Point Real-Time Kinematic PositioningY. Gao, M. Abdel-Salam, K. Chen and A. WojciechowskiDepartment of Geomatics Engineering2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4

Abstract. Autonomous point positioning was theoriginal aim of GPS. However, due to errors causedby satellite ephemerides, satellite clock corrections,ionosphere, troposphere, multipath and noise, theaccuracy of autonomous point positioning cannot bebetter than a couple of meters, even after SelectiveAvailability (SA) was turned off.

Nowadays, with the emergence of Point Real-Time Kinematic (P-RTK) positioning, a lot ofattention is again focused on standalonepositioning. This method arose from the availabilityof precise ephemeris and satellite clock corrections.The concept behind this method is to make use ofun-differenced code and carrier phase observationsalong with precise corrections to get the bestaccuracy out of GPS. The concepts, challenges,processing steps and results of the P-RTK systemare presented in this paper.

Currently at the University of Calgary, asoftware system has been developed that achievescentimeter to decimeter level accuracy with a singleGPS receiver. All components of P-RTK have beenincorporated in this software, which is portable anduser-friendly, containing many graphical interfacetools for analyzing data.

Keywords: GPS, Precise Point Positioning, RTK

1 Introduction

Current carrier phase RTK positioning atcentimeter level accuracy requires the combinationof observations from a minimum of two GPSreceivers. At least one of these serves as the basestation with known coordinates, and the othersserve as rover stations whose position coordinatesare to be determined relative to the base station(s).Drawbacks of this approach include the practicalconstraints imposed by the requirement thatsimultaneous observations need to be made at therover and base stations, and that the rover stationshould be in the vicinity of the base station(s),typically up to 20 kilometers.

The requirements for the deployment of basestation(s) and the spatially limited operating rangeof the rover stations have increased the operational

cost and logistical complexity in the field. As such,the full adoption of RTK technology has beenlimited in many applications.

This paper describes the concept of Point Real-Time Kinematic Positioning (P-RTK). This newRTK positioning approach is based ori un-differenced carrier phase data using a single GPSreceiver assisted by precise orbit and clockcorrections. Since P-RTK has no requirement forthe deployment of base station(s), it is a global RTKapproach capable of providing greater solutionconsistency with increased operational flexibility.P-RTK will not only enrich the positioning optionsavailable to users, but also provide better timetransfer capabilities and aid meteorologicaldisciplines.

The remainder of the paper is organized intofive sections. The P-RTK method is first describedin Section 2. Various models for un-differencedcarrier phase data processing are presented inSection 3. Challenges in point RTK are explored inSection 4. Numerical results are presented inSection 5 to assess the potential of the method andto identify problems that need to be solved in thefuture. Finally, several concluding remarks aregiven in Section 6.

2 Concept of Point RTK Positioning

Conventional Standard Point Positioning (SPP)that was initially designed for GPS is subject to theinfluence of all error sources. Major error sourcesinclude those introduced by broadcast orbits andclocks, as well as atmospheric effects. Since SPP isonly able to provide position solutions with anaccuracy level of several meters, it is not suitable asa positioning method for geodetic and surveyingapplications that require a positioning accuracy ofseveral centimeters.

However, the situation has changed with theadvent of precise orbit and clock correctionproducts with centimeter level accuracy, since thetwo errors associated with broadcast orbits andclocks can be significantly reduced. Once orbit andclock errors are removed from GPS observations,much higher positioning accuracy can be expectedeven when a single GPS receiver is used. The

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method that derives high precision positioningsolutions by processing un-differenced carrier phaseobservations from a single GPS receiver assistedwith precise orbit and clock corrections is calledPrecise Point Positioning (PPP) (Zumberge et al.,1997; Kouba and Heroux, 2000; Gao and Shen,2002). The word "precise" is used here todistinguish it from the conventional SPP method.

PPP has received increased attention in the pastseveral years within the GPS community due to itsgreat operational flexibility and accuracy promise.Furthermore, if ambiguity resolution of un-differenced carrier phase observations becomesfeasible, the PPP method will be ready to supportthe development of new RTK systems using asingle receiver, without the need for base station(s).This is P-RTK, which is discussed in this paper.

PPP relies on the use of precise orbit and clockcorrections. To date, there are many organizations,including the International GPS service (IGS),Natural Resources Canada (NRCan) and JetPropulsion Laboratory (JPL), which offer precisedata in both post-mission and real-time modes. IGScurrently provides precise satellite orbit and clockcorrection data with different precisions andlatencies: predicted corrections with accuracies of25cm and 5ns, available twice daily, rapidcorrections with accuracies of 5cm and 0.2ns, witha latency of 17 hours, and final corrections withaccuracies of less than 5cm and 0.1ns, and with alatency of 13 days (IGS, 2000). JPL and NRCanprovide real-time satellite orbit and clockcorrections over the Internet for testing purposes.

3 Observation Models in Point RTK

The observation equations for code and carrierphase measurements on the Li frequency (i = 1, 2)are shown in Equations (1) and (2).

P(Li) = p + c(dt - dT) + dorb + dtrop + dwnlLl

+ d,

) = p + c{dl-dT) + dorb

+ d

(1)

dtmp - dionlLi

where:P(Li) is the measured pseudorange on Li

(m);is the measured carrier phase on Li

(m);

p is the true geometric range (m);

c is the speed of light (m/s);dt is the satellite clock error (s);dT is the receiver clock error (s);dorij is the satellite orbital error (m);

dtrop

dion/LiX,

Nt

i>r(t0,Li)

$s(tQ,Li)

dmultlP{Li)

dmult/<t>(Li)

e(.)

is the tropospheric delay (m);

is the ionospheric delay on Li (m);

is the wavelength on Li (m);

is the integer phase ambiguity on Li

(cycle);is the initial phase of the receiver

oscillator;is the initial phase of the satellite

oscillator;is the multipath effect in the measured

pseudorange on Li (m);is the multipath effect in the measured

carrier phase on Li (m) and

is the measurement noise (m).

Note that the initial phase of the receiver andsatellite oscillators is commonly ignored inconventional double-difference RTK systems basedon carrier phase. If it is combined with the integerphase components into a single term, Equation (2)can be rewritten as:

-dt on I Li(3)

where 7V; is no longer an integer term.

In order to mitigate the ionospheric effect,which is the largest error source in GPS positioningafter SA was turned off, ionosphere-freecombinations are usually formed in dual-frequencyreceivers and have the following expressions:

= [/i2 • P{L\)- /22 • P(L2)]/[f,2 - f2

2]

dm,,l/P{Ll+L2)+e(P(Ll

f

(4)

= p + c{dt-dT) + dorh +dlwp +f1 - f-

(5)

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Applying corrections from precise orbit andclock products to Equations (4) and (5) results inthe following equations:

P1F =p-cdT)-

-s (P(Ll + L2))

= p - cdT

+ " mlt I<5>(L\+L2) '

l /P(L1+L2)

\N\-cf2N2

f2 f2

s ($(/,! + L2))

(6)

(7)

Based on Equations (6) and (7), the unknownsto be estimated include the position coordinates,receiver clock offset, troposphere and a combinedambiguity term. The above model has been used inthe observation model for PPP processing byZumberg et al (1997) and Kouba and Heroux(2001).

The integer property of the ambiguity cannotbe exploited based on Equations (6) and (7). Gaoand Shen (2001) have proposed a new codecombination to replace Equation (6) as follows:

=0.5[P(Li)

trop+Q.5X,N] (8)

imultlP(Li) + 0.5s

where / = 1 and 2.

Note that Equation (8) is still an ionosphere-free observable. A combination formed fromEquations (7) and (8) yields an alternativeobservation model for PPP. Different from themodel based on Equations (6) and (7), the newmodel is capable of estimating the ambiguitiesassociated with LI and L2 frequencies separately.This makes it possible to exploit the integerproperties of both LI and L2 ambiguities, which isessential for real-time kinematic positioning.

To facilitate high precision positiondetermination using P-RTK, a number ofunconventional error corrections have to be applied,including corrections for earth tides, satelliteantenna offsets and phase wind up. Thesecorrections are commonly ignored in conventionalRTK positioning because they can be canceled outby the carrier phase double-differencing procedurethat is implemented between satellites andreceivers.

4 Challenges in Point RTK

There are three major challenges associatedwith P-RTK: error mitigation, carrier phaseambiguity convergence, and ambiguity resolution.

The ionosphere is a conventional error thatcauses the largest absolute error in GPSobservations. Due to its dispersive nature, signalsfrom satellites suffer different delays on LI and L2.As such, they can be mitigated through standardionosphere-free code and carrier phasecombinations. The troposphere cannot be mitigatedin this manner due to its non-dispersive nature.However, it can be modeled or estimated along withother parameters.

Unconventional error sources must also betaken into consideration in order to implement a P-RTK system. Many of these errors have beenignored in the past because they are irrelevant,negligible, or are canceled out through differencing.However, in the case of un-differenced code andcarrier phase observations some of these errors donot cancel out and their sizes are relatively large,influencing the accuracy of the method.

These unconventional errors may be related tothe un-differenced observations, the precise data orthe standard GPS errors. Satellite antenna phasecenter, earth tide and ocean loading are examples oferrors related to precise data. The satellite antennaphase center correction is necessary for Block II/IIAsatellites because the phase centers and centers ofmass of these satellites do not coincide. Earth tideand ocean loading models are necessary becauseerrors associated with them can reach severaldecimeters. Similarly, a satellite phase windupcorrection is necessary since the error can reach halfa cycle.

In most cases, carrier phase ambiguities areconsidered as float terms, often requiring longconvergence times ranging from several tens ofminutes to several hours. Since convergence is acrucial issue for real-time applications, longconvergence times may prevent P-RTK fromfulfilling the operational requirements of suchapplications. As such, fast ambiguity convergencemethods and algorithms should be developed.

Integer ambiguities must be resolved to fullyrealize the accuracy of carrier phase observations. Ifinteger values of the carrier phase ambiguities canbe determined on-the-fly (OTF) over short timeintervals, the P-RTK method will be able to supportreal-time kinematic positioning at centimeter levelaccuracy. Currently, this accuracy level is onlyfeasible using double-difference RTK systems.

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New ionospheric-free observation combinationmodels are necessary because the conventionalmodel, which forms ionospheric-free code andcarrier phase combinations, does not allow for theexploitation of the integer property of theambiguities.

Estimation of integer ambiguities is difficultdue to the existence of non-zero initial phase offsetsassociated with the satellite and receiver oscillatorsand the level of residual errors that may exist in thesystem due to precise data and noise. Non-zeroinitial phase offsets are constant biases associatedwith the time and geometry of receiver-satellitelock, and are constant during an observed cycle slipfree satellite arc (Gabor, 2000; Teunissen, 1997).

5 Numerical Results and Analysis

A software package called P3 has beendeveloped at the University of Calgary for precisepoint positioning. The software can be used toassess the performance of different data processingmodels as well as the influence of different errorsources on positioning results.

Processing can be done in post-mission or inreal-time, and the program can be run in eitherstatic or kinematic mode. Two point positioningmodes are available: Single Point Positioning(SPP), which only makes use of codemeasurements, and Precise Point Positioning (PPP),which makes use of both code and phasemeasurements, as well as precise satellite orbit andclock corrections.

Processing in real-time requires real-timecorrections. Currently, the developers of thesoftware can obtain GPS-C data, which are real-time GPS wide-area corrections available fromNatural Resources Canada (NRCan). GPS-Cprovides enhanced real-time positioning byapplying corrections to user GPS receiverobservations (GPS-C ICD, 2001).

5.1 Static Results

2 are static dataprecise orbit and

Shown in Figures 1 andprocessing results using finalclock corrections from IGS, with accuracies of 5cmand 0.1ns, respectively. A one-day's data setcollected on February 15, 2003, at the IGS stationALGO was used with a sampling interval of 30s.Precise satellite coordinates were available every15min, and precise clock corrections every 5min.Due to the discrepancy in the sampling intervalbetween the GPS observations and the precise data,

the precise orbit and clock corrections wereinterpolated to 30s, equivalent to the samplinginterval of the GPS observations. It should be notedthat this may degrade the accuracy of the satelliteorbit, and in particular the satellite clockcorrections.

Positioning accuracy statistics are given inTable 1. The results indicate that the positionsolution converges to 2cm in 30min with a biasbelow lcm.

A—

. o jvaj

-1

1

518400 540000 561600 583200 604800

Fig. 1 Position Error Using IGS Final Products

583200 604800

Fig. 2 Clock, Troposphere, PDOP and Number ofSatellites Using IGS Final Products

Table 1. Static Positioning Accuracy Statistics Using IGSFinal Products

Mean (m)RMS (m)STD (m)

North0.0010.0050.005

East0.0020.0050.004

Up-0.0060.0190.018

5.2 Kinematic Results

In this section, the same IGS data set wasprocessed in kinematic mode. In addition, a two-hour data set (1 Hz) was processed in kinematic

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mode using real time GPS-C corrections. GPS-Cephemeris and satellite clock corrections areaccurate at the level of 20-25cm and l-2ns,respectively (GPS-C ICD, 2001).

As mentioned above in Section 5.1, precisedata had to be interpolated to 30s to be consistentwith the sampling interval of the GPS observations.As such, two solutions using IGS final products arepresented. Results of the first solution, using clockcorrections interpolated at 30s, are shown in Figures3 and 4, with statistics in Table 2. Results of thesecond solution, using the original 5min clockcorrections, are shown in Figures 5 and 6, withstatistics in Table 3.

^ 0

-0.5

0.5

I 0

5T8400 540000 561600 583200 604800

Fig. 5 Position Error Using Original IGS Final Products

Fig. 3 Position Error Using IGS Final ProductsInterpolated at 30s

g -800

-850

2.4

X2.3

Fig. 4 Clock, Troposphere, PDOP and Number ofSatellites Using IGS Final Products Interpolated at30s

Table 2. Kinematic Positioning Accuracy Statistics UsingIGS Final Products Interpolated at 30s

Mean (m)RMS (m)STD (m)

North0.0140.0540.052

East0.0010.0230.023

Up-0.0210.0980.096

-750

-850

2.4

518400 540000 561600 583200 604800

Fig. 6 Clock, Troposphere, PDOP and Number ofSatellites Using Original IGS Final Products

Table 3. Kinematic Positioning Accuracy Statistics UsingOriginal IGS Final Products

Mean (m)RMS (m)STD (m)

North0.0040.0160.016

East0.0000.0090.009

Up-0.0060.0360.035

Comparing Figure 3 and Table 2 with Figure 5and Table 3 shows the degradation of results afteran interpolation of clock coiTections from 5min to30s. Without interpolation the results converge veryquickly, as shown in Figure 5. After convergence,the bias is less than lcm, and the errors are below5cm. The results reflect the accuracy of IGS finalproducts. When using clock corrections interpolatedto 30s, convergence occurs after about 15min, withabout 5cm horizontal accuracy and 10cm verticalaccuracy. Along with station coordinate parameters,estimation of the troposphere and receiver clockoffsets are presented in Figures 4 and 6. These twoproducts have a demand from meteorological andtime transfer sectors.

Figures 7 and 8 show kinematic results usingGPS"C real-time corrections for a two-hour data

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set. A summary of the results is shown in Table 4.10cm accuracy in the horizontal direction and 30cmaccuracy in vertical direction is achievable with aconvergence time of 30min. The results of IGS dataand GPS"C data suggest that convergence dependson the accuracy of the precise data.

The results obtained with GPS"C correctionsmight be improved since the system is still in a testphase. With GPS'C real-time clock corrections at atwo-second interval, no accuracy degradationassociated with interpolation will occur. Moreover,GPS'C corrections can be widely used in real-timeapplications since they can be provided to users inCanada as well as to global users in real-time.

302-

Fig. 7 Position Error Using GPS'C Real-TimeCorrections

304200 306000 307800 309600

Fig. 8 Clock, Troposphere, PDOP and Number ofSatellites Using GPS-C Real-Time Corrections

Table 4. Kinematic Positioning Accuracy Statistics UsingGPS-C Real-Time Corrections

Mean (m)RMS (m)STD (m)

North0.0400.0620.048

East0.0020.0630.063

Up-0.0510.2800.275

6 Conclusions

The method of Point Real-Time KinematicPositioning (P-RTK) has been described in thispaper. Compared to the conventional RTK method,P-RTK can offer much greater operationalflexibility to help in the full implementation of RTKtechnology in many applications in the future.

Test results have indicated that P-RTKpositioning at centimeter to decimeter levelaccuracy is achievable. Using IGS final products, anaccuracy of several centimeters has been achieved.Using real-time GPS'C data, an accuracy below10cm is achievable in the horizontal components,and below 30cm in the vertical direction. Althoughthe results are promising, several challenges stillexist with respect to error mitigation, carrier phaseambiguity convergence, and ambiguity resolution.

Acknowledgement

Mr. Pierre Heroux from Natural Resources Canadais acknowledged for many valuable discussions.This research is partially supported by GEOIDE.

ReferencesGao, Y. and X. Shen (2002). "A New Method Of

Carrier Phase Based Precise Point Positioning",Navigation: Journal of the Institute of Navigation,Vol. 49, No. 2.

IGS, (2000). The IGS-2000 Annual Report.

Kouba, J and P. Heroux (2000). "GPS Precise PointPositioning Using IGS Orbit Products", GPSSolutions, Vol.5, No.2, Fall.

Michael J. Gabor (2000) "Characteristics OfSatellite-Satellite Single Difference WidelaneFractional Carrier Phase Biases". Proceedings ofION GPS 2000, Salt Lake City, UT, 19-22September.

Teunissen, P.J.G. and A. Kleusberg (Eds.) (1998).GPS for Geodesy. Second Edition, Springer-Verlag, New York.

Zumberge James. F., M. M. Watkins and F. H.Webb (1997). "Characteristics And ApplicationsOf Precise GPS Clock Solutions Every 30Seconds". Navigation, Journal of the Institute ofNavigation. 44(4), 449-456 winters 1997-1998.

GPS*C ICD, 2001, Natural Resources Canada.URL: www.cdgps.com/e-test/cdgps documents/%5B14%5D%20-%20ICD%20-%20GPSC%20-%20200 l-02-06.pdf

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