the ionospheric corrections are concerned, the proportion of SD ionospheric residuals within ±0.25 total electron content units (TECU) were 95.08%, 95.93%, 95.68%, and 96.98% for the F2-MAX, F2-ALT, F3-MAX, and F3-ALT, respectively. The centimeter-level accuracy of both the horizontal and vertical positioning errors can be achieved almost instantaneously in F3-ALT. This is attributed to the accurate and reliable SD NL ambiguity fixing based on the Gram–Schmidt approach.