# GNSS RTK Positioning Augmented with Large LEO Constellation
## 4 Performance of Different LEO Constellations Augmented GPS RTK  
To get a representative evaluation of LEO constellations augmented GPS RTK with different numbers of LEO satellites, the 'P339-P341' baseline scheme is taken as an example to analyze the augmentation performance based on the simulated polar-orbiting data of 60, 96, 192, and 288 LEO satellites. The simulation data for ground stations covered 2 h from 12:30 to 14:30 on January 1, 2017, and which has been initialized to recalculate every half an hour in RTK positioning processing. Thus there are four computations for one baseline in 2-h observations to ensure the reliability of positioning results by using the statistical averages, including the average convergence time, time to first fix (TTFF), fixing ratio, visible satellite number, position dilution of precision (PDOP) value, and baseline solution precision. In this context, the convergence time is defined as the time required to keep horizontal errors \begin{table}
\begin{tabular}{l l} \hline \hline
**Items** & **Models** \\ \hline Satellites & GPS (G); GPS+LEO (GL); \\ Estimator & Extended Kalman filter (EKF) \\ Observations & Pseudorange and carrier phase observations \\ Signal selection & GPS: L1/L2; LEO: L1/L2 \\ Sampling Interval & 1s \\ Elevation mask & 7.5\({}^{\circ}\) \\ Observation weight & Elevation dependent weight \\ Satellite orbit & Precise ephemeris by STK \\ Satellite clock & DD elimination or weakening \\ Receiver clock & DD elimination or weakening \\ Station coordinate & Estimated in kinematic mode \\  & Short baselines: Not estimated; \\ Ionospheric delay & Medium-to-long baselines: Estimated \\  & Short baselines: Not estimated; \\  & Medium-to-long baselines: Estimated \\  & Medium-to-long baselines: Estimated \\ Process noise of vertical iono.delay & \(10^{-3}\) m/sqrt(s) \\ Process noise of wet ZTD & \(10^{-4}\) m/sqrt(s) \\ Phase ambiguity & LAMBDA \\ Ratio & 3.0 \\ Code/carrier-phase error ratio & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Data processing strategy for LEO augmented GPS RTK (real-time kinematic). reaching and afterward remaining within 0.1 m. The fixing ratio reflects the ambiguity-fixed situation after the validation by the ratio-test with the critical value of 3, which can be defined as the ratio of the numbers of ambiguity-fixed epochs to the numbers of all epochs. The baseline solution precision is an indicator to directly judge coordinates accuracy in RTK solution, which can be expressed by the root mean square (RMS) errors after successfully fixing ambiguity in east, north and up baseline components. Moreover, the reference values of baseline components can be calculated in view of known station coordinates provided by NGS. Figure 3 presents the GPS RTK results with the augmentations of different LEO satellite numbers for 'P339-P341' baseline scheme. According to Figure 3a-c and the local enlarged Figure 4, the introduction of LEO satellites does contribute for accelerating the convergence of GPS RTK, no matter in the east, north and up components. And observing more LEO satellites yields shorter convergence time. And we can also see that the precision of the vertical component is obviously worse than that of horizontal components. Figure 3d, e show the corresponding numbers of visible satellites and the PDOP values of different constellation schemes, respectively. Depending on the location of ground tracking station, the average number of tracked GPS-only satellites is 7.13 and the PDOP value is 1.97. When combining with 60-, 96-, 192-, and 288-LEO constellations, the visible satellite numbers can correspondingly increase by 1.1, 3.4, 6.8, and 9.5, respectively. Meanwhile, the average PDOP values can also decrease to 1.76, 1.57, 1.31, and 1.1, respectively. Besides, the fast movement of the LEO satellite can be observed from the rapid variation of PDOP values.