# GNSS RTK Positioning Augmented with Large LEO Constellation
## 2 Methods
### Data Simulation  
In this study, we simulate the LEO and GNSS observations for ground tracking stations. And a new system is designed by introducing the LEO ranging satellites transmitting similar frequency signals on the L1 and L2 bands to the GPS. The data simulation is essentially the reverse process of positioning, so the receiver/satellite position and clock are all known values in observation simulation processing. On this basis, the satellite-to-receiver geometric range can be computed, and the various range errors are also calculated by observation models, all which are finally brought together to constitute simulating observations with the random noise [23]. Moreover, the atmospheric residuals cannot be ignored for medium-to-long baselines, the ionospheric and tropospheric delays are simulated strictly according to the existing models to make the simulation situation as close as possible to reality. The observation equations for the undiferenced (UD) pseudorange \(P\) and carrier phase \(\Phi\), respectively, can be expressed as:  
\[P_{r,f,i}^{s}=p_{r,i}^{s}-t_{i}^{s}+t_{r,i}+l_{r,f,i}^{s}+T_{r,i}^{s}+b_{f}^{s }+b_{r,f}+e_{r,f,i}^{s} \tag{1}\]  
\[\Phi_{r,f,i}^{s}=p_{r,i}^{s}-t_{i}^{s}+t_{r,i}-l_{r,f,i}^{s}+T_{r,i}^{s}+ \lambda_{f}N_{r,f}^{s}+B_{f}^{s}+E_{r,f}+e_{r,f,i}^{s} \tag{2}\]  
with the subscript \(f\) = 1, 2, and \(i\) = 1,..., denote the carrier frequency and the epoch number, respectively; the superscript \(s\) and subscript \(r\) refer to the satellite and receiver respectively; \(p_{r,f,i}^{s}\) and \(\Phi_{r,f,i}^{s}\) are code and phase observations which need to be simulated at epoch \(i\); \(p_{r,i}^{s}\) is the geometric distance between the phase centers of satellite and receiver antennas;\(t_{r,i}^{s}\) and \(t_{r,i}\) are the satellite and receiver clock offsets; \(l_{r,f,i}^{s}\) is the ionospheric delay at the frequency \(f\); \(T_{r,i}^{s}\) is the tropospheric delay; \(b_{f}^{s}\) and \(b_{r,f}\) are the code hardware delays for satellite and receiver signal, respectively; \(\lambda_{f}\) is the wavelength of frequency \(f\) and \(N_{r,f}^{s}\) is the integer ambiguity; \(B_{f}^{s}\) and \(B_{r,f}\) are the satellite-dependent and receiver-dependent uncalibrated phase delay; \(e_{r,f,i}^{s}\) and \(e_{r,f,i}^{s}\) denote the mixture of measurement noise and multipath error for code and phase observations, respectively. In terms of pseudorange and phase observation simulation, the primary concern is to calculate all the components on the right side of (1) and (2). Meanwhile, considering the atmospheric residuals are the main cause of DD ambiguity resolution, the ionospheric and tropospheric delays will be simulated based on the real observables. The ionospheric delay is related to the total electron content (TEC) which can be calculated by the global ionosphere maps (GIM) from Center for Orbit Determination in Europe (CODE). Particularly, the ionospheric-delay simulation of the LEO observation must multiply by one related coefficient based on GIM calculations due to the large difference with GNSS satellites in orbit altitude [24]. Regarding the tropospheric delays, which consist of the dry and wet components, can be calculated by zenith delay and corresponding mapping function for the ground tracking stations. The wet delay adopts the estimated values from multi-GNSS PPP resolution, and the dry delay is computed by the Saastamonien model together with the Global Mapping Function (GMF) [25; 26; 27]. Besides, the DD observation models are sufficiently insensitive to the clock biases, as well as the code biases and the phase delay of satellite and receiver, which can be eliminated or weakened through between-station and between-satellite single differenced (SD) equations [10]. But these errors in simulating process are still taken into consideration to ensure the simulated data much closer to real data.