PPP-RTK: Results of CORS Network-Based PPP with Integer Ambiguity Resolution
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Journal of Aeronautics, Astronautics and Aviation, Series A, Vol.42, No.4 pp.223 - 230 (2010) 223
PPP-RTK: Results of CORS Network-Based PPP with
Integer Ambiguity Resolution *
Peter J.G. Teunissen1,2, Dennis Odijk 1, and Baocheng Zhang **,1,3
1
GNSS Research Centre, Curtin University of Technology, Australia
2
Delft Institute of Earth Observation and Space Systems, Delft University of Technology, The Netherlands
3
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, P.R. China
ABSTRACT
During the last decade the technique of CORS-based Network RTK
has been proven capable of providing cm-level positioning accuracy for
rovers receiving GNSS correction data from the CORS network. The
technique relies on successful integer carrier-phase ambiguity resolution at
both network and user level: at the level of the CORS network, ambiguity
resolution is a prerequisite in order to determine very precise atmospheric
corrections for (mobile) users, while these users need to resolve their integer
ambiguities (with respect to a certain CORS network master reference
station) to obtain precise cm-level positioning accuracy. In case of Network
RTK a user thus needs corrections from the network, plus the GNSS data of
one of the CORS stations. In practice, there exists a variety of
implementations of the Network RTK concept, of which VRS, FKP and
MAC are best known [1, 2, 3, 4]. In this contribution we discuss a closely
related concept, PPP-RTK, and show its performance on two CORS
networks.
Keywords: GNSS, Rank defects, PPP-RTK, Ambiguity resolution
I. INTRODUCTION reparameterization process adopted for the
rank-deficiency elimination. After CORS network
In this contribution we describe PPP-RTK and show ambiguity resolution, the very precise ambiguity-fixed
some of its performance. In short, PPP-RTK works very network estimates are stored in a database, which is made
much like standard PPP, but achieves positioning available to RTK users. By forming certain combinations
accuracies comparable to Network-RTK. This high of these network parameters for correcting their
accuracy is due to the RTK ambiguity resolution single-receiver GNSS phase and code data, users can
capability at the user site. perform integer ambiguity resolution and realize cm-level
Our CORS-based Network PPP-RTK approach can positioning. In the paper the performance of this
be described as follows. The un-differenced GNSS CORS-based Network PPP-RTK concept will be
observations from the CORS network are processed demonstrated by means of two tests. The first test is
based on the reparameterized observation equations as based on the Hong Kong CORS Network, which is of
developed in Delft in the 1990s, see [5, 6, 7]. The relatively small size, with inter-station distances below
estimated network parameters include single-differenced 30 km. The second test is based on the GPS Network
(biased) receiver clocks, (biased) satellite clocks, (biased) Perth having inter-station distances up to 70 km.
phase and code instrumental delays, double-differenced
ambiguities, single-differenced zenith tropospheric delays II. PPP-RTK THEORY
and ionospheric model parameters. The biases of partially
aforementioned estimable parameters are mainly due to a In this section we give a brief outline of the theory
*
This paper was presented in “International Symposium on GPS/GNSS 2010”, Taipei, Taiwan on October 25-28, 2010, and is
recommended by the Symposium General Committee. Final manuscript is accepted on December 08, 2010.
**
To whom correspondence should be addressed, E-mail: b.zhang@curtin.edu.au224 Peter J.G. Teunissen Dennis Odijk Baocheng Zhang
of PPP-RTK. The theory is applicable to any GNSS and Since the redundancy is defined as the number of
for an arbitrary number of frequencies. observations (2mnf) minus the number of parameters
(2nf+2mf clocks, m ionospheric delays, mnf ambiguities),
2.1 Un-differenced Observation Equations plus the rank defect (f+nf+mf+m), the single-epoch
We start from the set of un-differenced carrier phase redundancy of our network model is
and pseudo range (or code) observation equations. For a
receiver-satellite combination r-s at frequency j, we have redundancy = f (m 1)(n 1)
[8]:
2.2 Null-space Identification
E ( rs, j ) lrs rs tr , j t,sj j I rs j M rs, j Before we can eliminate the rank defects we need to
(1) identify the null-space of the network’s design matrix.
E ( prs, j ) lrs rs dtr , j dt,sj j I rs
The null-space can be identified by showing which
parameter changes leave the observations invariant.
where rs, j and prs, j denote the phase and code
observable, l s
the receiver-satellite range, rs the slant The clocks
r
The phase and pseudo range observations remain
tropospheric delay, tr , j and t,sj the frequency invariant if we add an arbitrary constant, per frequency,
dependent, receiver and satellite phase clock errors, to the phase clocks and to the pseudo range clocks.
dtr , j and dt,sj the frequency dependent, receiver and Hence, the observations are invariant for the following
two transformations,
satellite pseudo range clock errors, I rs the (first-order)
slant ionospheric delays on the first frequency r , j 1 d r , j 1
( j j2 / 12 ) and M rs, j the (non-integer) ambiguity, 's x and d 's x (3)
, j 1 , j 1
with j the wavelength of frequency j . Note that all
clock errors are in units of range. The rank defect contribution is therefore 2f. These
If un-differenced observation equations are used, the rank defects can be eliminated in many different ways. A
design matrix of the network will show a rank defect. simple choice is to fix the (phase and pseudo range)
This rank defect can be eliminated through an appropriate receiver clocks of the first network station: t1, j and
reduction and redefinition of the unknown parameters. dt1, j .
Here we will follow the method of reparametrization as
developed in Delft in the 1990s. It is based on the theory The redefined clocks become therefore,
of S-transformations [5, 6, 7, 9].
s
Just for the purpose of illustrating the theory, we t r , j tr , j t1, j , t , j t ',s j t1, j
make some simplifying assumptions concerning the (4)
dtr , j dtr , j dt1, j , dt , j dt ',s j dt1, j
s
network. These assumptions do not reduce the general
applicability of our method, but for the present discussion
simplify the derivations somewhat. The ionosphere
The network is assumed to consist of n receivers Since every perturbation in the ionospheric delay
(r=1,...,n), tracking the same m satellites (s=1,…,m) on can be nullified by the satellite clocks, the observations
the same f frequencies (j=1,…f). The position of the also remain invariant under the transformation,
receivers and the position of the satellites are assumed
known. Thus the geometric range lrs is assumed known. t ,s j
We assume further that the network is sufficiently small, j
so that 1s 2s ns s and I1s I 2s I ns I s . dt ,s j x (5)
s j
Based on these assumptions, the network observation I 1
equations become
E ( rs, j lrs ) tr , j t ',s j j I s j M rs, j This gives an additional rank defect of m, which can
(2) be eliminated by fixing I s . This results in the redefined
E( p s
r, j l ) dtr , j dt ' j I
s
r
s
,j
s
satellite clocks
where t ',s j t,sj s , dt ',s j dt,sj s , i.e. the s
t , j t ',s j t1, j j I s , dt, j dt ',s j dt1, j j I s
s
tropospheric delay has been lumped with the phase and
pseudo range satellite clock errors. As will be shown, the (6)
rank defect of this system of network equations is
Ambiguities and satellite clocks
rank defect = f nf mf m A constant shift of all the ambiguities of satellite s
on frequency j can be nullified by an appropriate change
in the j-frequency satellite clock. Hence, the observationsPPP-RTK: Results of CORS Network-Based PPP with Integer Ambiguity Resolution 225
remain invariant under the transformation, With these minimum constraints, we obtain a
full-rank system of observation equations
s
t, j j E ( rs, j lrs ) tr , j t, j j M 11rs, j
s
M s 1 (12)
1, j x (7)
E ( prs, j lrs ) dtr , j dt, j
s
M s 1
n, j
in which the reparametrized phase and pseudo range
clocks and integer ambiguities are given as
This gives an additional rank defect of mf. They can
be eliminated by fixing the ambiguity of the first network
tr , j t t M 1 ,
station: M 1,s j . This results in redefined satellite clocks r, j 1, j j 1r , j
and redefined ambiguities, t t 's t I s M s ,
s
,j ,j 1, j j j 1, j
s
dt , j dt ', j dt1, j j I ,
s s
t, j t ',s j t1, j j I s j M 1,s j , M 1sr , j M rs, j M 1,s j (13)
s
(8) dt r , j dtr , j dt1, j ,
1s
M 1r , j M r , j M 1, j M r , j M 1, j
s s 1 1
Note that the ambiguity is now a between receiver,
single-differenced ambiguity.
The full-rank network system can now be solved
Ambiguities and receiver clocks
using the integer constraints of the ambiguities. This
Like above, a constant shift of all the ambiguities of
gives ambiguity resolved estimates for the satellite clocks,
receiver r on frequency j can be nullified by an
t, j and dt, j . When these precise estimates are passed
s s
appropriate change in the j-frequency receiver clock,
on to the user, the above given definition of these clocks
t r , j ensures that the ambiguities of the user are also integer
1 j and that therefore ambiguity resolution becomes able at
M 1r , j 1
x (9) the user side as well. This is the principle of PPP-RTK.
M 1mr, j 1 2.4 The Full-rank User System
To demonstrate this principle, note that for the
roving user ‘u’, a similar un-differenced, full-rank system
This gives an additional rank defect of (n-1)f, and of observation equations can be formulated. But in this
not of nf, since the parameter vector is identically zero
case, the satellite clocks, t, j and dt, j , are known (as
s s
for r=1. These rank defects can be eliminated by fixing
the ambiguity of the first satellite: M 11r , j . This results in provided by the network) and the range lus is unknown,
redefined receiver clocks and redefined ambiguities,
E ( us, j t, j ) lus tu , j j M 11us, j
s
tr , j tr , j t1, j j M 11r , j , M 11rs, j M 1sr , j M 11r , j (14)
E ( prs, j dt, j ) lus dtu , j
s
(10)
Note that the ambiguity is now a double-differenced The user can now resolve his integer ambiguities
ambiguity and therefore integer. M 11us, j and compute his ambiguity resolved position fast
and accurately.
2.3 The Full-rank Network System Note that other minimum constraints can be used,
We are now in the position to formulate and than the one we used, to eliminate the rank defect in the
interpret the full-rank, un-differenced network network’s design matrix. Each minimum constrained
observation equations. The S-basis or minimum solution can be transformed to another minimum
constraint set [5], [9], that we used to eliminate the rank constrained solution by means of an S-Transformation [9].
deficiency, is given by the set: Importantly, the user’s ability to solve for his unknown
parameters is not affected by the choice of minimum
t1, j and dt1, j for j 1,..., f constraints.
s In the next two sections we will apply the PPP-RTK
I for s 1,..., m principle to two CORS networks, one in Hong Kong
s (11)
M 1, j for j 1,..., f ; s 1,..., m (China) and one in Perth (Australia). Instead of assuming
M 1 for j 1,..., f ; r 1,..., (n 1) 1s 2s ns s and I1s I 2s I ns I s , as
1r , j was done in the previous section, we now assume the
atmospheric delays rs , I rs to vary from station to station.226 Peter J.G. Teunissen Dennis Odijk Baocheng Zhang
Although this will result in a different, and higher Full LAMBDA ambiguity resolution was done after
dimensioned, null space of the network’s design matrix, an ambiguity initialization time of 10 epochs (5 min).
our same method to overcome the various rank defects After initialization LAMBDA+FFRatio test were
has been applied, i.e. null space identification and rank executed every epoch. If a new satellite rises, its float
defect elimination through reparametrization. ambiguities are solved immediately, however their
The function of the network is to provide the user integers are only resolved after 60 epochs (30 min). Fig 2
with satellite clocks and interpolated ionospheric delays. shows the results of the Hong Kong network ambiguity
For the orbits, the precise IGS orbits are used. For the resolution. All integer ambiguity solutions were accepted.
network processing, a Kalman filter was used, assuming
the ambiguities time-invariant, while for the user an
15
epoch-by-epoch least-squares processing was used, thus
number of satellites
providing truly instantaneous single-epoch solutions. The 10
integer ambiguity resolution of both network and user 5
was based on the LAMBDA method [10], with the Fixed 0
500 1000 1500 2000 2500
Failure Ratio Test [11]. The critical values of this test epoch [30 sec]
were computed for a failure rate of 0.001. For the data 1
quality control we applied the recursive DIA procedure
ratio
for the detection, identification and adaptation of outliers 0.5
and cycle slips [12].
0
500 1000 1500 2000 2500
epoch [30 sec]
III. HONG KONG PPP-RTK
Figure 2 Hong Kong network: # tracked satellites
The used Hong Kong network is a small CORS (top) and ratio vs. threshold value of the
network of 4 stations with interstation distances ranging FFRatio test (bottom; ratio depicted in
from 9 to 27 km, see Fig 1. The dual-frequency blue and threshold in red).
(L1-L2-C1-P2) Leica GPS data has been collected over
24hrs on 5th March 2001 with 30 sec sampling rate.
3.2 Hong Kong User Processing
HKST
The corrections received by the user are the satellite
clocks and the interpolated ionospheric delays. Satellite
12km HKFN clocks for each epoch were added as user
pseudo-observations, with appropriate variance matrix.
19km Interpolation of the ambiguity-fixed network ionospheric
27km HKKT
delays to the location of rover HKKT was based on
17km Kriging.
Figure 3 shows the double-differenced (DD)
ionospheric delays based on fixed integer ambiguities for
HKLT the user station HKKT with respect to network station
9km
HKSL HKSL. As can be seen at the beginning and end of the
day the DD ionospheric delays are relatively small, but
Figure 1 Hong Kong network consisting of 4 increase to large values of about 50 cm for this 16-km
stations (blue triangle) plus user station baseline during the middle of the day. This is due to the
HKKT (red triangle) fact that the Hong Kong network is in a tropical region,
in which the ionospheric conditions are usually more
severe than at mid-latitudes. In addition, the GPS data are
3.1 Hong Kong Network Processing from 2001, close to the most recent maximum of the
The undifferenced standard deviations of solar cycle.
dual-frequency phase and pseudo range (in local zenith) The user processing settings differs in the following
were set at 2mm and 20cm, respectively; in addition all way from that of the network: no estimation of zenith
observations are weighted according to their elevation, tropospheric delay was applied and the undifferenced
with a cut-off elevation of 10 degrees. The ionospheric standard deviation of the
ionosphere-weighted model [6] was used, with an a-priori ionosphere-weighted model was set at 5mm. Moreover, a
ionospheric standard deviation of 50cm. For the truly epoch-by-epoch processing was applied. This
troposphere, we applied Saastamoinen tropospheric resulted in a user empirical ambiguity success rate of
corrections and relative zenith tropospheric delay (ZTD) 2864/2869 = 99.8%, i.e. for almost all epochs the
estimation; (correct) integer ambiguities passed the FFRatio test, see
Our network Kalman filter processing is Figure 4.
characterized as: Filter initialization based on first epoch; Figure 5 shows the float and fixed epoch-by-epoch
Dynamic models for ZTDs (time-constant), DD positioning results for PPP-RTK user HKKT. Horizontal
ambiguities (time-constant), ionospheric delays (random (East-North) and vertical (Up) position errors are
walk with standard deviation of 1 cm). depicted with respect to known ground-truth coordinates.PPP-RTK: Results of CORS Network-Based PPP with Integer Ambiguity Resolution 227
DD ionospheric delay [m]
0.5
0.4 IV. PERTH PPP-RTK
0.3
0.2
0.1
0
−0.1
−0.2
The used Perth network is a CORS network of 4
−0.3
−0.4
−0.5
stations with interstation distances of about 60 km, see
500 1000 1500 2000 2500
Fig 6. The dual-frequency (L1-L2-C1-P2) Trimble NetR5
residual DD ionospheric delay [m]
epoch [30 sec]
0.05
0.04
GPS data has been collected over 16hrs on 31th July 2010
0.03
0.02 with 10 sec sampling rate.
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
500 1000 1500 2000 2500
TORK
epoch [30 sec]
Figure 3 Ambiguity-fixed DD ionospheric delays for 59km
baseline HKSL-HKKT (16 km) before 56km
applying interpolated ionospheric network MIDL
corrections (top) vs. after correction 47km
(bottom)
CUT0
ROTT 46km
15
number of satellites
58km
10
5
0 WHIY
500 1000 1500 2000 2500
epoch [30 sec]
1
Figure 6 Perth network consisting of 4 stations
(blue triangles) plus user station CUT0
ratio
0.5
(red triangle)
0
500 1000 1500 2000 2500
epoch [30 sec]
4.1 Perth Network Processing
Figure 4 PPP-RTK positioning for Hong Kong user The same procedure and almost the same settings as
station HKKT: Number of satellites (top) vs. in the Hong Kong network were used. The difference
ratio (bottom; blue) and threshold value of being that full LAMBDA ambiguity resolutiion was now
FFRatio test (bottom; red). done after an initialization time of 60 epochs (10 min)
and if a new satellite rises, their integers are only
resolved after 210 epochs (35 min). Fig 7 shows the
float horizontal scatter fixed horizontal scatter results of the Perth network ambiguity resolution. All
1 std East: 0.049m 0.05 std East: 0.0058m integer ambiguity solutions were accepted.
std North: 0.051m std North: 0.0061m
0.5
North [m]
North [m]
0 0
15
number of satellites
−0.5
10
−1 −0.05
−1 −0.5 0 0.5 1 −0.05 0 0.05 5
East [m] East [m]
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
float vertical scatter fixed vertical scatter epoch [10 sec]
2 std Up: 0.15m 0.1 std Up: 0.019m 1
1 0.05
Up [m]
Up [m]
ratio
0.5
0 0
−1 −0.05 0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
epoch [10 sec]
−2 −0.1
500 1000 1500 2000 2500 500 1000 1500 2000 2500
epoch [30 sec] epoch [30 sec]
Figure 7 Perth network: # tracked satellites (top)
and ratio vs. threshold value of the
Figure 5 PPP-RTK epoch-by-epoch positioning for FFRatio test (bottom; ratio depicted in
Hong Kong user station HKKT: float blue and threshold in red).
horizontal scatter (top left) and float
vertical time series (bottom left) vs. fixed
horizontal scatter (top right) and fixed 4.2 Perth User Processing
vertical time series (bottom right). The corrections received by the user are the satellite
Computed standard deviations are clocks and the interpolated ionospheric delays. Figure 8
included in the graphs.228 Peter J.G. Teunissen Dennis Odijk Baocheng Zhang
shows the performance of the ionospheric delay float horizontal scatter fixed horizontal scatter
interpolation. It can be seen that the residual 1 std East: 0.18m 0.05 std East: 0.006m
double-differenced ionospheric delays are within ±5 cm. std North: 0.2m std North: 0.0074m
0.5
North [m]
North [m]
At the user site the same procedure and settings were
used as for the Hong Kong user, except that the 0 0
undifferenced ionospheric standard deviation was now −0.5
set at 1cm. The user single-epoch success rate turned out
−1 −0.05
to be 91.8%, which is somewhat lower than for the Hong −1 −0.5 0 0.5 1 −0.05 0 0.05
Kong case, due to the slightly larger ionospheric residuals. East [m] East [m]
However, if we replace the epoch-by-epoch processing, float vertical scatter fixed vertical scatter
2 std Up: 0.55m 0.1 std Up: 0.02m
by a batch processing of small windows of only 6 epochs
(= 1min), a user success-rate of 100% is achieved, see 1 0.05
Up [m]
Up [m]
Fig 9. This means that if the user would be using a 0 0
Kalman filter, a complete loss-of-lock could be
successfully re-initialized after only 6 epochs. The −1 −0.05
corresponding positioning results of the 6 epoch batch −2 −0.1
processing are shown in Fig 10. 10002000300040005000 10002000300040005000
epoch [10 sec] epoch [10 sec]
Figure 10 PPP-RTK positioning for Perth user
DD ionospheric delay [m]
0.5
0.4
0.3
0.2
station CUT0: float horizontal scatter (top
0.1
0 left) and float vertical time series (bottom
−0.1
−0.2
−0.3
left) vs. fixed horizontal scatter (top right)
−0.4
−0.5
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
and fixed vertical time series (bottom
residual DD ionospheric delay [m]
epoch [10 sec] right). Computed standard deviations are
0.05
0.04
0.03
included in the graphs.
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
undifferenced GNSS observation equations so as to
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
epoch [10 sec] eliminate the various rank defects. It was shown that
PPP-RTK works very much like standard PPP, but has the
Figure 8 Ambiguity-fixed DD ionospheric delays for quality of Network-RTK due to its ambiguity resolution
baseline TORK-CUT0 (65 km) before ability. Its excellent performance was demonstrated by
applying interpolated ionospheric network means of two test networks, one in Hong Kong and one
corrections (top) vs. after correction in Perth.
(bottom)
ACKNOWLEDGEMENT
15 This work has been done in the context of the ASRP
number of satellites
10 research project “Space Platform Technologies”. The data
5
of the Hong Kong network is kindly provided by the
Land Department of Hong Kong. The data of the Perth
0
500 1000 1500 2000 2500 3000
epoch [10 sec]
3500 4000 4500 5000 5500 Network is kindly provided by GPS Network Perth. The
1 first author is the recipient of an Australian Research
Council (ARC) Federation Fellowship (project number
FF0883188). All this support is gratefully acknowledged.
ratio
0.5
0
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