ArticlePDF Available

Exploring GNSS RTK performance benefits with GPS and virtual galileo measurements

Authors:
Article

Exploring GNSS RTK performance benefits with GPS and virtual galileo measurements

Abstract and Figures

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 dual-frequency 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 double-differenced measurements are introduced.
Content may be subject to copyright.
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 (CRC-SI), 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 dual-frequency 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 double-differenced 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 RTK-GPS techniques: single-base
RTK and network-based RTK (e.g. Lachapelle et al., 2002).
The limitation of single-base 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.
Network-RTK is a GPS positioning technique that
challenges the current constraints of single-base RTK (Rizos,
2002). In current Network-RTK implementations, the inter-
receiver distance is typically between 50 and 70km. In
addition to the distance factor, a common problem for both
single-base 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, distance-dependence of the
magnitude of measurement biases and the unreliability of
ambiguity solution are the two key problems in the existing
dual-frequency 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 double-differenced
(DD) phase measurements and amplification of phase
noises containing effects of multi-path errors.
Regional network-based RTK services at the cm-dm
accuracy level: The reference stations equipped with
triple-frequency receivers may be spaced up to a few
hundred kilometers, to provide differential and RTK
services over a region, typically statewide or country-
wide. Network-based atmospheric error corrections
may be provided for higher accuracy and/or to dual-
frequency users within the network coverage. Inverse
network-RTK 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
sub-metre accuracy levels: These can be provided with
ambiguity-resolved widelane signals over any distance
(but with satellites in common view). For the same
decimeter-level 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 dual-frequency
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 dual-frequency
receivers employing existing network-based RTK
techniques). The planned Galileo/GPS ground tracking
network of tens of stations can provide decimeter-level
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, GPS-based 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
base-rover distance, time-to-ambiguity 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 double-differenced
P
12
, L
1
and L
12
measurements, where the L12 stands for the
wide-lane 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 3-by-1 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 L1-E6, L1-E5A and L1-E5B.
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
“Two-Epoch Approach”. With the GPS-only 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 re-written 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 n-by-1
observation vector, A is n-by-3 matrix; δX is 3-by-1 state
vector; and B is n-by-p matrix, N is p-by-1 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 delta-position vector
of the current epoch jth with respect to the (j-M)th epoch,
which may be estimated from the double-differenced phase
measurements between the jth and (j-m)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 delta-position 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
Least-Squares 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 well-known Least-squares 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
Least-Squares 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 lane-based vehicle
navigation. The performance of a RTK system, consisting of
a base station/network, data links between receivers and
user-terminal, 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
Base-rover 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 epoch-by epoch. A relevant concept is
the inter-station distance in the network case. As shown
in Figure 1, the base-rover distance D is approximately
equivalent to 0.5774 times of the inter-station 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 base-rover
distance is allowed.
Figure 1. Relation of Inter-station distance S and equivalent
base-rover 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 First-Fix (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 Ambiguity-Fix (TTAF)” or
“initialisation time” of the RTK system. However, Time
to Ambiguity-fix 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 ambiguity-fixed 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, base-rover distance, time to
first-fix, 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
HD-RTK2
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 long-hour 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 first-fix (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 HD-RTM2
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 integer-fixed 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. Time-to-first-fix at 50%, 60%, 70%, 80%,90% and
95%, plotted against base-rover distances(baselines)
Figure 3. Illustration of Ambiguity Resolution Availability
variation Vs base-rover distances.
Figure 4. AR reliability variation Vs base-rover distances.
Figure 5. RTK positioning accuracy Vs base-rover 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 dual-frequency measurements.
A minimum of three double-differenced 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 ambiguity-fix. 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 single-epoch 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 South-North 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 P473-P472. 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 epoch-by-epoch 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 j-120, simultaneously. This solution was also
obtained epoch-by-epoch 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 un-fixed
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 un-fixed integers.
Table 2. Statistics of the RTK performance with GPS and
GPS + Virtual Galileo measurements from single epochs for
a baseline of P473-P472 (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 P478-P472 (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 P473-P478 (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)
P473-P478
P472-P478
P473-P472
Figure 7. RMS values of the double-differenced phase
measurements (ionosphere-free), 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 P473-P478 are much better than these
from shorter baselines. This implies that the base-rover
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 integer-fixed, double-differenced phase
measurements (ionosphere-free) in the three data sets.
The performance of the LAMBDA method, particularly the
de-correlation 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 de-correlation 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 Base-Rover
distance, Timeliness of RTCM messages, Time-to-first 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 dual-frequency 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 post-processing
RTK software package, and results from three RINEX dual
frequency data sets for different lengths of base-rover
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 base-rover 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.
REFERENCES
Feng, Y. (2005) Future GNSS Performance Predictions
Using GPS with a Virtual Galileo Constellation, GPS World,
March 2005, 46-52,
Feng Y and Rizos, C (2005), Three Carrier Approaches for
Future Global, Regional and Local GNSS Positioning
Services: Concepts and Performance Perspectives, The
Proceedings of ION GNSS 2005, 13-16 September 2005,
p2277-2287
Feng, Y and M Moody (2006), Improved Phase Ambiguity
Resolution Using Three GNSS Signals,
PCT/AU2006/000492 (http://www.wipo.int/pctdb), April
2006.
HandyNav (2005), HandyNav RTK Testing Report using
NovAtel Dual-frequency receivers.
Hatch, R (2006), “A New Three-Frequency, Geometry-Free
Technique for Ambiguity Resolution, Proceedings of ION
GNSS 2006, September 2006.
Lachapelle G, Ryan S, Rizos C (2002) Servicing the GPS
user, Chapter 14 in Manual of Geospatial Science
and
Technology, J. Bossler, J. Jenson, R. McMaster & C. Rizos
(eds.), Taylor & Francis Inc., 201-215.
O’Donnell, T Watson,J Fisher, St Simposon, G.,Brodin, E.
Bryant, and D Walsh, Galileo Performance, GPS World,
June 2003, 38-45.
Pratt, M., B Burke, and P Misra (1997), Single-Epoch
Integer Ambiguity Resolution with GPS Carrier Phase:
Initial Results, Navigation, Vol 32, No 4, Pp386-400.
Rizos C (2002) Network RTK research and implementation:
A geodetic perspective, Journal of Global Positioning
Systems, 1(2), 144-150.
Teunissen, P.J.G. (1995),The least-squares ambiguity
decorrelation adjustment: a method for fast GPS integer
ambiguity estimation. Journal of Geodesy, Vol. 70, No. 1-2,
pp. 65-82.
Teunissen, P.J.G, P J De Jonge and C.C. J.M Tiberious
(1997). Performance of the LAMBA Method for fast GPS
Ambiguity Resolution, Navigation, Vol 44, No 3, pp373-
383.
Wang J. (2000) Stochastic modelling for RTK GPS/Glonass
positioning, Navigation, Journal of the US Institute of
Navigation, 46(4), 297-305
Wang J., C. Satirapod & C. Rizos (2002) Stochastic
assessment of GPS carrier phase measurements for precise
static relative positioning, Journal of Geodesy, 76(2), 95-104
Xu, P.L (2001), “Random Simulation and GPS
decorrelation”, Journal of Geodesy, Vol 75, pp 408-423.
... The key point of precise positioning of the Global Navigation Satellite Syste) (GNSS) is the ability to mitigate (reduce) all the potential source errors and interference in the system. All errors in the GNSS observations caused by signal propagation, the environment around the receiver and the equipment of the recipient, must be mitigated [1][2][3][4]. ...
... number of independent positions can be constructed by subtracting 2 { , the carrier phase observation multiplied by the frequency ratio, from 1 { , which can be recorded as: ...
... The above Equations (5) and (6) eliminate the first-order ionospheric advance effect of the observation, which is widely used in the GNSS data processing [26][27][28]. The disadvantage of this linear combination is that noise from 1 { and 2 { measurements increases threefold and that ambiguities cannot be directly resolved as integers. ...
Article
Full-text available
The article presents results of verification of the kinematic measurements usefulness for precise real-time positioning RTK in the local reference system. These measurements allow for continuous RTK measurements in the event of temporary interruptions in radio or internet connections, which are the main reason for interruptions in RTK kinematic measurements and cause a decrease in the reliability and efficiency of this positioning method. Short interruptions communication are allowed during the loss of the key correction stream from the local RTK support network, so the global corrections obtained from the geostationary satellite are used. The aim of the article was to analyze the accuracy of measuring the position of moving objects. Practical conclusions were formulated according to the research subject, the presented mathematical models, the experiment and the analysis of the obtained results.
... Titik yang diukur ditentukan secara real time dari titik yang sudah diketahui koordinat sebelumnya menggunakan data fase. Metode penentuan posisi pun sangat beragam disajikan pada Gambar 2, metode RTK ini digunakan untuk mereka yang membutuhkan ketelian pada orde centimeter level (Feng & Wang, 2007). Sebuah sistem RTK terdiri dari jaringan CORS (Continuously Operating Reference Station) dan data link antara network server-stasiun referensi dan antara server dengan user-terminal (Feng & Wang, 2008). ...
Conference Paper
Full-text available
ABSTRAK Perangkat CHC X91+ ini dikembangkan dan diproduksi melalui Shanghai HuaCE Navigation Technology ltd. Perangkat CHC X91+ ini merupakan perangkat yang mampu menerima sinyal GNSS dengan 220 channel dan bisa menangkap sinyal dari satelit GPS, GLONASS, SBAS, Galileo dan Beidou bersamaan. Perangkat ini dirancang bisa digunakan untuk pengukuran posisi metode RTK (Real Time Kinematic). Berdasarkan spesifikasi teknis alat, penentuan posisi metode RTK, CHC X91+ menghasilkan koordinat horizontal dengan nilai RMS 8 mm + 1 ppm dan untuk posisi vertikal dengan nilai RMS 15 mm + 1 ppm. Penelitian ini bertujuan untuk mengkaji sejauh mana ketelitian pengukuran koordinat metode RTK, yang diperoleh dari hasil pengukuran GPS menggunakan receiver CHC X91+. Metode penelitian yang digunakan adalah membandingkan nilai hasil pengolahan data statik dibandingkan dengan hasil nilai RTK metode single, nearest dan network RTK. Uji coba alat dilakukan di kompleks Badan Informasi Geospasial tepatnya di atas Gedung Laboratorium Geodesi Pusat Jaring Kontrol Geodesi dan Geodinamika. Pengolahan data (post processing) menggunakan data CORS BAKO selama delapan hari dan data pengamatan statik di titik yang sama dengan pengamatan RTK nya. Hasil penelitian menunjukkan perangkat CHC X91+ menghasilkan ketelitian bagus untuk metode Network RTK dan single RTK dengan jarak base kurang dari 30 km dan solusi float untuk jarak base lebih dari 40 km. Ketelitian RMS perangkat CHC X91+ sesuai dengan spesifikasi teknis alat yang ada sehingga alat CHC X91+ ini dapat digunakan untuk pemetaan skala besar dan aplikasinya dapat digunakan untuk koreksi citra satelit resolusi tinggi.
... This ensures that the exact differences that result from measurements with and without Galileo satellite signals will be displayed. Taking into account the general assumption that correction errors up to 10 km from the base station are identical in both the rover receiver and the base station and the assumption that the baseline should not exceed 35 km to provide real-time online measurements, the following distances to the base station have been selected: 25 km, 35 km and 45 km [4]. The longest baseline has been selected over 35 km to ensure that measurements with baselines above 35 km could be performed using the Galileo signal. ...
Article
Full-text available
Two factors of the existing GNSS Real-Time Kinematic (RTK) positioning are as follows: distance-dependence and unreliable ambiguity resolution under bad observation conditions in cities or forests. Use of multi-frequency GNSS signals and systems could possibly redefine RTK services in LatPos, regionally and globally, and more redundant measurements from multiple satellite systems, such as NAVSTAR, Galileo, Glonass and BeiDou, can improve the performance of RTK measurement results in terms of accuracy, availability, reliability and time to fix. The benefits of multiple systems of GNSS services are as follows: 1) savings in the reference station infrastructure costs, and 2) improvement on RTK preciseness and reliability for the professional users. The paper aims at studying how the RTK system, using multiple satellite constellations, performs, adding Galileo signal measurements. Galileo measurements are observed using a field receiver and corrections received from LatPos base station network. Numerical analysis is performed using real-time corrections in field receivers, and results from collected RINEX data are compared by various computing schemes, such as L1/L2 and wide lane signals, NAVSTAR and NAVSTAR with Galileo measurements. The results have preliminary demonstrated the significant improvement using both GNSS satellite signals. Further improvements on the LatPos system have been introduced and the planned improvements shown.
... Teknologi GPS pun semakin berkembang dengan kemampuan mampu mendapat koordinat teliti secara real time atau sering disebut dengan metode penetuan posisi secara differensial Real Time Kinematic (RTK). Metode RTK ini digunakan untuk mereka yang membutuhkan ketelian pada orde centimeter level [6]. ...
... Odijk and Teunissen [8] proposed an analytical closed-form expression for the multifrequency ambiguity dilution precision. Feng [9,10] outlined a general modeling strategy for improved AR and positioning estimation using three or more phase and code ranging signals. Paziewski and Wielgosz [11] presented a method by using frequencies L1/E1 and L5/E5a combination to account for Galileo/GPS intersystem biases in precise satellite positioning. ...
Article
Full-text available
A novel method, which is based on the triple-frequency combination and Space-Based Telemetry, Tracking, and Command (STT&C) stations, is proposed in this paper. Considering BeiDou Navigation Satellite System (BDS) Geostationary Orbit (GEO) and Inclined Geostationary Orbit (IGSO) satellites as the STT&C facilities, firstly, we presented the BDS Medium Earth Orbit (MEO) satellites’ precise orbit determination scheme based on triple-frequency combination. Then, we gave the sufficient and necessary conditions about the visibility and the coverage rate calculation model of STT&C to BDS MEO satellite. And then we deduced the model of BDS MEO satellites precise orbit determination based on triple-frequency combination observations. At last, we designed the simulation calculation. The simulation results show that orbit determination of BDS MEO satellite based on STT&C station can be realized at all times. And most of the simulation period time, under the condition of the dm level orbit determination for GEO/IGSO satellites, the position accuracy of the relative orbit determination is better than 4 m, the horizontal accuracy of the relative orbit determination is within 2.5 m, and the vertical accuracy of the relative orbit determination is less than 3.5 m.
... Accuracy can be defined as how far the coordinates calculated during testing are from the true values (Feng and Wang 2007). As mentioned previously the coordinates of the GPS reference marks calculated during the static survey were considered as true values for all accuracy check. ...
... The Real Time Kinematics (RTK) Global Navigation Satellite System (GNSS) is based on a phase measurement of carrier electromagnetic wave and using of on-line corrections for accurate pseudorange measurement (Feng, 2007). On-line corrections can be received from a simple base station or network of base stations. ...
Article
In the paper is discussed using of RTK GNSS for a mobile robot navigation and georeferencing of currently measured data. There is introduced a concept of usage RTK GNSS equipment for 5DOF self-localization of mobile robot. A simple but robust method for navigation of mobile robot to waypoints in area is described. In a conclusion there are shown results of using RTK GNSS with proposed navigation method. Presented algorithms are prepared for autonomous mobile mapping projects - field measurements in outdoor environments.
... 72 The most important parameters to be considered when evaluating the performance of the RTK system include the following: the base-rover distance, time to ambiguity fix, ambiguity resolution (AR) reliability, RTK availability, and RTK accuracy. 73 Based on these parameters, practical field experiments have been carried out to evaluate the performance of the RTK GPS technique. For example, field tests were carried out to evaluate RTK performance in a highway stake-out project, under varying field scenarios and constraints, such as challenges due to signal interruptions and data latency. ...
Article
Full-text available
The main goal of our ongoing research is to design a low-cost continuous monitoring system for landslide investigation using the Reverse RTK (RRTK) technique. The main objectives of this paper are to review the existing Global Positioning System (GPS) tools and techniques used for landslide monitoring, and to propose a novel low-cost landslide monitoring technique using Reverse RTK GPS. A general overview of GPS application in landslide monitoring is presented, followed by a review of GPS deformation monitoring systems and some of the factors used for their categorization. Finally, the concept, principles and advantages of the proposed new landslide monitoring system are discussed.
Article
Full-text available
Multipath is one of the main causes of degraded position accuracy in the Global Navigation Satellite System (GNSS) because portions of the signals can be reflected by high buildings in dense urban areas. Multipath mitigation techniques based on hardware enhancement or signal processing help to improve GNSS accuracy for high-precision surveying. Geographic Information System (GIS) is also used in the signal propagation model to predict multipath effects. In addition to these existing approaches, we found that spatial statistical methods are useful in multipath mitigation because a unique spatial distribution of user positions can be produced by the multipath. In this paper, we present a spatial statistical method for mitigating multipath and improving the accuracy in GNSS positioning. Multipath tends to be associated with spatial outliers in simulated user positions (SUPs) and contributes little to the spatial clustering of SUPs. Using these spatial characteristics, we developed a method for identifying multipath satellites, which consists of the components of deviation distance, deviation load, and deviation ratio. Once the identified multipath satellites are excluded, a user position is determined using a mean spatial center of the SUPs from the remaining satellites. The effects of such multipath mitigation were validated by examining whether our method correctly identified multipath satellites and by comparing the position errors with and without the method. We demonstrated the applicability of our solution with a simulation experiment for Shinjuku, using a precise ephemeris for the Global Positioning System (GPS) and the orbital parameters for the proposed constellations of the GALILEO and the Quasi-Zenith Satellite System (QZSS).
Article
Based on original code and phase measurements to three or more ranging signals offered by future GNSS systems, this paper outlines geometry-free and geometry-based ambiguity resolution (AR) strategies for DD phase measurements and introduces the algorithms that improve estimation of zero-differenced (ZD) phase biases using a network of GNSS reference stations. Given three L-band frequencies, AR strategies can generally identify two best Extra-widelane (EWL: λ≥293cm, in this context) or Widelane (WL: 75cm ≤λ<293cm) virtual signals to allow for more reliable ambiguity resolution, thus supporting decimetre RTK positioning over baselines of hundreds of kilometres in length. Analysis shows that the success rates for these selected virtual signals, having minimal or near minimal ionospheric effects and low noise levels, can be easily over 90% with measurements from a signal epoch. The third virtual signal has to be a choice of Medium-lane (ML:19cm≤λ<75cm) or Narrow-lane (NL: 10cm ≤λ<19cm) signals, whose integer ambiguity should be performed with refined widelane measurements, to support centimetre RTK positioning over distances of up to a hundred kilometres or so. When the double-differenced (DD) integer ambiguities between multiple stations are resolved and fixed, constraints can be imposed to ZD measurements from all the stations to improve the ZD phase bias of each receiver over a short span of observations. Precise ZD measurements are fundamental to enhanced Precise Point Positioning. Numerical experiments using 24-hour dual-frequency GPS data from four US CORS stations, with spacing 21, 56 and 74km, were performed in order to demonstrate the performance benefits of some of the key algorithms proposed in this paper. Results confirm that the AR with the ionosphere-reduced NL signals, instead of the original L1 and L2 signals, performs better over longer baselines. For instance, the AR success rate for the 74km data set is improved from 88% to 93%. Introducing DD constraints between four stations and eight satellites, the phase-bias estimates of ZD measurements are improved by a factor of I. 6, demonstrating the performance potential of the methods for precise point positioning using a short time span of observations.
Article
Full-text available
The GPS double difference carrier phase measurements are ambiguous by an unknown integer number of cycles. High precision relative GPS positioning based on short observational timespan data, is possible, when reliable estimates of the integer double difference ambiguities can be determined in an efficient manner. In this contribution a new method is introduced that enables very fast integer least-squares estimation of the ambiguities. The method makes use of an ambiguity transformation that allows one to reformulate the original ambiguity estimation problem as a new problem that is much easier to solve. The transformation aims at decorrelating the least-squares ambiguities and is based on an integer approximation of the conditional least-squares transformation. This least-squares ambiguity decorrelation approach, flattens the typical discontinuity in the GPS-spectrum of ambiguity conditional variances and returns new ambiguities that show a dramatic improvement in correlation and precision. As a result, the search for the transformed integer least-squares ambiguities can be performed in a highly efficient manner.
Article
This paper provides an overview of the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) method for the estimation of integer GPS ambiguities. The method's performance is discussed, together with the theoretical concepts on which it is based. The method is based on the integer least-squares principle and requires no application-dependent restrictions or assumptions. The actual integer estimation is preceded by a decorrelation step in order to make it more efficient. Especially for short time spans, a large gain in efficiency is obtained. The decorrelation of the ambiguities enables one to refrain from any approximation as far as the shape of the search space is concerned; i.e., the search is performed within the ellipsoidal space induced by the covariance matrix of the float ambiguities. The decorrelated ambiguities also make it possible to scale the search space such that, to a large degree of accuracy, it contains only the k best vectors of integer ambiguities.
Article
 (i) A random simulation approach is proposed, which is at the centre of a numerical comparison of the performances of different GPS decorrelation methods. The most significant advantage of the approach is that it does not depend on nor favour any particular satellite–receiver geometry and weighting system. (ii) An inverse integer Cholesky decorrelation method is proposed, which will be shown to out-perform the integer Gaussian decorrelation and the Lenstra, Lenstra and Lovász (LLL) algorithm, and thus indicates that the integer Gaussian decorrelation is not the best decorrelation technique and that further improvement is possible. (iii) The performance study of the LLL algorithm is the first of its kind and the results have shown that the algorithm can indeed be used for decorrelation, but that it performs worse than the integer Gaussian decorrelation and the inverse integer Cholesky decorrelation. (iv) Simulations have also shown that no decorrelation techniques available to date can guarantee a smaller condition number, especially in the case of high dimension, although reducing the condition number is the goal of decorrelation.
Servicing the GPS user, Chapter 14 in Manual of Geospatial Science and Technology
  • G Lachapelle
  • S Ryan
  • C Rizos
Lachapelle G, Ryan S, Rizos C (2002) Servicing the GPS user, Chapter 14 in Manual of Geospatial Science and Technology, J. Bossler, J. Jenson, R. McMaster & C. Rizos (eds.), Taylor & Francis Inc., 201-215.
Single-Epoch Integer Ambiguity Resolution with GPS Carrier Phase: Initial Results, Navigation
  • M Pratt
  • P Burke
  • Misra
Pratt, M., B Burke, and P Misra (1997), Single-Epoch Integer Ambiguity Resolution with GPS Carrier Phase: Initial Results, Navigation, Vol 32, No 4, Pp386-400.
Performance of the LAMBA Method for fast GPS Ambiguity Resolution, Navigation
  • P J G Teunissen
  • C C J Tiberious
Teunissen, P.J.G, P J De Jonge and C.C. J.M Tiberious (1997). Performance of the LAMBA Method for fast GPS Ambiguity Resolution, Navigation, Vol 44, No 3, pp373-383.