# Improved PPP Ambiguity Resolution with the Assistance of Multiple LEO Constellations and Signals
## 2 Data and Methods  
### Constellation Design and Data Simulation  
In order to investigate the performance of LEO-augmented PPP AR, three types of LEO constellations were designed using Satellite Tool Kit (STK) software, which is commercial analytical software applied for the aerospace industry, developed by Analytical Graphics [25]. The satellite number was designed to be 60, 192, and 288, respectively. The altitude of the LEO satellite is 1000 km and the inclination is 90\({}^{\circ}\). The polar orbits, which pass over the north and south poles of the earth and have the capability of covering different parts of the earth's surface, were adopted here. Apart from the LEO observations, the GNSS observations were also simulated in this paper since the BDS and Galileo have not been fully completed. The GNSS constellation is designed based on their nominal parameter configurations [26; 27; 28; 29]. Therefore, the satellite number of GPS, GLONASS, Galileo, and BDS was set as 24, 24, 30, and 35, respectively. All satellites of GPS, GLONASS, and Galileo were distributed over a medium earth orbit (MEO), while BDS consists of five geostationary Earth orbiting (GEO) satellites, three inclined geosynchronous orbiting (IGSO) satellites, and 27 MEO satellites. Twenty-four GPS MEO satellites are distributed in six orbital planes, while the GLONASS, Galileo, and BDS MEO satellites employ a Walker 24/3/1 constellation geometry. The orbit altitudes of GPS, GLONASS, and Galileo satellites are 20,180, 19,100, and 23,220 km, respectively. For BDS, the orbit altitude of the MEO satellite is 21,528 km, while GEO and IGSO satellites distribute on the orbit of the same altitude (35,786 km) with different inclinations (0\({}^{\circ}\) and 55\({}^{\circ}\)). The simulated GNSS and LEO constellations are plotted in Figure 1. Compared to the GNSS satellites, the LEO satellites are much closer to the Earth's surface because of the lower orbit altitude. It can also be found that more LEO satellites can be observed in the high-latitude region than the low-latitude region because the polar orbit is employed. Figure 1: GNSS and designed LEO constellations. (**a**) GPS; (**b**) GLONASS; (**c**) Galileo; (**d**) BDS; (**e**) 60 LEO; (**f**) 192 LEO; (**g**) 288 LEO. Observations of the GNSS and LEO satellites can be simulated based on the original phase and pseudorange observation equations (\(L_{r,j}^{s}\) and \(P_{r,j}^{s}\)) as follows [30]:  
\[L_{r,j}^{s}=\rho_{rg}^{s}+c(t_{r}-t^{s})+\lambda_{j}(B_{r,j}-B_{j}^{s})+\lambda_{ j}N_{r,j}^{s}-I_{r,j}^{s}+T_{r}^{s}+\varepsilon_{r,j}^{s} \tag{1}\]  
\[P_{r,j}^{s}=\rho_{rg}^{s}+c(t_{r}-t^{s})+c(b_{r,j}-b_{j}^{s})+I_{r,j}^{s}+T_{r}^ {s}+\varepsilon_{r,j}^{s} \tag{2}\]  
where the indices \(s\), \(r\) and \(j\) refer to the satellite, receiver, and carrier frequency, respectively; \(t^{s}\) and \(t_{r}\) are the clock offsets of the satellite and receiver, respectively; \(c\) is the speed of the light; and \(b_{r,j}\) and \(b_{j}^{s}\) are the code hardware delay at the receiver and satellite sides, respectively. The receiver-dependent hardware delays are different from the individual systems and the difference is always called the inter-system bias (ISB) or inter-frequency bias (IFB, for GLOASS). \(I_{r,j}^{s}\) and \(T_{r}^{s}\) refer to the ionospheric delay and tropospheric delay, respectively; \(N_{r,j}^{s}\) is the phase ambiguity; \(B_{r,j}\) and \(B_{j}^{s}\) refer to phase delay of the receiver and satellite, respectively; \(\rho_{rg}^{s}\) is the geometric distance from the satellite to receiver; and \(\varepsilon_{r,j}^{s}\) and \(\varepsilon_{r,j}^{s}\) denote the sum of measurement noise and multipath error for the code and carrier phase observations, respectively. The process of data simulation is generating observations by the geometric distance from the satellite to receiver and a variety of errors. The static PPP was conducted before the simulation to obtain the receiver clock offset (\(t_{r}\)), the inter-system biases, and the zenith wet delay of the troposphere.