#### 3.2. RTOD Accuracy Analysis

Two solutions were generated from SATODS to examine whether the pseudo-ambiguity is able to reduce the LOS errors. The first solution was from the use of the ionosphere-free pseudo-range combination ${P}_{\alpha}$ only, and it is abbreviated as “pseudo-range based” for convenient expression. The other solution was obtained by processing both the ionosphere-free pseudo-range and carrier phase combinations, ${P}_{\alpha}$ and ${L}_{\alpha}$, and the pseudo-ambiguity was estimated in the filter process. To emphasize the use of the carrier phase data and the effect of the pseudo-ambiguity, this solution is abbreviated as “carrier-phase based”.

Figure 4 and

Figure 5 show the 3D RMS of the RTOD position and velocity errors, where the former is for the HY2A satellite in the interval 2012/001–2012/005, and the latter for the ZY3 satellite in 2012/032–2012/036. The label “All” is for the whole five-day interval. Both the 3D RMS of pseudo-range based and carrier-phase based solutions are presented.

**Figure 4.**
The accuracy (3D RMS) of real-time orbits for the HY2A satellite. (**a**) The position accuracy; and (**b**) the velocity accuracy.

**Figure 4.**
The accuracy (3D RMS) of real-time orbits for the HY2A satellite. (**a**) The position accuracy; and (**b**) the velocity accuracy.

**Figure 5.**
The accuracy (3D RMS) of real-time orbits for ZY3 satellite. (**a**) The position accuracy; and (**b**) the velocity accuracy.

**Figure 5.**
The accuracy (3D RMS) of real-time orbits for ZY3 satellite. (**a**) The position accuracy; and (**b**) the velocity accuracy.

For the HY2A satellite, the position and velocity accuracy of the pseudo-range-based solution are at 0.9–1.1 m and 0.8–1.0 mm/s, respectively, which are similar to GSFC, JPL and DLR’s accuracy level [

2,

3,

4,

5,

6,

7,

8,

9]. When the ionosphere-free carrier phases are used in the carrier-phase based solution, much better accuracy of 0.2–0.4 m for position and 0.2–0.4 mm/s for velocity, respectively, is achieved.

Slightly worse results are obtained for the ZY3 satellite. A position accuracy of 0.8–1.0 m and velocity accuracy of 0.8–1.0 mm/s are achieved with the pseudo-range based solution. A superior accuracy of 0.3–0.5 m for position and 0.3–0.5 mm/s for velocity, respectively, is realized with the carrier-phase-based solution.

The overall RMS statistics of the real-time orbit accuracy are summarized in

Table 3. For the HY2A satellite, the radial (R), along-track (A), cross-track (C), and total (3D) position accuracies of the carrier-phase based solution are better than those of the pseudo-range based solution by 73%, 64%, 64%, and 65%, respectively, and the 3D position accuracy of the carrier-phase based solution is 0.334 m. In particular, the radial accuracy of the HY2A satellite is 8.5 cm, which is better than 0.1 m. Similarly, for the ZY3 satellite, the performance improvements are 47%, 53%, 50%, and 51%, respectively, and the 3D position accuracy of the carrier-phase based solution is 0.407 m.

**Table 3.**
The overall statistics of real-time orbit accuracy.

**Table 3.**
The overall statistics of real-time orbit accuracy.
Accuracy (RMS) | HY2A | ZY3 |
---|

Pseudo-Range Based | Carrier-Phase Based | Pseudo-Range Based | Carrier-Phase Based |
---|

Position (m) | R | 0.312 | 0.085 | 0.348 | 0.186 |

A | 0.701 | 0.256 | 0.714 | 0.339 |

C | 0.547 | 0.197 | 0.255 | 0.128 |

3D | 0.942 | 0.334 | 0.835 | 0.407 |

Velocity (mm/s) | R | 0.624 | 0.226 | 0.741 | 0.360 |

A | 0.287 | 0.113 | 0.357 | 0.209 |

C | 0.538 | 0.181 | 0.304 | 0.142 |

3D | 0.873 | 0.311 | 0.877 | 0.440 |

The position errors of the pseudo-range based and carrier-phase based solutions on several days are shown in

Figure 6 and

Figure 7, where the former is for the HY2A satellite on 2012/001 and 2012/005, and the latter for the ZY3 satellite on 2012/032 and 2012/035. As can be observed, the error curves for the radial, along-track and cross-track directions, and the 3D position error, for both solutions, present some periodicities. For each solution, the periodicities of orbit error curves are the results of combined effects of many periodic factors, such as the periodicity of the GPS orbit and clock offset error, and the periodical error of dynamic models. However, the variation amplitudes of radial, along-track, cross-track, and 3D position errors of the carrier-phase-based solutions are all notably reduced compared with those of the pseudo-range-based solutions. After the filter process is stabilized, the 3D position errors of the carrier-phase based solutions are all less than 1.0 m, with a RMS value of 0.2–0.5 m. The dynamical models for both solutions are exactly identical, so the results from

Figure 6 and

Figure 7 demonstrate that the estimated pseudo-ambiguity in the carrier-phase based solution has absorbed a large part of the GPS orbit and clock offset errors and, thus, greatly reduce the orbit determination error.

One should note that, the carrier-phase based solution is more dependent on the quality of GPS carrier-phase data. Some exceptions, such as frequent phase cycle-slips or frequent tracking-losses of GPS satellites, may affect the absorption effect of the pseudo-ambiguity parameter on the GPS broadcast ephemeris error, leading to less significant improvement in the orbit accuracy, when compared with the pseudo-range based solution.

However, the HY2A and ZY3 experimental data sets only cover two five-day intervals, which may lead to an over-optimistic conclusion on the orbit accuracy of the carrier-phase based solution in this study. In order to further demonstrate the orbit accuracy improvement using this novel algorithm, more space-borne GPS data sets from other satellites are tested. This includes two 30-day intervals, from 2010/001 to 2010/030 and from 2013/030 to 2013/060, respectively, for Europe’s MetOp-A [

18] and GRACE-A [

20] missions. The altitudes of 820 km for MetOp-A and 460 km for GRACE-A are similar to those of China’s HY2A and ZY3, respectively.

**Figure 6.**
Position errors of real-time orbits for the HY2A satellite. (**a**) Position errors on 2012/001; and (**b**) position errors on 2012/005.

**Figure 6.**
Position errors of real-time orbits for the HY2A satellite. (**a**) Position errors on 2012/001; and (**b**) position errors on 2012/005.

**Figure 7.**
Position errors of real-time orbits for the ZY3 satellite. (**a**) Position errors on 2012/032; and (**b**) position errors on 2012/035.

**Figure 7.**
Position errors of real-time orbits for the ZY3 satellite. (**a**) Position errors on 2012/032; and (**b**) position errors on 2012/035.

The daily position and velocity accuracies (RMS) of RTOD results in the radial, along-track and cross-track directions, and 3D position of the carrier-phase-based solution for HY2A and MetOp-A are shown in

Figure 8. The accuracies of ZY3 and GRACE-A are shown in

Figure 9. The label “All” is for the whole 30-day interval. Comparing subgraph (a) with (c), and (b) with (d) in

Figure 8, it can be observed that daily 3D orbit accuracy (RMS) with China’s HY2A satellite is better than 0.4 m, while the daily 3D RMS with MetOp-A is between 0.3–0.5 m.

Figure 9 shows that the orbit accuracies of ZY3 are only slightly worse than those of GRACE-A. These two figures demonstrate that the RTOD accuracies with China’s HY2A and ZY3 missions are at the same level with those of MetOp-A and GRACE-A. Although the quality of China’s space-borne GPS receiver is different from that of Europe’s receivers, it is believed that the pseudo-ambiguity parameter plays a major role in achieving accurate carrier-phase based orbit solutions. Therefore, it can be generally concluded that the method presented in this paper can provide the carrier-phase based orbit accuracy of 0.3–0.5 m for position and 0.3–0.5 mm/s for velocity.

**Figure 8.**
Orbit accuracy of the carrier-phase based solutions for HY2A and MetOp-A. (**a**) Position accuracy of HY2A; (**b**) velocity accuracy of HY2A; (**c**) position accuracy of MetOp-A; and (**d**) velocity accuracy of MetOp-A.

**Figure 8.**
Orbit accuracy of the carrier-phase based solutions for HY2A and MetOp-A. (**a**) Position accuracy of HY2A; (**b**) velocity accuracy of HY2A; (**c**) position accuracy of MetOp-A; and (**d**) velocity accuracy of MetOp-A.

**Figure 9.**
Orbit accuracy of the carrier-phase based solution s for ZY3 and GRACE-A. (**a**) Position accuracy of ZY3; (**b**) velocity accuracy of ZY3; (**c**) position accuracy of GRACE-A; and (**d**) velocity accuracy of GRACE-A.

**Figure 9.**
Orbit accuracy of the carrier-phase based solution s for ZY3 and GRACE-A. (**a**) Position accuracy of ZY3; (**b**) velocity accuracy of ZY3; (**c**) position accuracy of GRACE-A; and (**d**) velocity accuracy of GRACE-A.

#### 3.3. Effect of Pseudo-Ambiguity

Orbit results with higher precision are obtained using the carrier-phases compared with those using only the pseudo-ranges. On one hand, this noteworthy performance benefits from the fact that the ionosphere-free dual-frequency carrier phases ${L}_{\alpha}$ have a ranging noise far lower than that of the ionosphere-free dual-frequency pseudo-ranges ${P}_{\alpha}$. On the other hand, credits should be attributed to the introduction and estimation of the pseudo-ambiguity, which absorbs a large part of the LOS error caused by the GPS broadcast orbit and clock offset error.

Figure 10 shows two LOS error curves on day 2012/001 between HY2A and GPS satellite. The green curve is the “true” total LOS error curve computed using the GPS broadcast orbit and clock offset, and the IGS high-precision GPS orbit and clock products. The red one is the estimated LOS error in the form of the pseudo-ambiguity (excluding the true ambiguity) using the carrier-phases. To separate the estimated LOS error from the pseudo-ambiguity, the true ambiguity in the pseudo-ambiguity is computed and excluded using the IGS high-precision orbit and clock products via post-processing. As can be observed, the two curves follow a similar pattern, indicating that the LOS error caused by the GPS broadcast orbit and clock offset errors is well absorbed through the estimate of the pseudo-ambiguity. Comparisons of the two LOS error curves in two tracking arcs are presented in

Figure 11. In the 1st arc, the two curves do not agree well because the filter process is not stabilized. This means the LOS error is not well absorbed, resulting in less accurate RTOD orbits. Within the fourth tracking arc, although there is an ephemeris switch, the two error curves follow a similar pattern, because the filter process has been stabilized.

**Figure 10.**
Comparison of LOS errors caused by the broadcast ephemeris and estimated in the pseudo-ambiguity.

**Figure 10.**
Comparison of LOS errors caused by the broadcast ephemeris and estimated in the pseudo-ambiguity.

**Figure 11.**
Comparison of LOS errors in a tracking arc. (

**a**) The first tracking arc of

Figure 8; and (

**b**) the fourth tracking arc of

Figure 8.

**Figure 11.**
Comparison of LOS errors in a tracking arc. (

**a**) The first tracking arc of

Figure 8; and (

**b**) the fourth tracking arc of

Figure 8.

Figure 12 shows the RMS of the original total LOS error of HY2A about every GPS satellite in the whole interval 2012/001–2012/005 and the RMS of its residual errors after the pseudo-ambiguity is estimated. The label “All” represents the overall statistics about all tracked GPS satellites. As can be seen clearly, for every GPS satellite, the original LOS error is at 0.5–2.0 m level, but it is reduced to 0.1–0.4 m level due to the absorption effect of the pseudo-ambiguity. Apparently, a large part of the LOS error has been absorbed by the pseudo-ambiguity. In fact, according to overall statistics labeled “All”, the residual error after the estimate of the pseudo-ambiguity is only about 25% of the original LOS errors , which is a primary contributing factor for the realization of the decimeter level (0.3–0.5 m) RTOD.

**Figure 12.**
RMS comparison of original and residual LOS error.

**Figure 12.**
RMS comparison of original and residual LOS error.