# Practical Use of Starlink Downlink Tones for Positioning
## 4 Multi-Epoch Positioning
### Results  
We will investigate based on experimental data whether the frequency error state can absorb the frequency errors due to the satellite orbit error and the satellite clock drift. We collected 15 min of data in the 11,324-11,326 MHz band using the acquisition tool presented in Section 2.2, which we post-processed using the tracking tool presented in Section 2.3, to obtain tone frequency shifts that are provided to the multi-epoch positioning algorithm. The position was initialized 165 km away from the actual position of the recording location. The initial user clock drift and the initial frequency states are set to zero. The user time has an error of a few seconds. Figure 12 gives the number of captured satellites as a function of time, and the elevation angles of the satellites from the observer's point of view. We can see that the number of satellites captured reaches six, and that the satellites are mostly tracked above an elevation angle of 28\({}^{\circ}\) with a few satellites passing over the recording location (i.e., elevation angles near 90\({}^{\circ}\)). There are a few periods with no measurements at all. Using the acquisition tool of Section 2.2, the number of satellites actually tracked in this collection is therefore less than a fifth of the number of geometrically visible satellites. Figure 13 shows the Doppler position dilution of precision (DPDOP), a measure of satellite geometry. It is computed based on the observation matrix, \(H\), and reflects the effect of the satellite geometry on the position state accuracy. The better the geometry of the satellite, the lower its value, and vice versa. A multi-epoch interval duration of m = 50 s is considered, and the DPDOP was calculated for different Doppler measurement rates  
Figure 12: (**a**) Number of tracked satellites as function of time; (**b**) satellites’ elevation angles at measurement epochs. between 1 Hz and 10 Hz. It can be seen that the DPDOP increases significantly with the measurement rate. This is because the satellites' positions became too close, which led to unfavorable geometry. Strong peaks are observed for rates above 1 Hz, which is the consequence of a singularity in the observation model. The measurement rate at the input of the algorithm is therefore fixed at 1 Hz in order to have measurements referenced to satellite positions that are sufficiently spaced to avoid singularity problems in the algorithm. For a rate of 1 Hz, during periods when more than two satellites are available, the DPDOP is between 500 and 3500. Its value peaks during periods of low satellite visibility (near 250 s and 670 s, as can be seen in Figure 12a). The algorithm is run by trying different values of the multi-epoch interval (m = 10 s, 30 s and 50 s) in order to assess the effect of this parameter on the positioning accuracy. The positioning results as well as the DPDOP are shown in Figure 14.  
The DPDOP increases as the number of stacked measurements decreases. In particular, for m = 10s, the algorithm often faces singularity problems due to the small number of satellites captured at the same time. After convergence, the position error stabilizes at \(\sim\)1 km (3D) and is the least accurate for the tested values of m. This is due to the fact that the number of measurements accumulated over m = 10 s is not sufficient to better solve the unknowns. For m = 30 s, the DPDOP as well as the positioning result improves compared to m = 10 s. The position error after convergence is \(\sim\)375 m (3D), and the convergence time is \(\sim\)330 s. For m = 50 s, the position error, 705 m (3D), is worse than in the case of m = 30 s despite a better DPDOP. This is because in this case, the assumption of a constant frequency  
Figure 14: (**a**) Doppler position dilution of precision; (**b**) Position error magnitude.