# Performance Analysis of the Korean Positioning System Using Observation Simulation
## 3 Simulations  
### Observation Simulation  
Since KPS is a planned regional satellite navigation system, there are no actual data. Therefore, a software package for observation simulation is essential to analyze the performance of KPS. In this study, we adopted the simulation package in the Positioning and Navigation Data Analyst (PANDA) software developed by [PERSON] [17] for KPS/GPS simulation and orbit and clock determination, as well as the PPP package developed by [PERSON] and [PERSON] [22] for KPS/GPS validation. It is assumed that KPS uses the same frequency as GPS. The GPS correction and observation models are applied directly to KPS. The observation equations for dual-frequency measurements can be expressed as follows [23]:  
\[P_{1}^{sys}=\rho+c\cdot(dt_{r}-dt^{s})+I_{1}+T+\varepsilon_{P_{1}} \tag{1}\]  
\[P_{2}^{sys}=\rho+c\cdot(dt_{r}-dt^{s})+\frac{f_{1}^{2}}{f_{2}^{2}}I_{1}+T+ \varepsilon_{P_{2}} \tag{2}\]  
\[L_{1}^{sys}=\rho+c\cdot(dt_{r}-dt^{s})-I_{1}+T+\lambda_{1}N_{1}+\varepsilon_{ L_{1}} \tag{3}\]  
\[L_{2}^{sys}=\rho+c\cdot(dt_{r}-dt^{s})-\frac{f_{1}^{2}}{f_{2}^{2}}I_{1}+T+ \lambda_{2}N_{2}+\varepsilon_{L_{2}} \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**GNSS Parameter**} & \multicolumn{3}{c}{**GNSS System**} \\  & **GPS** & \multicolumn{2}{c}{**KPS**} \\ \hline
**Orbit type** & MEO & GEO & IGSO \\
**Number of satellites** & 24 & 3 & 4 \\
**Semi-major axis** & 26,587 km & 42,164 km & 42,164 km \\
**Inclination** & 55\({}^{\circ}\) & 0\({}^{\circ}\) & 43\({}^{\circ}\) \\
**Eccentricity** & 0 & 0 & 0.075 \\
**Constellation** & Four satellites in 6 orbital planes & 78\({}^{\circ}\)E, 128\({}^{\circ}\)E, 178\({}^{\circ}\)E & RAAN \(\sim\) 218\({}^{\circ}\), 128\({}^{\circ}\), 38\({}^{\circ}\), 308\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The orbit parameters and designed constellations for the Global Positioning System (GPS) [21] and Korean Positioning System (KPS). where \(P\) and \(L\) represent the code and carrier-phase measurements, respectively. The superscript \(sys\) indicates the GPS or KPS system, and the subscripts 1 and 2 indicate the frequency. \(\rho\) is the distance, including the geometric distance from the satellite to the ground receiver, antenna phase center offsets (PCO), and phase center variations (PCV) of the satellites and receivers, the relativistic correction, and tidal effects. \(c\) represents the speed of light in a vacuum. \(dt_{r}\) and \(dt^{s}\) denote the satellite and receiver clock errors that are estimated as white noise. \(I_{1}\) is the ionospheric delay, and \(T\) represents the tropospheric delay. \(N\) is the integer ambiguity parameter. \(\varepsilon_{P}\) and \(\varepsilon_{L}\) denote the code and carrier-phase observation noise, which have a standard deviation of 1.0 and 0.005 m with zero-mean Gaussian distribution. They are somehow simplified since the observation noise is usually elevation-dependent. In the data processing stage, we use the elevation weights, not the equal weights for observations. \(f_{1}\) and \(f_{2}\) are the frequencies for L1 and L2 signals. \(\lambda_{1}\) and \(\lambda_{2}\) are the wavelengths for L1 and L2 signals. The International GNSS Service (IGS) provides the absolute antenna phase center correction model, including the PCO and PCV [24]. The PCO and PCV corrections for GPS satellites from the igs14.atx are applied for observation simulation, while those for KPS satellites are assumed to be zero. The PCO and PCV corrections are dependent on the GNSS signal frequencies. In this study, those for ground receivers are assumed as the same value as the GPS signal frequencies. Moreover, the white noise model was applied for simulation of the KPS/GPS satellite and receiver clock offsets. The tropospheric delay is still a significant error source in GNSS precise positioning. The tropospheric delays at ground stations are corrected by the a priori Saastamoinen model [25] and the global mapping function [26].