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Exploring GNSS RTK Performance Benefits
with GPS and Virtual Galileo Measurements
Yanming Feng, School of Software Engineering and Data Communications, Queensland University of
Technology, Australia
Jinling Wang, School of Surveying and Spatial Information Systems, The University of New South Wales, Australia
BIOGRAPHY
Dr Yanming Feng is Associate Professor with School of
Software Engineering and Data Communications,
Queensland University of Technology, Australia. He is also
a Research Leader within Cooperative Research Centre for
Spatial Information (CRCSI), responsible for future
regional GNSS studies. He has acquired significant research
experiences in areas such as satellite orbit determination, real
time kinematic positioning, GPS water vapour estimation
and three carrier ambiguity resolutions (TCAR) etc in the
past 20 years. Dr Feng is also Chair of IAG Working Group
4.5.4 for “Multiple Carrier Ambiguity Resolutions Methods
and Applications”.
Dr Jinling Wang is a Senior Lecturer in the School of
Surveying & Spatial Information System at the University of
New South Wales. He is a member of the editorial board for
the international journal GPS SOLUTIONS, and Chairman
of the international study group on pseudolite applications in
positioning and navigation within the International
Association of Geodesy's Commission 4. He was 2004
President of the International Association of Chinese
Professionals in Global Positioning Systems (CPGPS), He
holds a PhD in GPS/Geodesy from Curtin University of
Technology, Australia.
ABSTRACT
Two key limiting factors in the existing GPS real time
kinematic (RTK) positioning are the strong distance
dependence and unreliable ambiguity resolution under
degraded observation conditions. While use of multiple
frequency GNSS signals could possibly redefine future
GNSS RTK services locally, regionally and globally, more
redundant measurements from multiple satellite systems,
such as GPS, Galileo and Glonass, can improve the
performance of RTK solutions in terms of accuracy,
availability, reliability and time to ambiguity fix. The direct
benefits of these advancements for all scales of GNSS
services are 1) significant savings in reference station
infrastructure and operational costs, and 2) potential
improvement on RTK availability and reliability for most of
the professional users in the future.
This paper aims to study how the RTK system with multiple
satellite constellations works and performs, using GPS and
virtual Galileo dualfrequency measurements. Virtual
Galileo measurements are obtained from the same set of
GPS data, but starting from the epoch in 1 or 2 hours later.
To proceed, generic linear equations for Ambiguity
Resolution (AR) and position estimation are outlined for
GPS and Virtual Galileo measurements first and various
performance parameters for ambiguity resolution (AR) and
RTK solutions are defined generally in order to more
comprehensively evaluate a RTK system. Numerical
analysis is performed using the research version of a post
processing RTK software package, and results from three
RINEX dual frequency data sets are given against various
computing schemes, such as L1/L2 and widelane signals,
GPS and virtual Galileo measurements. The performance
results have preliminarily demonstrated the significant
improvement due to the doubled number of measurements.
Further improvements on the success rates are anticipated if
refined modelling strategies for systematic and random
errors in the doubledifferenced measurements are
introduced.
INTRODUCTION
GNSS positioning solutions may be classified into several
different types, depending on (1) the types of measurements
used in the estimation, (2) data epochs or data arcs required
to create a set of solutions, and (3) the number of receivers
involved in the operation. Single point positioning (SPP)
produces navigation solutions with pseudorange
measurements from a single receiver and a single epoch.
Precise Point Positioning (PPP) solutions are obtained using
both code and phase measurements from a single receiver,
for a period of tens of minutes to hours before the centimetre
level accuracy can be achieved, regardless of kinematic or
static user status. Differential GPS solutions are based on
code measurements from a single epoch as well, but using
the differential corrections from a reference station or
network. Real Time Kinematic (RTK) positioning makes use
of carrier phase measurements in the differential positioning
mode, ideally, from a single epoch. Practically, multiple
epochs or a short period of observations are often involved to
achieve a desirable reliability for the RTK solutions.
There are two types of RTKGPS techniques: singlebase
RTK and networkbased RTK (e.g. Lachapelle et al., 2002).
The limitation of singlebase RTK is the distance between
reference and user receivers, due to the impact of distance
dependent biases such as orbit error, and ionospheric and
tropospheric signal delay. This has restricted the inter
receiver distance to about 20km or less if rapid Ambiguity
Resolution (AR) is desired (i.e. of the order of a few
seconds), depending on the activity of the ionosphere.
NetworkRTK is a GPS positioning technique that
challenges the current constraints of singlebase RTK (Rizos,
2002). In current NetworkRTK implementations, the inter
receiver distance is typically between 50 and 70km. In
addition to the distance factor, a common problem for both
singlebase and network RTK is the reliability and
availability of RTK solutions, which depends on successful
AR within a shortest possible period. In normal operations a
commercial RTK system can only offer a certain level of
RTK availability. In obstructed urban areas the availability
can be much lower.
In general, distancedependence of the
magnitude of measurement biases and the unreliability of
ambiguity solution are the two key problems in the existing
dualfrequency based RTK positioning techniques.
Use of multiple frequency GNSS signals could possibly
redefine future GNSS RTK services locally, regionally and
globally. Possible future GNSS services have been identified
(Feng & Rizos, 2005, Feng and Moody 2006, Hatch, 2006):
• Local RTK services at the cm accuracy level: The
concept is to simply use a single reference station to
serve an area of tens to 100km in radius, achieving cm
level accuracy in real time. The dominant error is the
residual tropospheric effect in the doubledifferenced
(DD) phase measurements and amplification of phase
noises containing effects of multipath errors.
• Regional networkbased RTK services at the cmdm
accuracy level: The reference stations equipped with
triplefrequency receivers may be spaced up to a few
hundred kilometers, to provide differential and RTK
services over a region, typically statewide or country
wide. Networkbased atmospheric error corrections
may be provided for higher accuracy and/or to dual
frequency users within the network coverage. Inverse
networkRTK strategies would enable the instantaneous
estimation of tropospheric delays for the reference
stations and user terminals, thus allowing further
extension of the network spacing
• Global differential GNSS services at the decimetre to
submetre accuracy levels: These can be provided with
ambiguityresolved widelane signals over any distance
(but with satellites in common view). For the same
decimeterlevel accuracy solutions, the smoothing (or
averaging) time is reduced to the level of seconds to a
few minutes, a factor of about 10 less than the
observation period required using dualfrequency
receivers with current global differential GPS services.
The direct benefits of the above advancements for all three
scales/levels of GNSS services are significant savings in
reference station infrastructure and operational costs. As a
result, a single reference station may be set up to provide
RTK services citywide, and a state or country may set up
regional GNSS services with much fewer number of the
receiver numbers required today (i.e., for dualfrequency
receivers employing existing networkbased RTK
techniques). The planned Galileo/GPS ground tracking
network of tens of stations can provide decimeterlevel
accuracy positioning services regionally or globally (e.g., via
additional real time communication links, and using RTCM
3.0+ messages).
The benefits of more redundant measurements from multiple
satellite systems may be the improvement on the
performance of AR and RTK solutions. Unlike code based
navigation solutions, simulation studies offer little
knowledge about the performance of AR and RTK solutions.
However, GPSbased virtual Galileo phase measurements
may be used. Virtual Galileo measurements are obtained
from the same set of GPS data, but starting from the epoch in
1 or 2 hours latter (Feng, 2005). The performance of a RTK
system may be described by various performance parameters
for its AR and position solutions, which typically include
baserover distance, timetoambiguity fix, AR availability
and reliability, RTK availability and accuracy and so on.
This paper gives a more systematic review for RTK
performance characteristics and then explores GNSS AR and
RTK performance improvement with GPS and virtual
Galileo measurements in terms of the various RTK
performance parameters. In the following sections, we first
outline generic linear equations for AR and position
estimation using GPS and Virtual Galileo measurements.
This is followed by the definitions and numerical examples
of various performance parameters for ambiguity resolution
(AR) and RTK solutions in order to comprehensively
evaluate a RTK system. In the forth section, numerical
experiments are performed a research version of a RTK
software package and results from several RINEX dual
frequency data sets for different baselines are presented and
analysed according to various computing schemes, such as
widelane and L1/L2 frequency, GPS only and GPS and
virtual Galileo measurements. Finally, the conclusions about
the performance improvement due to the use of GPS and
Galileo measurements are summarised.
LINEAR EQUATIONS FOR AMBIGUITY
RESOLUTIONS WITH GPS AND VIRTUAL
GALILEO MEASUREMENTS
We begin with the linearised observation equation for a GPS
RTK positioning problem with pseudoranges P
1
(or C/A) and
P
2
and carrier phases L
1
and L
2
provided by any dual
frequency GPS receivers:
gps
L
L
p
gps
gpsgpsgps
N
N
X
A
A
A
L
L
P
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
12
1
12
2
1
1212
1
12
1
12
0
00
ε
ε
ε
λλ
λδ
δ
δ
δ
(1)
where
12
P
δ
,
1
L
δ
and
12
L
δ
are the residual vectors between
the observed and computed ranges for doubledifferenced
P
12
, L
1
and L
12
measurements, where the L12 stands for the
widelane combination and P
12
is (f
1
P
1
+f
2
P
2
)/(f
1
+f
2
); A is the
design matrix with respect to the user position states;
X
δ
is
the 3by1 user state vector;
1
N
and
2
N
are the integer
ambiguities for L1 and L2 respectively; λ
1
and λ
12
are the
wavelengths for L1 and L12 carriers; and finally, ε
p12
, ε
L1
,
ε
L12
, are the noise vectors for P
12
, L
1
and L
12
measurements,
respectively. To achieve better performance, more efficient
combination may be used in (1) instead of L1 and L12 (Feng
and Moody, 2006).
For the Galileo constellation, an equation similar to (1) is
obtained, but the carrier phase L
2
would represent E6 or E5A
or E5B. Therefore, we would have the following linear
equations:
gal
L
L
p
gal
galgalgal
N
N
X
A
A
A
L
L
P
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
12
1
12
2
1
1212
1
12
1
12
0
00
ε
ε
ε
λλ
λδ
δ
δ
δ
(2)
where all the notations are the same as those for Equation
(1), but the wavelength λ
12
, would vary, depending on the
choice from L1E6, L1E5A and L1E5B.
For performance analysis of the combined GPS and Galileo
constellation, Feng (2005) proposed the Virtual Galileo
Constellation (VGC) approach, which allows the
performance benefits of the dual GPS/Galileo constellation
to be demonstrated using the current GPS constellation. The
concept is to combine the GPS measurements data sets
recorded at two epochs separated by a few hours, and
compute the state parameters  three positional parameters
and two sets of ambiguity parameters  for the two different
epochs. Hence the approach may also be referred to as the
“TwoEpoch Approach”. With the GPSonly constellation
one can obtain the observation equations at two different
epochs, the jth and (j+2h) th, which are similar to Equations
(1) and (2), assuming we use the data collected at two static
stations at both epochs;
j
L
L
p
j
j
j
jj
N
N
X
A
A
A
L
L
P
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
12
1
12
2
1
1212
1
12
1
12
0
00
ε
ε
ε
λλ
λδ
δ
δ
δ
(3)
hj
L
L
p
hj
hj
j
hjhj
N
N
X
A
A
A
L
L
P
2
12
1
12
2
2
1
2
1212
1
22
12
1
12
0
00
+
+
+++
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
ε
ε
ε
λλ
λδ
δ
δ
δ
(4)
If the numbers of GPS and virtual Galileo satellites in view
are n
1
and n
2
respectively, there are 3(n
1
1)+3(n
2
1)
measurements, 3 position parameters, and 2(n
1
1) +2(n
2
1)
ambiguity parameters in Equations (3) and (4), which have
the similar structure as Equations (1) and (2). Therefore it is
possible to ‘virtually’ combine the real GPS data collected at
two epochs separated by a certain period of time to
demonstrate the positioning performance of the GPS and
Galileo constellations. For convenience, Equations (1) and
(2) or (3) and (4) are rewritten as follow as:
jjjjjj
NBXAY
ε
δ
δ
+
+
=
(5)
12
)(;0)(
−
==
jjj
WVarE
σεε
(6)
where the subscript j represent the jth epoch; δY is nby1
observation vector, A is nby3 matrix; δX is 3by1 state
vector; and B is nbyp matrix, N is pby1 ambiguity vector;
in which n=3(n
1
1) and p=2(n
1
1) with GPS only, n=3(n
1

1)+3(n
2
1) and p=2(n
1
1)+2(n
2
1) with both GPS and virtual
or real Galileo measurements.
The above equations are provided for each measurement
epoch, implying that the state and ambiguity parameters are
estimated and fixed with the measurements at the current
epoch only, which yields desirable kinematic positioning
solutions without imposing the assumptions of phase
measurement continuity and sample intervals. To enable the
use of data over more epochs to improve the reliability of
ambiguity resolution, we introduce the state transition
equation:
mjmjjj
XXX
−−
=
∆
+
δ
δ
δ
),(
(7)
where the
),( mjj
X
−
∆
δ
is defined as the deltaposition vector
of the current epoch jth with respect to the (jM)th epoch,
which may be estimated from the doubledifferenced phase
measurements between the jth and (jm)th epochs. In order
to consider the effects of the variation in the design matrix A
on the solution, the estimation
),(
ˆ
mjj
X
−
∆
δ
is obtained
accumulatively as follows
∑
=
−+−−
∆=∆
m
i
ijijMjj
XX
1
),1(),(
ˆˆ
δδ
(8)
Substitute (7) into (5) and replace the deltaposition vector
with its estimation (8), we can use the measurements over
the most recent m epochs for estimation and search of
ambiguity integers at the current epoch, assuming that at
least 3 ambiguity parameters remain unchanged over the
period of m epochs.
Alternatively, we can introduce constraints to the integer
parameters that have affirmatively been identified being
unchanged, or known, over the periods of m epochs to the
current. As a result, the linear equation (5) is added with a
conditional equation, i.e.:
jjj
jjjjjj
ZNC
NBXAY
=
++=
ε
δ
δ
(5’)
where C
j
is the matrix with elements of ones or zeros for the
conditional equations, Z
j
is the known constant vector. The
LeastSquares estimation and integer search will be based on
the (5’) and (6) instead of (5) and (6). This process is more
applicable if more than one satellite systems are used.
Modelling is always the first key process for a RTK system.
The functional model (5) builds the functional relations
between all the observations and all states, including rover
position, ambiguities and ionospheric and tropospheric
biases. This includes the processes of combining and
differencing measurements, imposing constraints such as
known coordinates and integers, applying ionosphere and
troposphere corrections etc. The stochastic models (6) give
the statistical knowledge or assumptions on the residual
errors and measurement noises, such as zero mean white
noise, correlations between the measurements of different
epochs (Wang, 2000; Wang et al., 2001)
The next key process is to complete ambiguity resolution
and position estimation following one of the AR methods,
such as the wellknown Leastsquares ambiguity
decorrelation adjustment (LAMBDA) method by Teunissen,
(1995, 1997), Minima Search (LMS) method by Pratt et al
1997). Any improved version of the methods will be an
additional advantage. The process basically consists of a
LeastSquares estimator and an Integer Search Engine. The
estimator provides approximate real values for both state
parameters and ambiguity parameters and their covariance
matrix for AR integer search, and position estimation after
the ambiguities are fixed to their integer values. The integer
search engine performs statistical search over the potential
ambiguity candidates to find and validate the best set of
integer candidates.
Implementation of efficient multipath mitigation approaches,
quality control and quality assurance procedures in the above
modeling and estimation processing is also important. It is
fairly the case that the success of a RTK system depends on
detailed processing techniques. Some software systems
implement a more efficient integer search algorithm, whilst
others are superior in deterministic and/or stochastic
modeling. The most successful ambiguity resolution
software takes good care of the detailed elements, this being
especially true in the current GPS case, where only L1 and
L2 carriers are available for use in ambiguity resolution.
RTK PERFORMANCE CHARACTERISTICS
One may have to use different performance characteristics to
evaluate different types of GNSS solutions, to address
different application requirements. For instance, the
parameters such as accuracy, availability, continuity and
integrity have been well defined to describe the performance
of code based single point navigation solutions and
differential GPS solutions for aviation navigation. Code
based SPP and differential navigation are the simplest and
robust positioning modes, to provide metre to tens of meter
accuracy. As identified, RTK positioning is a much more
complicated and vulnerable technique, aiming to achieve the
accuracy as high as centimetres with as few as possible data
epochs in real time for any user dynamics. Therefore a
greater care has to be taken to characterise the performance
to address the concerns of its professional positioning users,
such as surveying, data acquisitions, machine automation in
precision agriculture, mining and future lanebased vehicle
navigation. The performance of a RTK system, consisting of
a base station/network, data links between receivers and
userterminal, may be analysed from both system service
providers’ and users’ perspectives. The service provider is
referred to as the operator of the continuously operating
reference network which comprises a minimum of one
reference station, a data processing centre and data links
between the network processing centre and user receivers to
provide or deliver the differential corrections to user
terminals. The user terminal is generally equipped with a
GNSS receiver, and a communication device and a user
control/interface unit where RTK solutions are integrated or
interfaced into a particular application. From the service
provider’s points of view, the following factors are of
concern
•
Baserover distance: the maximum radius over which a
base station can serve effectively with RTCM messages,
allowing the users within the coverage to receive the
RTCM messages within certain latency and obtain its
RTK solutions epochby epoch. A relevant concept is
the interstation distance in the network case. As shown
in Figure 1, the baserover distance D is approximately
equivalent to 0.5774 times of the interstation distance
S. The distance limitation is mainly caused by the strong
dependence of the ionospheric biases on the separation
of two receivers. Other less sensitive error factors are
residual troposphere errors (after modelling corrections)
and relative orbital errors. The system performance is
considered more desirable if a longer baserover
distance is allowed.
Figure 1. Relation of Interstation distance S and equivalent
baserover distance D=0.5774S
•
Timeliness of RTCM message: the time latency of the
latest RTCM message available for users with respect to
the user time. Users will need to predict the ranging
corrections to the most current time when the user
terminal produce RTK solutions. This latency is the sum
of the delays caused by processing at the base
station/network centre and raw transmissions from the
base stations to network centre, and messages from the
centre to users, typically one to a few seconds. The
parameter is obtainable from statistical results for a
given operational environment and communication
links.
The above two factors will directly affect the performance of
the RTK solutions at user ends. From a user’s perspective,
the following characteristics are of interest:
•
Time to FirstFix (TTFF). This is referred to the time
period or number of secondly sampling measurement
epochs required to resolve or fix all ambiguity integers
at a sample epoch. In some literature, this parameter is
known as “Time To AmbiguityFix (TTAF)” or
“initialisation time” of the RTK system. However, Time
to Ambiguityfix is more suitable for more general
situations where all the integer parameters are resolved
and fixed independently at each epoch, involving
measurements from single or multiple most recent
epochs. It is most desirable if the RTK system always
fix all integers using data of the current epoch.
•
AR Availability. This instantaneous performance
characteristic is defined as the percentage of DD
integers: the total number of fixed DD integers over the
total number of DD integers over a continuously
operating session. The fixed integers are these that have
passed the validation tests in the integer search process.
The problem is that the validation process may include
incorrect integers and exclude correct integers.
• AR Reliability is defined as the percentage of DD
integers: the total correctly fixed DD integers over the
total number of DD integers. In some studies, the AR
success rate was defined the correct integer number with
respect to the total integers.
• RTK availability (in term of accuracy) is defined as
the percentage of time during which the RTK solutions
are available at a certain accuracy using the ambiguity
integers fixed phase measurements.
•
RTK availability (in terms of AR reliability) may be
defined as the percentage of time, of which the integer
are correctly fixed for all integers at each epoch,
assuming all the ambiguityfixed solutions will produce
the required centimetre accuracy.
•
RTK accuracy is defined as the degree of conformance
of an estimated RTK position at a given time to a
defined reference coordinate value (or ‘true’ value)
which
is obtained from an independent approach,
preferably at higher accuracy.
These parameters may be used to evaluate the performances
of a RTK system from different perspectives, although
variations and modifications to these definitions are still
possible. Of these parameters, baserover distance, time to
firstfix, AR reliability, RTK availability and RTK accuracy
are of most concerns. Few existing commercial RTK
systems provide sufficient performance parameters in their
specifications. However, to users, more detailed performance
information would help their decision to choose a desirable
RTK system to meet performance requirements. Fortunately,
we are able to access to some testing results obtained from
HDRTK2
TM
, a commercial RTK developed by HandyNav
Inc (HandyNav, 2005), allowing us to demonstrate the
concepts of the above performance parameter and how the
different parameters are obtained for through experiments
and are related each other. Using NovAtel OEM4 receivers,
the test GPS data were collected for 16 static baselines over
2 to 45 kilometres and for 60 hours to 400 hours. To reflect
the performance of the system in real world situation more
comprehensively, each longhour data set was processed
separately every 300 seconds, producing over 250,000 sets
of results and solutions. Therefore the results can definitively
demonstrate performance characteristics of the tested RTK
system.
Figure 2 shows the time to firstfix (TTFF) at different
percentages plotted against the baselines. It is observed that
the longer the baseline, the longer the TTFF at a given
percentage, or the shorter the TTFF, the higher the
percentages for a given baselines. Figure 3 illustrates the AR
availability varying with baselines, while Figure 4 plots the
AR reliability against the baseline length, showing the
difference and similarity between the two indicators. We see
that AR reliability is not necessarily worse than AR
availability. But, in general, the longer the baseline, the
lower the AR availability and AR reliability. The RTK
accuracy in horizontal and vertical is illustrated in Figure 5,
confirming the formal horizontal and vertical accuracy
within 1cm+0.5ppm and 2cm+1ppm respectively (All these
four figures are presented by courtesy of HandyNav). In
summary, with these extensive experimental results, we
conclude that the HDRTM2
TM
system can provide instant
RTK solutions for distance of up to 20 km, and ambiguity
fixed solutions for distance of up to 50km. The AR
reliability of the fixed solutions is above 99% for 20km
baselines and 98% for 50 km baselines. The position
accuracy for integerfixed solutions is 1cm+0.5ppm
(horizontal) and 2cm+1pmm (vertical). It is believed that
these performance specifications would be more convincing
from users perspective.
Figure 2. Timetofirstfix at 50%, 60%, 70%, 80%,90% and
95%, plotted against baserover distances(baselines)
Figure 3. Illustration of Ambiguity Resolution Availability
variation Vs baserover distances.
Figure 4. AR reliability variation Vs baserover distances.
Figure 5. RTK positioning accuracy Vs baserover distance.
EXPERIMENTAL RESULTS FROM GPS VIRTUAL
GALILEO MEASUREMENTS
SPP performance may be studied with simulation data and
real GPS data, such as O’Donnell et al (2003) and Feng
(2005). But, RTK performance is preferably based on the
real data, due to the dependences of the solutions on specific
observation environment and software algorithms. Based on
the RTK performance characteristics examined above, the
experimental studies now are to demonstrate how the
performance of a RTK system improves using multiple
satellite systems. This is achieved using GPS and virtual
Galileo dualfrequency measurements.
A minimum of three doubledifferenced phase measurements
is required to resolve ambiguity and obtained ambiguity
fixed coordinates. More redundant phase measurements
provide stronger geometric constraints between double
differenced phase measurements, thus enhancing the
availability and reliability of ambiguityfix. Theoretically,
this improvement may be more evident for longer baselines
and when the AR is performed instantly with measurements
from single epochs. Therefore, the following experimental
results will focus on a fewer performance parameters such as
AR reliability and RTK availability and RTK accuracy,
based on singleepoch ambiguity resolutions.
Table 1 Three GPS data sets on January 1, 2007 used in
experimental analysis: source: US Continuously Operating
Reference Stations (http://www.ngs.noaa.gov
/CORS). All
are SouthNorth baselines.
CORS
Station
ID
Base
Rover
Distance
Data
Period:
GPST
Sample
Interval
Total
# of
epochs
Mark
Angle
P473
P472
22.5 km 00:00:00
23:59:45
15 sec 5760 15 deg
P472
P478
38.5 km 00:00:00
23:59:15
15 sec 5758 15 deg
P473
P478
56.82km 00:00:00
23:59:45
15 sec 5760 15 deg
Table 2 summarises the statistical results from the first data
set for the baseline P473P472. The data were processed
using a research version of the QUT kinematic GPS
positioning software using the standard LAMDA ambiguity
search method. The software resolves integer ambiguities for
two virtual combinations using single and multiple data
epochs, and recovers the correct integers on L1 and L2
simultaneously. For the given data set, two different
solutions were with GPS data and GPS+virtual Galileo data,
respectively. GPS solutions start from GPST 00:00:00
resolving L1 and L2 ambiguities epochbyepoch to the last
epoch at GPST 23:59:45. There were a total of 5760 epochs.
GPS+virtual Galileo solutions starting from GPST 01:00:00,
resolving L1 and L2 ambiguities for the current epoch j and
the epoch j120, simultaneously. This solution was also
obtained epochbyepoch until 23:59:45. There were a total
of 5640 epochs. The results show the improvement on all
three major performance indicators: AR reliability and RTK
availability and RTK accuracy.
With the same data processing strategy, we obtained the
statistical results from the second and third data sets as
outlined in Tables 3 and 4 respectively. The results once
again confirm the improvement on all the performance
indicators with additional virtual Galileo measurements,
although the extent of improvement is different in terms of
AR reliability and RTK availability based on correct integer
solutions and the RTK availability in terms of accuracy of ±5
cm. To demonstrate this performance improvement, Figure 6
compares the easting errors between GPS measurements and
GPS + virtual Galileo measurements for the third set. It is
clearly observed that the GPS RTK solutions appear
generally nosier and contain the effects of a fairly number of
larger unfixed integers, whilst the GPS/virtual Galileo RTK
solutions contains mostly of the effects of smaller unfixed
integers. In other words, the magnitudes of the integer errors
in the dual system case are smaller than these obtained in the
single system case. These observations may be useful for
further enhancement of the AR performance.
0 1000 2000 3000 4000 5000
1.5
1
0.5
0
0.5
1
1.5
Time Epoch (15 second)
Easting error (metre)
GPS
GPS+Virtual Galileo
Figure 6. Illustration of RTK position errors including the
effects of wrong integers, for GPS and GPS+virtual Galileo
measurements respectively, showing the GPS RTK solutions
affected by a fairly number of larger unfixed integers, whilst
the GPS/virtual Galileo RTK solutions contains mostly of
the effects of smaller unfixed integers.
Table 2. Statistics of the RTK performance with GPS and
GPS + Virtual Galileo measurements from single epochs for
a baseline of P473P472 (NA=Not Available)
GPS Measurements
Only
GPS+Virtual Galileo
Measurements
P12,
L12
P12,
L12, L1
P12,
L12
P12,
L12, L1
Total Epochs
5760 5760 5640 5640
RTK available
epochs
3400 4014 4149 3947
Total DD phase
measurements
36326 72566 70781 141400
Total fixed DD
integers
30657 61827 64591 125858
AR
Reliability
84.39% 83.82% 91.25% 89.02%
RTK Availability
(AR reliability)
59.03% 69.69% 73.59% 70.01%
RTK availability
(Accuracy 5 cm)
NA 84.51%
86.42%
53.77%
NA 93.90%
92.93%
66.88%
RTK accuracy
(STD in E,N U,
in metre)
NA 0.0197
0.0175
0.0201
NA 0.0183
0.0156
0.0219
RTK accuracy
(RMS in E,NU,
in metre)
NA 0.0197
0.0175
0.0201
NA 0.0183
0.0156
0.0219
Table 3. Statistics of the RTK performance with GPS and
GPS+Virtual Galileo measurements from single epochs for a
baseline of P478P472 (NA=Not Available)
GPS Measurements
Only
GPS +Virtual
Galileo
Measurements
P12,
L12
P12,
L12, L1
P12,
L12
P12,
L12, L1
Total Epochs
5758 5758 5638 5638
RTK available
epochs
3167 3614 4080 3705
Total DD phase
measurements
36243 72489 70623 141246
Total fixed DD
integers
29790 61629 63982 124843
AR
Reliability
82.19% 85.02% 90.60% 88.38%
RTK Availability
(AR reliability)
54.98% 62.74% 72.38% 65.71%
RTK availability
(Accuracy within
±5 cm: E, N, U)
NA 76.14%
85.78%
48.54%
NA 85.67%
92.66%
60.39%
RTK accuracy
(STD in E,N U in
metre)
NA 0.0190
0.0177
0.0192
NA 0.0184
0.0164
0.0213
RTK accuracy
(RMS in E,N,U
in metre)
NA 0.0190
0.0183
0.0192
NA 0.0183
0.0169
0.0213
Table 4. Statistics of the RTK performance with GPS and
GPS+Virtual Galileo measurements from single epochs for a
baseline of P473P478 (NA=Not Available)
GPS Measurements
Only
GPS+Virtual Galileo
Measurements
P12,
L12
P12,
L12, L1
P12,
L12
P12,
L12, L1
Total Epochs
5760 5760 5640 5640
RTK available
epochs
3738 4706 4778 5354
Total DD phase
measurements
36226 72452 70590 141180
Total fixed DD
integers
31453 67408 67782 138885
AR
Reliability
86.82% 93.04% 96.02% 98.37%
RTK Availability
(AR reliability)
64.90% 81.70% 84.72% 94.77%
RTK availability
(Accuracy within
±5cm in E, N, U)
NA 89.48%
96.15%
90.40%
NA 97.15%
99.77%
98.71%
RTK accuracy
(STD in E,N, U
in metre)
NA 0.0093
0.0115
0.0186
NA 0.0072
0.0080
0.0156
RTK accuracy
(RMS in E,N,U
in metre)
NA 0.0095
0.0118
0.0203
NA 0.0079
0.0088
0.0185
0 1000 2000 3000 4000 5000 6000
0
0.01
0.02
0.03
0.04
0.05
0.06
Time epochs (15 second)
RMS for DD measurements (m)
P473P478
P472P478
P473P472
Figure 7. RMS values of the doubledifferenced phase
measurements (ionospherefree), showing overall data
quality of different data sets. The overall RMSs for the three
data sets in Table 1 are 0.0266m, 0.0265m and 0.0110m in
order.
We must realise that the above performance results are
obtained purposely to reflect the benefits of an additional
satellite system under the same circumstances and do not
represent the overall performance of the RTK system in use.
In fact, we have intentionally removed some modelling
process like known integer constraints and advanced
stochastic modelling procedures, which can significantly
change the AR and RTK performance results.
We also notice that the overall RTK performance results for
the longer baseline P473P478 are much better than these
from shorter baselines. This implies that the baserover
distance does not necessarily limit the performance
improvement of the RTK system used. Instead, the quality of
the data is likely the major factor for the performance of the
RTK system under the given testing environment, where the
effects of the ionosphere and troposphere on ambiguity
resolution are relatively insignificant, although they would
still be stronger over the longer baselines. Figure 7 compares
the Root Mean Squares (RMS) values of in position
estimation using the integerfixed, doubledifferenced phase
measurements (ionospherefree) in the three data sets.
The performance of the LAMBDA method, particularly the
decorrelation performance could be potentially a problem in
the case of large number of integer parameters as shown by
Xu (2001). It has been evident from the computing
experiments that the integer searching takes much longer
time when processing GPS+Virtual Galileo measurements,
where the integer numbers are doubled, typically, 30 to 40,
instead of 15 to 20. Improving decorrelation performance
deserves further research efforts, to address the needs for the
combination of GPS and Galileo systems in the future. On
the other hand, more redundant phase measurements allow
integer parameters to be partially fixed in the searching
process. This would also improve the situation.
CONCLUDING REMARKS
The paper has contributed to the definitions of various
Ambiguity Resolution (AR) and Real Time Kinematic
(RTK) performance characteristics, including BaseRover
distance, Timeliness of RTCM messages, Timetofirst fix,
AR availability, AR reliability, RTK availability and RTK
accuracy etc. These characteristics enable more
comprehensive assessments to the performance of a RTK
system or some particular algorithms. Thanks to HD
RTK2
TM
, we have demonstrated the above performance
parameters and how different parameters are related each
other through extensive experimental results from GPS data
collected for 16 static baselines over 2 to 45 kilometres and
for 60 hours to 400 hours.
The paper has also investigated how the RTK system with
multiple satellite constellations works and performs, using
GPS and virtual Galileo dualfrequency measurements.
Virtual Galileo measurements are obtained from the same set
of GPS data, but starting from the epoch in 1 or 2 hours later.
Generic linear equations for Ambiguity Resolution (AR) and
position estimation has been outlined for using GPS and
Virtual Galileo measurements. Numerical analysis has been
performed using a research version of QUT postprocessing
RTK software package, and results from three RINEX dual
frequency data sets for different lengths of baserover
distance are given against various computing schemes, such
as L1/L2 and widelane signals, GPS and GPS and virtual
Galileo measurements. The performance results have
demonstrated indeed the improvement due to the doubled
number of measurements in almost all the performance
parameters, although the degree of improvement may vary
from one data set to another. The key factor for performance
of the RTK system in use is the quality of the data, and
which is less sensitive to the baserover distance under
certain circumstances. It is also noted that improving de
correlation performance deserves further research efforts in
order to address the ambiguity search issues for integrated
GPS/Galileo RTK systems in the future.
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