To assess the impact of the LEO augmentation in large-scale ambiguity-float PPP-RTK positioning, real GNSS observations and simulated LEO observations were used in this contribution. This section is split into two parts. The first part introduces the measurement setup of the network and user stations for the GNSS-based real data processing, and the second part explains the simulation of LEO signals.
GNSS Real Data
CORS stations were used for the PPP-RTK processing in this study. As shown in Fig.
1, the network and user stations are distributed from central to eastern China, including 22 network stations (blue stars) and 19 user stations (red dots). The network is around 2,500 km in the north–south direction, and around 1,300 km in the east–west direction. All stations are equipped with geodetic-grade multi-GNSS receivers and antennas, i.e., with the receiver types ComNav PDB38 (ComNav Technology, Shanghai, China) and Unicore UR380 (UniCore Communications, Beijing), and the antenna type HX-CGX601A (Harxon Corporation, Shenzhen, China).
The GPS signals on L1C (1,575.42 MHz) and L2P (1,227.6 MHz), and the BDS signals on B1I (1,561.098 MHz) and B3I (1,268.52 MHz) on February 1, 2022, from 1:03:00 to 5:29:30 in GPS time (GPST) were used for the processing. The sampling interval was 30 s. The user processing started 3 h after the network processing to allow for the convergence of the network solutions. The user processing was performed in hourly sessions with the starting windows shifted by 30 s each time for all tested user stations. The user session was set not to exceed the end of the test period. The network processing was only computed once and not recomputed for each hourly user session. In total, nearly 800 hourly sessions without data gaps were considered in the statistical analysis. Both the network and the user processing were performed using the real-time orbital products provided by the National Centre for Space Studies (CNES) (
Kazmierski et al. 2018). The elevation mask was set to 10°. The user coordinates
were estimated in two modes:
•
static mode, with constrained as constant in time; and
•
kinematic mode, with no temporal constraint set to .
Fig.
2 shows the percentages of the numbers of used satellites per epoch for all network and user stations during their corresponding processing periods. The BDS geostationary (GEO) satellites were not considered in the processing due to their lower orbital accuracies than the MEO and inclined geosynchronous satellite orbit (IGSO) satellites with their very slow geometry changes to the Earth (
Lv et al. 2020).
From Fig.
2, it can be observed that seven to nine GPS satellites are used most of the time, and around five to nine BDS MEO and IGSO satellites are usable on their selected two frequencies. The higher yellow bar at the BDS satellite number of six than that of seven and above is caused by the missing satellite orbits of C11 and C12 in the CNES real-time orbits on the test day.
Simulation of the LEO Satellite Signals
In this contribution, the navigation-oriented LEO satellite constellation CentiSpace (
Yang 2019) was used for the simulation of LEO satellite signals. CentiSpace plans for circular orbits and follows the Walker constellation (
Walker 1984). It has currently registered two orbital layers at the ITU as shown in Fig.
3, i.e., Layer A (green) follows the Walker Delta constellation, with 120 satellites on 12 orbital planes, with an orbital height of 975 km and an inclination of 55°; Layer B (red) follows the Walker Star constellation with 30 satellites on three orbital planes, with an orbital height of 1,100 km and an inclination of 87.4° (
Wang et al. 2022a).
The CentiSpace satellites will broadcast GNSS interoperable phase and code signals on L1 (1,575.42 MHz) and L5 (1,176.45 MHz). As such, the dual-frequency phase (
) and code signals (
) are simulated using
where
and
= ground-truth coordinates of the stations and the LEO satellite orbits, respectively.
The wet part of the ZTDs
were taken from the estimated ZTDs from the GPS/BDS-combined PAR-enabled network solutions containing all stations used in this study, allowing 1 h of convergence time before the start of the processing period in this study. The estimated ZTDs were further smoothed with a sliding window of 10 min and mapped onto the line-of-sight (LOS) direction with the Ifadis mapping function
, forming the simulated wet tropospheric delays in the LOS direction. The estimable receiver clock errors (
) were estimated independently for the LEO constellation due to the involvement of
(Table
1); thus, the values of the simulated receiver clock errors here (
) will not influence the processing and were set to zeros. The satellite clock errors (
) and the ionospheric delays (
) were estimated as independent parameters in this study without applying any models. Their values will not influence the processing either and were set to zeros (
Wang et al. 2022b).
The LEO satellite clocks were assumed to be estimated aligned with the time system of the GPS satellite clocks. For the LEO satellite observations, the receiver phase () and code hardware biases () were simulated as random-walk processes with a temporal constraint of , and the LEO satellite phase () and code hardware biases () were simulated as random-walk processes with a temporal constraint of . The LEO satellite ambiguities were set to zeros.
Here,
and
represent the other phase- and code-related terms that were assumed to be corrected in the O-C terms, respectively. They included the satellite and receiver and satellite antenna sensor offsets, their PCO and PCV corrections, the hydrostatic part of the tropospheric delays, the relativistic effects, the solid Earth tide corrections, and the phase windups (only for phase). The
and
terms’ values will not influence processing as long as the same corrections are considered in the computed terms. The simulated phase noise (
) and code noise (
) followed the elevation-dependent weighting function in Eqs. (
8) and (
9), with the zenith-referenced phase and code standard deviations set to 0.002 and 0.2 m, respectively, due to possible multipath whitening and, accordingly, the smaller influences of the multipath biases.
Considering the disturbances of other mismodeled biases, such as the influences of the larger real-time orbital errors (to be discussed subsequently) and the simulated ZTDs biased from their true values, during the processing, the zenith-referenced standard deviations were also set to 0.003 and 0.3 m for phase and code, respectively, as mentioned in the “Processing Strategy” section. In practice, the LEO satellite signal noise behaviors with the whitened multipath effects under different measurement environments is an interesting topic to be studied when these signals are available in the future.
To generate the O-C terms for the processing, the computed part of the phase (
) and code observations (
) are formulated as follows:
where
= ground-truth coordinates of the network stations and the a priori coordinates for the user stations; and
= LEO satellite orbits introduced in the processing, which are not equal to the true orbits (
Hauschild et al. 2016;
Montenbruck et al. 2021).
The introduced LEO satellite orbits were assumed to be real-time orbits, which can be, e.g., predicted in short-term based on high-precision postprocessed LEO satellite orbits using GNSS measurements tracked onboard. Fig.
4(a) shows the root-mean square (RMS) of the orbital prediction errors for the 811-km Sentinel-3B as an example of LEO satellites, based on more than 8,000 prediction samples collected in 2019 and 2020. Due to the air drag effects that are difficult to be perfectly modeled, the along-track errors are higher than the errors in the other two directions when extrapolating the orbits with dynamic models, leading to sub-decimeter (dm) to dm-level prediction errors within 30 min.
The CentiSpace orbits are higher than the Sentinel-3B orbits with smaller influences of the air drag. One can thus expect an orbital prediction behavior not worse than those shown in Fig.
4(a). Considering the half-hour prediction errors of Sentinel-3B, real-time LEO satellite orbital errors with RMS of 2.3, 6.9, and 2.1 cm were considered in the radial, along-track, and cross-track direction, respectively. Considering that dynamically estimated or extended LEO satellite orbits often show periodic behaviors at the level of the orbital periods due to model deficiencies, the orbital errors for satellite
in direction
, denoted as
, were simulated with sine functions as follows:
where
is set to the CentiSpace orbital periods, i.e., around 1.74 and 1.79 h for Layers A and B, respectively;
and
= time of the day and the start of the day in GPST, respectively; amplitude
= around 1.42 times the wanted RMS of the orbital errors in each direction, i.e., about 3.3, 9.7, and 2.9 cm in the radial, along-track, and cross-track directions, respectively; and
= phase shift. For different LEO satellites on the same layer, a phase shift of
is generated to distinguish the orbital errors of different satellites and in different directions, where
is a random number between 0 and 1,000, and
, 1, 2 indicating the three directions.
The orbital errors simulated in the preceding paragraphs were added to the true orbits to form the introduced orbits
in Eqs. (
14) and (
15). The same orbits are introduced in the network and user processing, so that a significant part of the orbital errors can be reduced within a regional network.
Based on the increased signal strength and whitened multipath effects of the LEO satellite signals compared with those of GNSS satellites, the LEO satellite signals are considered to be more resistant to interferences, more capable of penetrating obstacles, and less influenced by mismodeled multipath biases. Thus, the elevation mask was set to 5°, i.e., lower than the 10° set for the GNSS signals.