# Initial Assessment of the LEO Based Navigation Signal Augmentation System from Luojia-1A Satellite
## 4 In-Orbit Performance Evaluation Results
### Handling Error Sources of the Ranging Measurements
How to integration the Luojia-1A ranging signals and the legacy GNSS signals is requires substantial effort due to the presence of these biases. With more visible LEO satellites, once the receiver clock bias becomes estimable, then the LEO satellites would be treated as a new satellite system in data processing. In order to validate the correctness of measurement, we calculate the measurement residuals by correcting the biases from the empirical models or external information. The observation model for Luojia-1A signals can be expressed as:  
\[ \begin{split} v_{p,i}=P_{i}-(\rho-dt_{s}-rel+dt_{r}+I_{i}+T)\\ v_{L,i}=\lambda_{i}\phi_{i}-\lambda_{i}N_{i}-(\rho-dt_{s}-rel+dt_ {r}-I_{i}+T)\end{split}  \tag{8}\]  
where \(P_{i},L_{i}\) are the pseudorange, carrier phase measurement on \(i\)th frequency expressed in meters. \(\rho\) is the geometry distance, \(dt_{s}\), \(dt_{r}\) are the satellite clock and receiver clock, respectively. \(I_{i}\) denotes the ionospheric delay on \(i\)th frequency and \(T\) is the troposphere delay. _rel_ refers to the relativistic effect, which is the sum of the general and the special relativistic effect. \(\lambda_{i}\) is the wavelength of the \(i\)th frequency. All of these terms are expressed in meters. \(\phi_{i}\) and \(N_{i}\) are the carrier phase measurement and ambiguity on \(i\)th frequency expressed in cycles. \(v_{p,i}\) and \(v_{L,i}\) are the residuals of the pseudorange and carrier phase measurement. The PCO and PCV are omitted in the measurement noise analysis. In practice, the residuals are computed as follows: the geometry distance is computed from the known ground antenna position and the Luojia-1A broadcast ephemeris. The satellite clock is corrected according to the broadcast ephemeris as well. Due to the clock drift and the frequency drift, the satellite clock bias for the Luojia-1A reaches up to \(1\) us, which corresponds several hundreds of meters, hence must be compensated. The general relativistic effect and special relativistic effect are corrected by the empirical model The total relativistic effect on GPS satellite is around \(\Delta f=-4.5\times 10^{-10}\times f\), while for the Luojia-1A is about \(\Delta f=-2.5\times 10^{-10}\times f\). The receiver clock is corrected by the clock bias estimated from GPS/Bedou observations. The magnitude of the receiver bias depends on the clock steering strategy. For the real-time clock steering, the receiver clock impact is at meter level. For the ground receiver used in the experiment, the clock bias is less than 30 m. The inter-system bias of the ground receiver is not derivable. Hence it is calibrated during the data processing. The Klobuchar model [70] and the Saastamoinen model [71] are used to correct the ionosphere and troposphere delay. Since there is no suitable ionosphere model for the LEO ranging signals, the Klobuchar model is adopted in the initial assessment stage. The ionospheric model for the LEO ranging signals should be refined in the future. The ionosphere delay and the troposphere delay are also elevation dependent errors, whose impact on the ranging signals is tens of meters at maximum. Then the satellite-specified hardware biases and the ambiguities are treated as a constant parameter and removed as well. For the positioning applications, these constant biases will be automatically assimilated into the receiver clock bias. As a result, they will not impact the positioning performance. After correction, the residuals of the pseudorange and the carrier phase measurement are presented in Figure 12. The standard deviation of the pseudorange residuals estimated with the raw measurements is 3.62 m, which includes the remaining orbit error, clock error, and the atmospheric errors. The figure indicates that the residuals are systematically biased at low elevation angle case, it is likely to be the remaining ionosphere residuals. The figure also shows there is a systematical constant offset, which is a combination of the hardware biases and inter-system bias.