# BeiDou Augmented Navigation from Low Earth Orbit Satellites
## 3 Strategy for Data Simulation and Procession
where the subscript \(i\) denote the number of frequency with frequency value \(f_{i}\). The subscript \(r\) and the superscript \(s\) denote receiver and satellite, respectively. \(\rho_{r,i}^{s}\) and \(\rho_{r,i}^{s}\) are code and phase range at frequency \(i\) which need to be simulated. \(R_{r}^{s}\) denotes satellite-to-receiver distance. \(dt_{r}\) and \(dt^{s}\) denote receiver and satellite clock. \(T_{r}\) and \(I_{r}^{s}\) denote tropospheric and ionospheric delay. \(b_{r,i}\) and \(b_{i}^{s}\) indicate the biases for code in the receiver and satellites sides at frequency \(i\), while \(B_{r,i}\) and \(B_{r}^{s}\) are the corresponding biases for phase. These biases terms are omitted in the simulation, resulting in no biases between different observations and system. The term \(\lambda_{i}\) denotes wavelength and \(N_{r}^{s}\) means integer ambiguity. \(\delta_{PCO,i}\) and \(\delta_{PCV,i}\) are the phase centre offset (PCO) and phase centre variation (PCV) at frequency \(i\), respectively. \(\delta_{rel}\) and \(\delta_{wind-up}\) are the relative effect and phase wind-up. \(\delta_{mid,\rho_{r,i}^{s}}\) and \(\delta_{mid,\rho_{r,i}^{s}}\) are the multipath error for code and phase measurement. The terms \(\varepsilon_{\rho_{r,i}^{s}}\) and \(\varepsilon_{\rho_{r,i}^{s}}\) denote the code and phase observation errors, respectively. The simulation is mainly to calculate all the components on the right side of observation equations. The satellite-to-receiver distance is computed with the receiver and satellite positions. Specifically, this distance is associated with the mass centers of the satellite and the receiver. Thus, this distance needed to corrected by the PCO and PCV of receiver and satellite. The receiver clock error of the ground tracking stations are with uncertainty about 10 us (3000 m), while the satellite clock are interpolated from the precise clock error file. The ionospheric delay is ignored as we use the ionosphere-free combination observations, and the second and third ionospheric delays are too small to consider in the data processing. For the tropospheric delay, we use the Saastamoinen model [27] and global mapping functions (GMF) [28] for the ground tracking network. Moreover, relativistic correction, phase windup correction, and tidal displacement are all taken into consideration. The ambiguities were set as zero. For GNSS satellites, the nominal yaw-steering attitude mode is used [21], which the LEO's attitude is assumed along the along-track, cross-track, and radial direction. The noises for code and phase are set as 1 m and 2 mm. Both of the LEO and GNSS satellites used the same frequency for the simulation. Table 4 summarizes the configuration. Once the observations are generated, PPP are performed for the station to investigate the accuracy and convergence time achieved by BeiDou-3 only and BeiDou-3/LEO combined solutions. In this case, the epoch-wise station coordinates, epoch-wide receiver clock, hourly troposphere zenith delays, and ambiguity are estimated. As no inter-system bias was simulated, this term were not be estimated when BeiDou-3 is combined with LEO for PPP [29]. The ionosphere-free linear combination was used, with 2 mm observational standard deviation applied in the weighting of both the BeiDou-3 and LEO phase observations, and 1 m for both the BeiDou-3 and LEO code observations. Once the 3D positioning accuracy within ten continuous epochs is less than 10 cm, the positioning is treated as convergence. Figure 1: The satellite distribution of the BeiDou-3 and LEO constellation.