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The calibration of GPS antennasis of the utmost importance inGPS baseline determination, es-pecially when millimeter pre-

cision is required for applications such as themonitoring of engineering structures or forGPS attitude determination. For these ap-plications, even correcting for the mean phasecenter is not enough. To fully meet the pre-cision requirements of these applications, thephase-center variation must also be takeninto account.

In relative GPS positioning, both anten-nas are set up and centered on the two endsof the baseline that is to be measured. Thegeometric center of each antenna is used todetermine the offset in relation to the geo-detic point above which the antenna is in-stalled. However, a GPS receiver determinesthe coordinates of the antenna’s electricalphase center. The phase center is definedas being the point where the satellite signalis collected. The offset between the meanphase center and the geometric center ofan antenna can range from a few millimetersto several centimeters.

The observation error due to the offsetbetween the instantaneous and the geomet-ric phase centers is represented by the amountof two projections on the antenna-satellitevector (see Figure 1): (1) the mean phase-cen-ter error (eFm), and (2) the phase-center vari-ation (eFr) that is the difference between themean phase center and the instantaneousphase center.

This article presents two methods for thecalibration of GPS antenna phase centers.The first one is carried out in the field in arelative mode using an antenna-support cal-ibration beam. The second method relies onobservations made in an anechoic chamber.A comparison of both methods in relativemode was performed with regard to the com-ponents of the mean phase center and thephase-center variations. The results showthat the differences between the two meth-

ods do not exceed 2 mil-limeters, even thoughthey use completely dif-ferent approaches.

Calibration BeamThe first GPS antennacalibration method in-vestigated in our studyhas been carried out inrelative mode using acalibration beam onwhich a pair of antennasis mounted. That is tosay that a slave antennais calibrated with ref-erence to another oneconsidered as the refer-ence (master) antenna.

The calibration beam was built by themetrology-geodesy laboratory of UniversitéLaval. It is made of aluminum, with an ap-proximate length of one meter, and it has twoantenna mounts with forced-centering bolts.The distance between the centers of the two

mounts was accurately measured in the lab-oratory using an interferometer. A theodo-lite is permanently installed at the center ofthe beam in order to define its orientationwith a back-sight toward another geodeticpoint. The sighting axis of the theodolite isperpendicular to the axis formed by the

ANTENNA CALIBRATIONINNOVATION ���

Calibrating Antenna Phase CentersA Tale of Two MethodsBoussaad Akrour, Rock Santerre, and Alain Geiger

OFTEN OVERLOOKED,THE ANTENNA IS A VITAL COMPONENT OF ANY GPS RECEIVER. WITHOUT IT, Areceiver simply would not work.The job of the antenna is to convert the miniscule energy in electromag-netic waves received from GPS satellites into an electrical current that can be processed by the receiv-er.The receiver then determines the coordinates of the antenna – or, more precisely, it determines thecoordinates of the electrical phase center of the antenna.Typically, the phase center is near the physi-cal center of the antenna and for many low-accuracy GPS applications, the exact location of the phasecenter is immaterial. However, for demanding applications such as monitoring the deformation of theEarth’s crust, assessing the stability of bridges and buildings, for machine control, and for attitudedetermination, the relationship between the electrical phase center and the physical structure of theantenna is crucial – it should be known to the millimeter level.But try as they might, antenna designers have yet to create an antenna whose phase center is absolutelystable with respect to the physical structure of the antenna – there is always some movement of thephase center as the elevation angle and azimuth of an arriving electromagnetic wave changes. Themovement can amount to millimeters or more. Such variation will contribute an error to positions com-puted using low-noise carrier-phase measurements. However, with the appropriate set-up, it is possibleto measure the phase-center variation and calibrate an antenna.The mean phase center and its varia-tion can then be used in software used to process collected carrier-phase data.In this month’s column, we investigate two techniques for calibrating GPS antennas and examine howwell they agree. — R.B.L.

¶ FIGURE 1 Geometry of GPS antenna phase center

forced-centering bolts of both antennamounts. To ensure the beam is level in bothlongitudinal and transversal directions, twotubular bubble levels are perpendicularlymounted on the beam.

The main advantage of the calibrationbeam over a baseline formed by two arbitrarygeodetic reference points is conveniencewhile maintaining a high accuracy in deter-mining the beam azimuth.

The calibration of both antennas, in rela-tive mode, is carried out by using two geo-detic points, as follows. The calibration beamwas installed on a pillar (PEPS) on the Uni-

versité Laval campus and oriented perpen-dicularly to the direction to another pillar(NORD) by means of the beam’s theodolite.Figure 2 illustrates the calibration beam in-stallation on the PEPS pillar.

The length of the calibration beam (LB)is known from laboratory measurements andthe beam is leveled and oriented in the fieldso that the three-dimensional baseline com-ponents DNB, DEB, and DhB, are known.

Temperature Compensation. Thelength of the beam was measured in the lab-oratory at a temperature of 20°C. During anantenna calibration session, the beam lengthgets shorter or longer depending on the am-bient temperature. If the ambient tempera-ture exceeds 20°C, the nominal length of thebeam is longer than when calibrated in thelaboratory. When the temperature is below20°C, the nominal length is shorter. Know-ing the thermal expansion coefficient of alu-minum (23 parts per million per °C), thebeam length variation and its correctedlength are given by the following equations:

DLB = 23 3 10-6 3 LB 3 Dt ( 1 )LBc = LB + DLB ( 2 )whereDLB is the beam-length variation

resulting from aluminum thermal expansion (meters);

Dt is the difference between the field and calibration temperatures —field minus laboratory (°C);

LB is the beam length during labora-tory calibration (meters);

LBc is the beam length corrected for thermal expansion (meters).

For example, for an outside operatingtemperature of -20°C, the beam lengthwould be shorter than that in the laboratoryby almost 1 millimeter.

Antenna Tests. We have tested our rel-ative calibration technique on two antennas.The first one was a conventional geodeticL1/L2 dual-frequency antenna. We use oneof these antennas as the reference antennain all of our investigations. The second an-tenna was a choke-ring type. We calibratedone of the geodetic antennas and a choke-ring antenna twice; once in 1995 and againin 2000 using the calibration beam. Theywere also calibrated in an anechoic chamberin 1999. These calibration tests were part ofa wider research effort related to monitor-ing of engineering structures with single-frequency receivers. Accordingly, the anten-nas were calibrated only at the L1 frequency.

We carried out two observation sessionsof 24 hours each for both of the antennas.These sessions were performed during twoconsecutive days but delayed by 4 minutesper day, since the satellite sky distribution isrepeated with a period close to a sidereal day(23 hours, 56 minutes). During the first ob-servation day, both antennas were orientedso that a physical mark on the antenna pointedto magnetic north. The second day, they wereturned to magnetic south. This scenario cov-ers all parts of the superior hemisphere of theantennas. Indeed, for mid-latitude sites, theinclination of the GPS satellite orbits causesa “shadow” area in the sky where it is not pos-sible to make GPS observations.

The sampling interval was 1 minute, andno mask angle was used. Yet, the post-pro-cessing was carried out with an imposed el-evation mask angle of 15º. Two geodetic-classreceivers were used for the calibrations and,to compensate for the beam-length variations,temperature measurements were taken auto-matically with an electronic thermometer.

We processed the data collected by thereceiver in between-receiver, single-difference mode using the DETECSAT2software developed at the Centre for Re-search in Geomatics at Université Laval.This processing approach allows us to calculate the relative mean phase-centercomponents of the antennas. These compo-nents are obtained by averaging the resultsof the two GPS observation sessions (an-tenna orientations to magnetic north andsouth). The components of the relative meanphase center (DN, DE, Dh) in the local ge-odetic reference frame are given by these

¶ Calibration beam set up on a pillar

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���INNOVATION ANTENNA CALIBRATION

¶ FIGURE 2 Calibrationbeam installation

What Is Antenna Phase Center?The electrical phase center of an antennais the apparent source of electromagneticradiation when the antenna is used fortransmitting. If the source is an idealpoint source, the phase center is the cen-ter of the radiating spherical wavefronts(of equal phase). According to the reci-procity theorem, the spatial response ofan antenna is exactly equivalent to itsradiation pattern. So, when the antennais used for receiving, the phase center isthe effective collecting point of the radia-tion – the point at which electromagneticwave energy is transferred to the anten-na. For a GPS receiving antenna, thephase center is the point to which thereceiver’s phase measurements actuallyrefer. Since a real antenna is not an idealpoint source, its equiphase contours willnot be perfectly spherical, and hence thecenter of curvature may vary with theazimuth and elevation angle of an arriv-ing signal.—R.B.L.

www.gpsworld.com FEBRUARY 2005 GPS World 51

equations:DN = DNB - DNGPS ( 3 )DE = DEB - DEGPS ( 4 )Dh = DhB - DhGPS ( 5 )The components DNB, DEB, DhB and

DNGPS, DEGPS, DhGPS are the known north,east, and vertical components of the calibra-tion-beam baseline, and those determinedby the GPS observations, respectively. Asthe length of the calibration beam is veryshort (about one meter), the relative errorsin the GPS results due to tropospheric andionospheric refraction and errors in the satel-lite ephemerides are completely eliminatedin the single-difference processing.

For each GPS observation epoch and foreach satellite, the single-difference residu-als are also calculated. These residuals con-tain information about the relative phase-center variation with respect to the relativemean phase-center values. The GPS obser-vation residuals also absorb the observationnoise and any multipath errors. The calibra-tion site was selected in order to minimizemultipath effects.

Anechoic Chamber CalibrationThe second GPS antenna calibrationmethod presented in this article is basedon the measurements taken in an anechoicchamber using a GPS signal simulator. Ananechoic chamber is an enclosure rangingin size from a few meters to tens of meterson a side used for the testing of antennas andother radio-frequency (RF) devices. The in-terior walls of the chamber are covered withRF-absorbing material that reduces signalreflections or “echoes” to a minimum. Thismethod is performed in absolute mode; that

is, each antenna is calibrated separately. Thecalibration of our antennas was carried outin the anechoic chamber of the Departmentof Electrical Engineering, University of NewBrunswick, in August 1999.

During this calibration, the antenna un-dergoes two rotations (variations of the “azimuth” and the elevation angle) per stepof 5° each. At each antenna position, the “azimuth,” the elevation angle, the gain ofthe antenna and the signal phase are meas-ured and recorded by a computer, whichmanages the whole calibration process. Foreach antenna, a total of 1387 observationsare recorded.

The observed phase measurements arereported in a coordinate reference framecentered at the antenna rotation point dur-ing calibration. X and Y axes are directed re-spectively towards the “north” and “east”marks on the antenna. The Z axis is perpen-dicular to the antenna ground plane. Thecoordinates (xi, yi, zi) of a point defined bythe phase measurement (converted into mil-limeters) are given by these equations:

xi = ri cosEi cosai = ri sinzi cosai ( 6 )yi = ri cosEi sinai = ri sinzi sinai ( 7 )zi = ri sinEi = ri coszi ( 8 )whereEi is the elevation angle (measured

during calibration);zi is the zenith angle (90° - Ei);ai is the “azimuth” of the phase meas-

urement counted from the antenna Nmark (measured during calibration);

ri is the phase measurement converted into millimeters.

The problem amounts to a sphere reso-lution, which means finding the radius andthe center coordinates from all the points (xi,yi, zi). The sphere is named “best-fit sphere.’’The general equation of a sphere in Carte-sian coordinates is given by:

(9)The parameters x0, y0, z0, R0 represent the

mean phase-center coordinates and the ra-dius of the best-fit sphere. These parame-ters and their respective precisions are de-termined by a least-squares adjustment. Theobservation residuals obtained from thisprocess contain information about the phase-center variations.

Phase-Center VariationsFor the beam calibration, since the observa-tion (satellite) sky distribution is non-ho-mogenous, the phase-center variations’ de-pendence on azimuth and elevation angle isdescribed with a spherical harmonics devel-opment using the following formulation:

(10) and (11)whereeFr is the phase-center variation

(least-squares residual) in a given direction (input for spherical harmonics modeling);

a and z are the azimuth and zenith angle of the residual;

pn,m is the standardized Legendre polynomial of the first kind;

yn,m and y*n,m are the standardized spherical harmonics;

an,m and bn,m are the standardized spherical harmonic coefficients (pa-rameters to be determined);

n and m are the spherical harmonic de-gree and order.

The an,m and bn,m coefficients are then esti-mated from the “observables” eFr(a,z) withequations (10) and (11). These coefficients

Phase-center components

(mm) DE DN Dh

Beam(September -1.5 0.5 29.6

1995)

Beam(January -0.9 1.0 30.4

2000)

AnechoicChamber(Aug 1999) -3.1 0.4 30.9

TABLE 1 Comparison of the L1 relative mean phase-center components (conventional geodetic and choke-ring antennas) for an elevation mask angle of 15° Calibration

method Beam Anechoic chamber

Semi-axis a b c a b c

Length (mm) 1.3 0.9 2.5 0.6 0.6 1.2

Azimuth (°) 175 85 178 0 90 180

Elevation (°) 0 0 90 0 0 90

TABLE 2 Size and shape of the confidence ellipsoids (with a probability level of 95%) of the relative mean phase centers resulting from the two calibration methods (for an elevation mask angle of 15°)

allow us subsequently to take into accountthe phase-center variations, with respect tothe mean phase center, as required for pre-cise GPS applications.

Comparing The Two MethodsComparison of calibration results via the

calibration beam method and the anechoicchamber method was performed in relativemode. It was carried out on components ofthe relative mean phase center as well as onrelative phase-center variations.

Mean Phase Center. The calibrationof the summer 1995 and winter 2000 ses-

sions on the beam show a high repeatabil-ity of the results, even though the observedweather conditions differed significantly (re-spectively 27ºC and -17ºC). The respectivecomponents resulting from the calibra-tion method on the beam and in the ane-choic chamber agree within 2 millimeterseven though both methods use completelydifferent approaches (see Table 1). Table 2shows the precision of calibration results ofthe relative mean phase center determina-tion of the two calibration methods.

Phase-Center Variations. The com-parison of the phase-center variations re-sulting from the two calibration methods isperformed by the relative residual patterns.These patterns depend on the direction (az-imuth and elevation angle) of the observa-tions carried out during the calibration.

Figure 3 illustrates the curves of the iso-values (in millimeters) of the relative phasecenter variations of the geodetic and choke-ring antennas from the adjustment of anechoic chamber observations for an ele-vation mask angle of 15°.

To compare with the pattern of thephase-center residuals resulting from theobservations in the anechoic chamber, thebeam calibration residuals are initially modeled with a development in sphericalharmonics, as described earlier. Then, a reg-ular grid is generated with the determinedmodel, every 5° in azimuth and elevationangle (same grid as for the anechoic cham-ber results).

Figure 4 shows the curves of the isoval-ues of the relative phase-center variationsfor the two antennas, from the beam cali-bration (September 1995) for an elevationmask angle of 15°. The curves are plottedfrom the relative residuals model developedwith the spherical harmonics of degree 6and order 4 (a total of 43 coefficients).

A comparison of the curves of Figures 3and 4 reveals a general similarity of the twopatterns. The figures show two hollows cen-tered at the azimuths -120° and 120° and15° elevation angle. They present also a peakof 4 millimeters centered at the azimuth -20° and 40° elevation angle. Figure 5 showsthat the difference between both sets ofcurves is less than two millimeters in all di-rections.

Synergy. The calibration of GPS an-

¶ FIGURE 4 L1 phase-center variations (in millimeters) with respect to the relative mean phase cen-ter of conventional geodetic and choke-ring antennas resulting from the beam calibration (September1995) and modeled with spherical harmonics (n=6, m=4) for an elevation mask angle of 15°

¶ FIGURE 3 L1 phase-center variations (in millimeters) with respect to the relative mean phase cen-ter of conventional geodetic and choke-ring antennas resulting from the anechoic chamber calibrationfor an elevation mask angle of 15°

¶ FIGURE 5 Difference (in millimeters) of the relative L1 phase-center variations resulting from thebeam calibration and anechoic chamber techniques for an elevation mask angle of 15°

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���INNOVATION ANTENNA CALIBRATION

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tenna phase-center variation in an anechoicchamber is expensive, not widely available,and must be performed by an electrical en-gineering specialist. While this techniqueprovides absolute calibration of GPS an-tenna phase-center variation, the relativecalibration procedure using a calibrationbeam, as presented here, is much less ex-pensive and can be performed by GPS usersthemselves. With this method, informationabout phase-center variation is relative to amaster GPS antenna. However, if the mas-ter antenna has been calibrated in absolutemode in an anechoic chamber, the absolutephase-center calibration of the second (slave)GPS antenna can then be inferred.

ConclusionsWe have compared two methods for the calibration of relative GPS antenna phasecenter: on a calibration beam and in an anechoic chamber. The relative mean phasecenters and the phase-center variations havebeen evaluated with the two calibration

methods for a pair of geodetic antennas. Theresults show that the differences between thetwo methods do not exceed 2 millimeters,though they use completely different ap-proaches.

The plots of the relative antenna phasecenter variations determined by the twomethods of calibration also show a great sim-ilarity. The values of the relative phase cen-ters varying between -7 millimeters and 5millimeters. The magnitutde of the variations shows the need to take them intoaccount when correcting GPS phase meas-urements in high-precision works such asthe monitoring of engineering structuresand for GPS attitude determination.

AcknowledgmentsFinancial support for this research came fromthe Natural Sciences and Engineering Re-search Council of Canada, the CanadianNetwork of Centres of Excellence in Geo-matics (GEOIDE, Project ENV#14), theFaculty of Forestry and Geomatics and the

Centre for Research in Geomatics at Uni-versité Laval. The authors gratefully ac-knowledge Professor Colpitts’ team in theDepartment of Electrical Engineering at theUniversity of New Brunswick for the GPSantenna calibrations in their anechoic cham-ber. This article is based on the paper “Com-paraison de méthodes de calibrage du cen-tre de phase d’antennes GPS” whichappeared in the journal Geomatica, Vol. 57,No. 4, 2003, pp. 411-417. We thank the ed-itor for his kind permission to publish anEnglish-language version of this paper. c

ManufacturersThe conventional geodetic antenna was anAshtech (now Thales Navigation, SantaClara, California) 700228A; the choke-ringantenna was an Ashtech 700936C, manufac-tured by Dorne and Margolin Company(now Edo Corporation, New York, NewYork ). Two Ashtech Z-XII geodetic-class receivers were used for the calibrations.

BOUSSAAD AKROUR holds an engineer diploma in geodet-ic sciences from the National Center of Space Techniquesof Algeria (1990), as well as M.Sc. (1998) and Ph.D.(2002) degrees in geomatics sciences from UniversitéLaval in Quebec City, Canada. He has twelve years’ experi-ence in the fields of monitoring engineering structuresand GPS. He is currently working in GPS applications andresearch at the Canadian Coast Guard in Quebec City.

ROCK SANTERRE is a professor of geodesy and GPS inthe Department of Geomatics Sciences, and a member ofthe Centre for Research in Geomatics at Université Laval.He holds a Ph.D. in geodesy from the University of NewBrunswick. He has twenty years’ experience in GPS posi-tioning, and holds two patents related to GPS equipment.

ALAIN GEIGER is a professor at the Swiss FederalInstitute of Technology (ETH) in Zürich. He lectures onsatellite geodesy and navigation in ETH’s Institute ofGeodesy and Photogrammetry. He is a member of theSwiss Geodetic Commission and the InternationalAssociation of Geodesy, president of the Swiss Institute ofNavigation, and professeur associé at Université Laval.

The “Innovation”column featuresdiscussions about advances inGPS technology and its applica-tions as well as the fundamentalsof GPS positioning.The column iscoordinated by RICHARDLANGLEY of the Department of

Geodesy and Geomatics Engineering at the University ofNew Brunswick, who appreciates your comments andtopic suggestions.To contact him, see the “Columnists”section on page 2 of this issue.

Further ReadingPrevious GPS World articles on antennasand antenna calibration:

“A Primer on GPS Antennas” by R.B.Langley, GPS World, July 1998, pp. 50-54.“Characterizing the Behavior of GeodeticGPS Antennas” by B.R. Schupler and T.A.Clark, GPS World, February 2001, pp. 48-55.“Calibrating Antenna Phase Centers : TheBlock IIA Satellite” by G.L. Mader and F.M.Czopek, GPS World, May 2002, pp. 40-46.

Further information on the spherical harmon-ics approach in characterizing antennaphase-center variations:

“Determination of Antenna Phase CenterVariations Using GPS Data” by M.Rothacher, S. Schaer, L. Mervart, and G.Beutler in Proceedings of the IGSWorkshop, Special Topics and NewDirections, Potsdam, Germany, May 15-18, 1995, pp. 205-220.

“A New Approach for Field Calibration ofAbsolute Antenna Phase CenterVariations” by G. Wübbena, F. Menge, M.Schmitz, G. Seeber, and C. Völksen inProceedings of ION GPS-96, the 9thInternational Technical Meeting of theSatellite Division of The Institute of

Navigation, Kansas City, Missouri,September 17-20, 1996, pp. 1205-1214.

Information on the effects of antenna phase-center variations on GPS measurements:“Influence of Phase Center Variations onthe Combination of Different AntennaTypes” by A. Geiger in Proceedings ofGPS’90, the Second InternationalSymposium on Precise Positioning withthe Global Positioning System, Ottawa,Canada, September 3-7, 1990, pp. 466-476.

“Étude des effets de la variation des cen-tres de phase des antennes GPS” by M.Bourassa, Master’s thesis, Départementdes sciences géomatiques, UniversitéLaval, Québec, Canada, 105 pp.

Further information on the anechoic cham-ber calibration procedure:

“GPS Antenna Design Characteristicsfor High Precision Applications” by J.M.Tranquilla and B.G. Colpitts, presented atthe American Society of Civil EngineersSpecialty Conference GPS 88,Engineering Applications of GPS SatelliteSurveying Technology, Nashville,Tennessee, May 11-14, 1988 and pub-lished in Journal of SurveyingEngineering, Vol. 115, No. 1, February1989, pp. 2-14.