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Development of an Analysis Centre
Software for multi-GNSS precise orbit and
clock estimation
For Further Information:
Email:tao.li@ga.gov.au
Ph:+61 2 6249 9510 Web: www.ga.gov.au
Introduction
To enable real-time precise Global Navigation Satellite Systems (GNSS) positioning,
more effort are encouraged to be participated in International GNSS Service (IGS) to
deliver high quality satellite orbits and clocks products for GPS, GLONASS, BeiDou and
Galileo.At Geoscience Australia, we are currently developing an Analysis Centre
Software (ACS) package for multi-GNSS, multi-frequency data processing. The software
package when complete, will support both real-time and post-processing of multi-GNSS
data in either network or precise point positioning (PPP) modes.When released, the
package will be freely available under an open source license. At present, the software
uses zero-differenced, dual-frequency ionosphere-free code and phase observations
from GPS L1&L2, GLONASS L1&L2, BeiDou B1&B2and Galileo E1&E5a in an
Extended Kalman Filter (EKF) to estimate precise orbits and clocks products based on a
global network.The next phase of development will focus on implementing full multi-
frequency data analysis in an un-differenced, un-combined approach with ambiguity
resolution.
Mathematical models
In the current implementation, the zero-differenced, ionosphere-free models are used to
eliminate the first order effect of the ionosphere. The mathematical models are given as:
(1)
(2)
with 𝑃and 𝐿being the code and phase measurements between satellite 𝑠and receiver
𝑟. Symbol “𝐼𝐹” indicates the ionosphere-free linear combination. Geometric distance 𝜌is
supposed to be known with receiver positions from the IGS SINEX solution and
approximated satellite positions, e.g., from the broadcast navigation file.Then unknown
parameters are the satellite orbit adjustment ∆𝑥!, biased receiver clock error 𝑑𝑡",$% ,
biased satellite clock error 𝑑𝑡!,$% ,the slant troposphere delay 𝜏"
!and the float ambiguity
term 𝑀"
!.For the other corrections, such as tidal corrections, antenna phase centre
offsets and variations, relativity effect, phase-wind-up corrections, differential code
biases, they are all considered where applicable.
The above mathematical models are applied to the four global constellations. The
GLONASS inter-frequency bias is safely ignored. Given the mathematical model and the
stochastic model, Kalman filter is used to estimate the unknown parameters in equations
(1) and (2), as follows:
(3)
(4)
with (3) and (4) being the measurement update equation and time update equation,
respectively. The measurement update equation consists of the code and phase
measurements in equations (1) and (2), with the corresponding design matrix 𝐻.𝑥is the
state vector with unknown parameters.𝑣is the residual with variance-covariance matrix
𝑅.Φis the transition matrix.𝜀is normally distributed with variance-covariance matrix as
𝑄. A pivot receiver is constrained to 0to remove the rank deficiency in equation (3).
Among the unknown parameters, the orbit adjustments and the ambiguities are set as
constant. The receiver clock is changing each epoch and the single point positioning
clock solution is used as initial value for each epoch, with the process noise as 10m. The
satellite clock is relatively stable and a smaller process noise is used. In summary, the
Kalman filter process noise are given as:
Table 1: Kalman filter process noise for the state parameters
Experimental results
One global GPS network with 88 stations are used and their locations are provided in
Figure 1. The dates span from DOY 195-201 in 2019.An elevation cut-off of 10-degree
is applied.The IGS daily SINEX file are used to fix the receiver positions. Cycle slip
detection has been enabled.In the first epoch, a least-squares estimation is carried out
to derive the initial estimates and their variance-covariance matrix.From the second
epoch, Kalman fitler is used to demonstrate the real-time capability.
The orbit solutions are compared against GFZ solution and the results are given on the
left side of Figures 3for GPS, GLONASS and Galileo.The orbits 3D RMS are around
5cm for GPS, 9cm for GLONASS and 9cm for Galileo.Their radial, along track and
cross track mean RMSs are detailed in the figure, with all directions within 6.5cm. It
should be mentioned that some satellites are not healthy (e.g, G04, E18)and they are
not considered. On the right side, the satellite clock standard deviations (STD) are given
for each constellation, using the second half of the day’s results. A single-differencing
between two satellites is applied to remove pivot receivers clock error.On average, in
tested cased, GPS clock has an mean STD of up to 0.1ns, excluding PRN G26. A large
STD for G26 is probably due to mis-modeling in the measurements or orbit SRP model
between GFZ and ours.GLONASS and Galileo STDs are at the level of 0.15ns.
Conclusions
Precise satellite orbit and clock estimation using the Australian Analysis Centre Software
is almost at comparable level comparing against the other ACs, with some
improvements to work on.We will continue to refine the results by ambiguity resolution
and then implement the un-differenced, un-combined algorithm.
Tao Li (Tao.Li@ga.gov.au), Thomas Papanikolaou, Tzupang Tseng, Simon McClusky, Michael Moore
𝐸 𝑃",$%
!− 𝜌"
!= 𝛿∆𝑥!+ 𝑐 𝑑𝑡",$% −𝑑𝑡!,$% + 𝜏"
!
𝐸 𝐿",$%
!− 𝜌"
!= 𝛿∆𝑥!+ 𝑐 𝑑𝑡",$% −𝑑𝑡!,$% + 𝜏"
!+ 𝑀"
!
𝑧&= 𝐻𝑥&+ 𝑣&~(0, 𝑅)
𝑥&= Φ𝑥&'( + 𝜀&~(0, 𝑄)
Orbit adjustments / Ambiguities
0
Receiver clocks
10m
Satellite clocks
10m
Troposphere zenith wet delay
1mm/ 30s
In Figure 2, it shows the carrier
phase residuals for the GPS
orbit estimation on DOY 199.
There are approximately 2
million observations processed
and the standard deviation
(STD) is 9mm, which implies
3mm for the un-combined
observations. Also need to
mention there are afew
undetected cycle slips as well,
which requires further effort to
refine the data cleaning. Figure 2: phase residual for DOY 199
Figure 1: Global network used for precise orbit and clock estimation
Figure 3: GPS, GLONASS and Galileo orbit and clock performance
In Figure 4, a static PPP solution for station
ALIC is provided from 12:00pm on DOY
199 as an example. The 3D positioning
accuracy is at ~1cm level comparing with
IGS weekly solution, which implies the orbit
and clock solutions can be used for PPP.
Figure 4: Static PPP for ALIC station