AN INTRODUCTION

SYSTEM AND

SOME

TO THE GLOBAL

GEOLOGICAL

POSITIONING

APPLICATIONS

T. H. Dixon

JetPropulsion Laboratory Divisionof EarthandPlanetarySciences Pasadena,California

Abstract.Receiversequippedto measuredualfrequency carrierphasesignalsfrom satellites of theGlobalPositioning System(GPS)havebeencapable,underspecialconditions,of determiningrelative horizontalpositionsamong stationsseparated by oneto a few hundredkilometerswith a precisionof one to severalmillimeterssincethe early 1980s. The major obstaclesto making this capability routine,extendingit to all partsof theglobe,andextending it to longerstationseparations, havebeenequipmentcost,

data acquisitionand analysisare havinga significantimpact on studies of near-fault crustal deformation and earthquakeprocesses,until recently the province of conventional terrestrialgeodetictechniques.The enhanced satelliteconstellation, improvedmodels,and establishment of global tracking networkshave extendedseveral millimetershorizontalpositioningcapabilityto stationseparationsof 1000km or morein virtuallyall partsof the world. Thisenablesstudyof new classesof tectonicproblemsthat limitations in the GPS satellite constellation,arduousdata previouslywere difficult to attackwith any geodetictechanalysis,uncertainties in satelliteorbits,uncertainties in nique.Examplesincludea completekinematicdescription propagationdelaysassociatedwith variabletropospheric of ongoingcrustaldeformationin broad, complexcontiwatervapor,and difficultiesin resolvingcarrierphasecy- nentalplate boundaryzones,and measurementof relative cle ambiguities. Recentimprovements haveoccurredin all plate motionat convergentboundarieswhere global modtheseareas.The increasingeaseandreducedcostof GPS elsmaybe poorlyconstrained.

INTRODUCTION

extendingthe range and accuracyof GPS measurements are emphasized. I haveaimed for broadcoverageof most With the adventin the early 1980sof a satellite-based of therelevanttopicsandan intuitiveratherthancomplete navigationsystemknownastheGlobalPositioning System or rigorous treatment, giving more derailed references (GPS) operatedby the U.S. Departmentof Defense it whereappropriate. becamepossiblefor a user with the properreceiverto obtain almost instantaneousthree-dimensionalposition informationaccurateto severalmeters.With the comple- A BRIEF HISTORY OF THE GPS PROGRAM tion of the satelliteconstellationin the early 1990s this capabilitywill be extendedto virtually all parts of the The spacesegmentof GPS is a constellationof satellites globe.This is a remarkableachievement and buildson a in highEarthorbitequippedwithpowerfulradiofrequency number of technologicaladvancesin the last several transmitters andhighlystableatomicclocks.Easton[1978] decades. Even more remarkable is the fact that. with careful reviewsthe majordevelopments leadingto thiscapability. attention toexperiment configuration anddat•analysis itis In 1967 an early prototypeof a GPS satelliteknown as possibleto obtain relative position data 3 orders of Timation 1 was launchedinto low Earth orbit (-900 km magnitudemore precise than the design level of the altitude) as part of a military test program in satellite system.This enhancedperformanceallows for measure- navigation. Weighingabout40 kg and consuming only 6 ment of crustalstrainand fault motionratesin just a few W of power,it carrieda UHF transmitterslavedto a stable )rears.

quartz clock, witha frequency driftofseveral parts in 10•

This paper reviews fundamentalprinciplesof GPS, discussessome geologicaland geophysicalapplications and their accuracyrequirements,and considersimplications for GPS experimentdesign.Recent developments

per day. Additional proof-of-conceptsatellitesfollowed,

culminating10 yearslater in the NavigationTechnology Satellite (NTS) 2, very similar to subsequentGPS satellites.NTS-2 was launched into a 20,300-km-altitude

Copyright1991 by theAmericanGeophysical Union.

Reviews of Geophysics, 29, 2 / May 1991 pages249-276

8755-1209/91/91RG-00152

$05.00

Papernumber91RG00152 ß 249

ß

250 ß Dixon: THE GLOBAL POSITIONING SYSTEM

29, 2/REVIEWS OF GEOPHYSICS

orbit, weighed440 kg, consumed400 W of power, and transmittedtwo L band (-1.2 and 1.5 GHz) timing and ranging signalsbased on a sophisticatedcesium clock,

oneanotherin theskyprovidecorrelated (redundant) range information,an effect known as geometricdilution of precision(GDOP). If observationsare limited to four witha frequency driftof lessthantwopartsin 10•3per satellitesby receiverdesign,thesegeometriceffectscanbe day. One year after NTS-2, the first "Block 1" GPS satellitewas launched,part of the operationaltestphaseof the GPS program. By 1990 the Block 1 constellation includedsix functioningsatelliteslaunchedbetween1978

minimized by choosingsatelliteswhich maximize the volumeof a tetrahedron, definedby the pointsof intersection on a unit spherecenteredon the user, of vectors betweenthe satelliteandthegroundreceiver. and 1985. It is alsopossible to obtaindistanceinformation (strictly It hasbeenrecognizedfor sometimethathigh-precision speaking,the changein distance)throughphasemeasuregeodetic measurementscould be made by exploiting mentson the carrier signal itself, keepingtrack of the signalsfrom artificial satellites[e.g.,Prestonet al., 1972; numberof cycles after signal acquisition.Assuming MacDoran, 1979; Counselmanand Shapiro, 1979]. The perfectclocks,andignoringpropagation effects, Block 1 constellationhasprovensatisfactoryfor developing andrefiningexperimentdesignandanalyticalconcepts and for initiating a number of high-precisiongeodetic = (v,/œ) (n+ 0)) (2) monitoringprograms.The first Block 2 satellite,with a numberof improvementsrelative to its forebears,was wheren is the numberof integercarrierwavelengths at launchedin February1989.As of thiswriting,a totalof 10 signalacquisition(initially unknown),t• is the phasein Block 2 satellitesare in operation.A total of 21 Block 2 )• isthewavelength, f isthefrequency, andv, isthe satellitesplus threesparesare plannedto be in operation cycles, phase velocity (the importance of distinguishing v, and by the end of 1992. They will orbit at an altitudeof about velocity, vg,will become apparent). Sincethe 20,000 km in six orbital planes with 12-hour periods, group wavelengthof the carrieris considerably shorterthanthat enablingsimultaneous observationof four or more GPS of the lower frequencycode modulations(Table 1), the satellitesin virtuallyall partsof theglobe.

resulting"distance" measurement, thoughambiguous by the initial numberof wavelengths,is considerably more

POINT

POSITIONING

WITH

GPS

RangeMeasurement An observeron Earth can uniquelylocatehis position by determiningthe distancebetweenhimself and three satelliteswhoseorbital positionsare alreadyknown.With GPS, distanceinformation is basedon the travel time x of a

satellite signal, obtained by measuringthe difference

between thetransmit (t•)andreceive (tr)timesat theGPS receiverof a specialrangingcode,describedin the next section.If we ignore transmissionmedia effects on the speedof light c andany timing(clock)errors,thenthetrue

,,,-ooioo tho,, o pseudorange measurementand is one of the keys to high-precision GPS measurements [Bossleret al., 1980;Counselman and Gourevitch,1981;Remondi,1985]. Carrier phaseis not measureddirectly, as this would requirevery high samplingrates.Rather,the signalis mixed ("heterodyned")with a signal generatedby the receiver's internal clock (local oscillator) and, after band-passor low-pass filtering, the resulting lower frequency"carrierbeat phase"is sampled.Most current generationreceiversaccomplishthis "downconversion"

with electronics that includeextensiveanalogcircuitry. Some newer generationreceivershave largely digital architecture, reducingproductioncost, size, and power rangep between satelliteandreceiveris justC(tr -- rs). consumption and enablingdigital samplingof the carrier Errorsin receiveror satelliteclocksarepresentin therange phasesignalwith only minimalpreprocessing [Melbourne, estimate,which for this reasonis referredto as pseudo1990].

rangeR, definedmorepreciselyas

TABLE 1. Summaryof GPS SignalCharacteristics

R = p + c(Atr - Ats+ Atp)

(1)

Carriers

Code Modulations C/A

whereAtr isthereceiver clockoffsetfrom"true"(GPSsys-

L1

L2

P

(L1 only)

tem) time (we ignore any other receiver-induced errors), (carrier) 1.57542 At•is thesatellite clockoffset,andAt isthedelayassoci- Frequency

ated withallother error sources, rn•ainly duetoatmos-

phericpropagation effects.Informationfrom a fourthsatel-

liteallows a first-order clockcorrection (Atr - Ats),andap-

or chiprate

GHz

1.2276 GHz

10.23 1.023 MHz MHz

19.0cm

24.4cm

~30m ~300m

(codemodulation)

Wavelength

proachesdiscussed below can be appliedto estimateand correct for At, enabling meter-levelpositioningunder

idealconditionS.

For analysispurposes we considerthe pseudorange or Observation geometryaffectsthequalityof theresulting phaseparameters in termsof what the receiveractually three-dimensional position.Satellitesthat appearcloseto sees(the "observable"), explicitlyaccounting for major

29, 2 / REVIEWSOF GEOPHYSICS

Dixon'THE GLOBALPOSITIONINGSYSTEMß 251

error sources.For example,in simplifiedform the phase wherem• andm2 are the angularfrequencies (m = 2•rf) observable, sometimes called integrated Doppler or associated withtheL1andL2carriers andA• andAcare accumulated deltarange,canbe written(in unitsof cycles) the relative amplitudesof the P and C/A codes. The amplitudeof the C/A signalis higherto facilitateinitial for a singlereceiver-satellite pairas signalacquisition.One effect of the PRN code modula(3) tionson the carriersis to spreadthe energyof the P code •(t) =-f'c(t) + v(t) + a signal+10.23 MHz aroundthe carriercenterfrequency; wherethedelayx(= p/c) is thegeometric delaydueonlyto becauseof this wide bandwidththe GPS signal is often thesatellite-station geometry,ignoringpropagation effects, referred to as "spread spectrum."Figure 1 illustrates v represents errors("noise")includingpropagation effects, schematicallyhow a carrier signalis biphasemodulated and a is a constant,representing a combinationof the with a PRN code.A receiverwith knowledgeof the code carrierphasecycleambiguityand an initial phaseoffset structure and an internal clock can recover an estimate of the codesequence and between satellite and receiver oscillators.Equation (3) signaltransittime by cogenerating ignorestime-dependent clockerrorsin both the satellite performinga crosscorrelationbetweenthe receivedsignal thetimedelaynecessary andreceiver;also,the variouseffectsrepresented by v and andits internalcode,determining a mustbe considered in moredetail.King et al. [1985]and to matchthe two sequences. Leick [1990] give a completederivationof the phase P-CODE CHIP 1 2 3 4 5 6 7 8 9 1011121314-.. observable equations. SEQUENCE C/A CODE VALUE

SignalStructure The followingdiscussion is abbreviatedfrom Spilker [1978]. GPS satellites transmit two L-band carrier

frequencies, each modulatedby severallower frequency signals(Table1). Thecarriers(L1 at 1.57542GHz andL2 at 1.22760 GHz) are coherentmultiplesof a 10.23-MHz atomicclock,a stableoscillatorthat providesa frequency standardon each satellite(L1 = 154 x 10.23 MHz, L2 = 120 x 10.23 MHz). The clock frequencyis actually set slightlylowerthanthisnominalfrequencyto accountfor

C/A CODE SCHEMATIC

1 1 1 1

P-CODE VALUE

1 0

shifted90ø (Figure1). TheL2 carrieris normally modu-

1

1 1

1

0 0

0 0...

1

1 0

1 0

0 1 0

0

1 1"'

0

P-CODE SCHEMATIC

1 "CHIP"=9.77x 10'8 sec -

--

54 L1 CYCLES

i

i

i

i

I

I

I

I

I

I

I

I

I

I

I

,

I

I

I

I

I

I _

relativisticeffects [McCaskill and Buisson,1985] so that

an observeron the ground"sees" 10.23 MHz almost exactly. The L1 carrier has two components.The "inphase" componentis modulatedby the precision(P) code. A lower frequencycoarse/acquisition (C/A) code is modulated in quadrature,i.e., on the samecarrier frequency

1 1

+1

MODULATED

CARRIER (C/A)

I SHh=Tr

I

I

ß = P-CODE

TRANSITION

I

= C/ACODE TRANSITION

lated only with the P code. All threecarriersalso are modulatedwith a low bit rate (50 Hz) data streamtrans-

Figure 1. Schematicillustration of biphase and quadrature

mittingsatellitehealth,ephemeris, andotherhousekeepingmodulationof the L1 carrier signal (1575.42 MHz) by the P information.These codes,P(t), C(t), and D(t), can be (10.23 MHz) and C/A (1.023 MHz) pseudorandom noise(PRN) consideredas squarewaves with valuesof +1 and are codes. Small circles on modulated carriers denote phase termedpseudorandom noise (PRN) codesbecausethey inversion points.Note90ø phaseshiftbetween P andC/A havesufficientlylongrepeattimesthattheyappearrandom modulations.In this illustrationthereis only one carriercycleper to a user without knowledge of code structure.For P codechip.In actualitythereare 154L1 cyclesperP codechip. example,theP coderepeattimeis 37 weeks.Eachsatellite Receiverdesignaffectsthe type of observablethat can is assigned a unique1-weekportionof thecode,and,since the number of active satellites will not exceed 24, each be extractedfrom the spectrumof GPS signals.There are in commonuse satellite can be uniquely identified by a PRN number currentlytwo basicreceiverarchitectures for high-precision geodesy, code correlating (as described corresponding to thecodeportiontransmitted. above) and codeless, where only the carrier phase observWe canrepresent L 1 andL2 signalsas Sl(t) =A•,t P(t)D(t) cos(tot t) + AcC(t)D(t) sin (tot t)

S2(t) = A•,2P(t)D(t) cos(0•2t)

(4)

able is available. Code-correlatingreceiverscan recover ("reconstruct") the phaseobservableas a by-productof the correlationprocess.Recoveryof carrierphasewithoutcode knowledgerequiresa nonlineardetectionschemesuchas signalsquaring.In effect,the signalis multipliedby itself,

(5)

making the original phaseinversions(equivalentto an

252 ß Dixon: THE GLOBAL POSITIONING SYSTEM

29, 2/REVIEWS OF GEOPHYSICS

amplitudechangeof +1) unity,givingthesecondharmonic positionat the time of signal transmission, and carrier of the carrier with no code modulations at double the phasecycle ambiguities. The remainderof this paperis originalfrequency,or half the wavelength.The advantage devotedto relative(as opposedto poin0 positioning and of the codelessapproachis that the high-precision part of associated conceptsthat enable high-accuracygeodetic the signal (the carrier phase) can be utilized without measurements with GPS. Relative positioninginvolves knowledgeof the classifiedP code, which may not be simultaneous observationof a group of satellitesby a availablein the future.Disadvantages includereductionof network of groundreceivers.Three-dimensionalvectors, signal-to-noise ratio, possiblyimportantunder marginal termed baselines, are def'med between all stations in the observingconditionssuchas periodsof high ionospheric network, relative to one or more fixed stations whose activity,andreducedeffectivewavelength,makingcarrier positionsare known a priori. The combinationof simulphasecycle ambiguityresolutionmore difficult. Also, P taneous networkobservations andtheanalyticaltechniques code data, which are otherwise useful for clock designed to accommodate suchdataenablesusto eliminate synchronization, data editing, and, dependingon data or greatlyreducethe errorslistedabove,resultingin the quality,resolutionof carrierphasecycle ambiguities,are millimeter-to centimeter-level positiondatawe requirefor obviouslyunavailable.Some civilian receiversemploy a mostgeologicalandgeophysical applications. hybridapproach,with code-correlating capabilityon L1 to recoverthe nonclassified C/A codefor clock synchroniza- Frequency Standards, Time,and ReferenceFrames tion and navigationinformation,anda codeless channelto One key to high-precisiongeodesy with GPS is recover the second harmonic of L2. simultaneous satelliteobservations by a numberof ground Block 2 satelliteshave additionalsecurityfeaturesthat receivers.Simultaneityin this case is defined quite will affect civilian users. "Selective availability" (SA) stringently;in 1 millisecond(ms) a stationat mid-latitudes reducespoint positioningaccuracyto about 100 m by moves more than 30 cm to the east as a result of Earth reducingthe accuracyof thebroadcast ephemeris, altering rotation,a GPS satellite,orbitingat about3 km/s,moves3 the clock epoch,and ditheringthe clock frequency,thus m, and a pseudorange signal propagates300 kin. Ulaffecting both code-correlatingand codelessreceivers. timately,we will relateobservations at widely separated "Antispoofing"(AS) will be activatedperiodicallyfor test groundstationsto better than a microsecond(gs), alpurposesand consistsof encryptionof the P code; this though,as we shallsee,physicalclocksynchronization at would not affect codelessreceivers.Most aspectsof SA anythingnear this level is unnecessary. We nevertheless and AS will not causeseriousimpacton high-precision require a precise time definition and measurement geodeticapplications.However, if activated,AS would capability,a methodfor eliminatingclock errors,and the limit high-precisiondynamic applications,and one abilitytorelatewithgreatprecision thepositions of ground important aspect of SA (clock dithering) is discussed receiversanywhereon the Earth to satellitesin orbit. The below. following discussionis summarizedfrom King et al. [1985],Lambeck[1980, 1988], andLeick [1990]. BothGPS satellitesandreceivershaveprecise"clocks," HIGH-PRECISION GEODESY: RELATIVE i.e., high-frequency,highly stableoscillators.A receiver POSITIONING might employa quartzoscillator,a mechanicalresonator that exploits the frequency-selective propertiesof the Uncertaintiesin a GPS point positionmay be several piezoelectric effect, with a fractionalfrequencystability meters to several tens of meters, althoughMalys and Af/fof about 1 partin 10•øperday.MostGPSsatellites Jensen [1990] recently reported point position uncer- employ higher-qualityrubidium or cesium frequency tainties of about 1 m using data from a speciallycon- standards, where atomicresonance phenomena basedon figuredglobalexperiment.One sourceof uncertaintyin a the energy difference between two states def'me the GPS point positionis the inherentimprecisionof the P "clock."Forexample, thecesiumclock[Essen andParry, code group delay measurement,meter level for most 1956] is basedon the splittingof the groundelectronic receivers,althoughat least one recentmodelachievesa stateof cesium133,depending on whetherthe spinof the severalcentimeterprecisionwith just severalminutesof unpairedvalenceelectronis parallelor antiparallelto the averaging[Melbourne, 1990]. For this reasonwe make nuclearspin.The transition betweenthesetwo hyperfme phase measurementson the carrier itself [Bossleret al., levelshasa frequencyof 9,192,631,770Hz andis thebasis 1980;Remondi,1985], with an inherentprecisionof a few for the currentlyaccepted(SI) definitionof the second millimeters or better. Major remaining error sources (moreon thisbelow). include clock biases (in both the satelliteand ground Onemeasure of clockstabilityis theAllan(twosample) receiver, althoughgroundreceiversare likely to have variance[Allan,1966].FollowingThompson et al. [1986], larger biases),the atmosphere,includingthe frequency- if thefractional average frequency deviation jr(t) fromthe dispersiveionosphereand the nondispersive troposphere, nominal frequency f0 overa timeintervalx is bothof whichaffect signalvelocityand thusour estimate of satellite-receiverdistance, uncertaintiesin the satellite [(t) = [{(t + x) - d•(t)]/2gfox (6)

29, 2/REVIEWS OF GEOPHYSICS

Dixon: THE GLOBAL POSITIONING SYSTEMß 253

then theAllan variance O'2A is

O'2n (1;) = /2

(7)

mentedby leap seconds.Thusthe differencebetweenGPS time andUTC increases by integralseconds, roughlyone per yearsince1980whentheyweresetequal.

We will need to relate the orbital positionsof the GPS satellites, computed in a celestial (--inertial) reference average. TheAllanstandard deviations (N/-•a 2) ofseveralframe,to the locationsof the groundstations,definedin an commonclocksare plottedin Figure2. Note the long-term Earth-fixed(terrestrial)referenceframe. This requires stability of cesium clocks, making them attractivefor preciseknowledgeof the Earth's orientationin inertial satelliteapplications,the short-termstability of quartz space. Earth orientation exhibits a rich spectrum of oscillators,making them adequatefor groundreceivers, temporal variation. Aside from intrinsic geophysical which can be periodically synchronizedwith satellite interest, we neeA to understand,measure, and correct for signals,and the exceptionalstabilityof hydrogenmasers, these effects in order to relate inertial and terrestrial makingthemthebestchoicefor groundreferencestations. referenceframes.In termsof causeswe can distinguish externaltorquesactingon theequatorialbulgeof theEarth (forcedmotion),associatedwith relative orbital motionof the Earth and moon about the Sun, and free motion, associated with the responseof the Earth to internalmass redistribution and correspondingangular momentum exchanges in the Earth system,includingthe hydrosphere and atmosphere.Effects include changesin spin axis directionwith respectto inertial space(precessionand nutation),spin axis changeswith respectto the Earth's

where the angular brackets denote the infinite time

crest(polarmotion),and changesin spinrate. Following Soversand Border [1990], it is convenient to consider the transformation from a terrestrial to inertial reference frame

as a series of rotation matricescorrespondingto these majoreffects. -16

0

1

2

3

4

5

6

7

Precession (P) is a large-amplitude (-23.5ø) slow

circularmotionof the pole with a periodof 26,000 years, Log• (sec) causedby lunar-solarattractionon the equatorialbulge. Figure 2. Frequencystability of "best commercial"clocks, Accurate models are describedby Kaplan [1981] and measured by thesquare rootof theAllanvariance (oa,equation Melbourneet al. [1983, 1985]. Nutations(N) are smaller (7)), asa functionof timeintervalx. Notetheshort-term stability (maximumamplitude9 s of arc), more rapid oscillations of quartzcrystaloscillators andthelong-termstabilityof Cs and superimposed on thisprecession, withperiodsfrom9 days H Maser clocks. Rb, Cs, and H Maser data from Allan et al. to 18.6 years.They are excitedby externaltorques,but [1989].QuartzcrystaldatafromHellwig [1979]. their responseincludesa free component.Nutation series are available from the 1980 International

Now that we can measure time, let's define it more

precisely. Sidereal "time," the kind many people are familiar with, is basedon the Earth's irregularrotation about its axis and is no longer usedas a time standard. Ephemeristime (ET) is more regular and exploitsthe periodicnatureof the orbitalmotionsof the Earth, moon, and Sun. Although no longer in use, ET is related to currenttime standardsbecausethe atomic (SI) secondis defined equal to the ET second,in mm defined as a fractionof the tropicalyear 1900.Universaltime (UT1) is a form of siderealtime, correctedfor someirregularities [Aokiet al., 1982]. Coordinateduniversaltime (UTC) is an internationallyacceptedstandardbasedon atomic time, roughly synchronizedto UT1 by adding"leap seconds" whenrequired.Over the lasttwo decadesthisadditionhas averagedaboutone secondper year,reflectingthe change of the Earth's rotationrate aboutits axis with respectto gravitationalorbitalmotion(thebasisof theET, andhence atomic,second).GPS satellitesbroadcast time signalsthat are synchronizedwithin 1 gs of UTC but are not incre-

Astronomical

Union (IAU) theoryof nutafions[Kaplan, 1981; Seidelman, 1982], basedon the theoryof Wahr [1981]. Additional annual and shorter-periodterms are now known from very long baselineinterferometry(VLBI) observations [Herring et al., 1986]. Correctionsto the standard precession-nutation model can be incorporatedin a perturbationmatrix, f•. Polar motionrepresentsboth free and forced responses.In addition to contributionsfrom externaltorques,possibleinternalexcitationmechanisms include large earthquakes [Slade and Yoder, 1989], aseismicdeformation,motionsof the atmosphereand hydrosphere,and core-mantlecoupling. The dominant periodis 14 months(theChandlerwobble),with additional annualand shorter-period motions,causingthe pole to wander in a circle less than a few tens of meters in

diameter.Pole positionis describedby two coordinates, x and y (correspondingtranslationmatrices X and Y), representing rotationsabouttwo orthogonalaxeslying in theequatorial plane(1 cmof polarmotioncorresponds to a rotationof about 0.3 millarc seconds(mas). The time-

254 ß Dixon: THE GLOBAL POSITIONING SYSTEM

29, 2/REVIEWS OF GEOPHYSICS

varyingrotationrate of the Earth, U, and corresponding centerof masswith a precisionof about5 cm or better changesin length of day (LOD), whose annual and [Smithet al., 1985;Tapleyet al., 1985].ColocatedVLBI shorter-period variationscouldonly be measuredafter the and SLR sites define the center of mass correction for a invention of atomic clocks in the mid-1950s, can be VLBI reference framewithcomparable precision. represented by the differencebetweenatomictime (UTC) The completetransformation from a positionvex:torin a and UT1. Time seriesof LOD or UT1-UTC showstrong terrestrial reference framerr toaninertialreference frame (several millisecondsamplitude) annual and semiannual r• isgivenby periodsassociatedwith seasonalchangesin wind circulation. Shorter-periodfluctuations(2 week and lunar rI = •PNUXY(co¾+ rcm+ A•o•+ Aoc+ Apd+ Aatm) (8) monthly) are due to tidal effects, while longer-period "decadefluctuations"may reflectcore-mantleinteractions whereot is an empiricalcoordinate framescalingfactor. (seereviewsby Wahr [1988], Lainbeck[1988], and Hide Note thatfor localGPS networks,tidal and atmospheric and Dickey [1991]). Values for both polar motion termscan be ignored,and the precisionrequirements for (precision~1 mas)andUT1-UTC (precision~0.1 mswhen the otherparameters may be lessstringentthanfor large averagedover a day) come from VLBI observationsof apertureor globalnetworks. quasars[Carter et al., 1985]. Satellitelaserranging(SLR) [Cohenand Smith, 1985] providescorresponding preci- Eliminating ClockErrors:Singleand Double sionsof approximately 2-4 masand0.2 msaveragedover Differencing severaldays[Smithet al., 1985;Tapleyet al., 1985].Lunar A GPSpointpositionis limitedto meter-level accuracy laser ranging (LLR) providesroughly comparabledata by a combinationof uncertaintiesin clocks, orbits, and [Dickey et al., 1985]. Sensitivityanalysesindicatethat atmospheric effects.However,for a relativepositioning systematicerrors in GPS orbit and baselineestimates experiment involvinga networkof groundstationswithina causedby theselevels of uncertaintyin Earth orientation specificregion,theseerrorsarecommonmodeor nearlyso are negligible[Lichtenand Border, 1987; Dixon et al., and can be eliminatedor greatlyreducedin subsequent 1991c]. data processing[Bossler et al., 1980]. Consider two In summary,spin axis changeswith respectto inertial receiverssimultaneouslyobservingtwo satellites.There space(precession and nutafion)are largelydue to external are four geometricrangeequationsand a maximumof 16 torques(forcedmotion),canbe largein amplitude,andcan observation equations, sinceeachrangecouldbe estimated be modeledquitewell. Polarmotionis a spinaxischange with botha pseudorange andcarderphaseequation,each with respectto the Earth's crest, consistsof both forced at two frequencies. For a single-frequency pseudorange and free motions,is much smallerin amplitudeand less measurement, R (equation(1)), denotingsatellites(id) by easily modeled.UT1-UTC likewise cannotbe modeledat superscripts andreceivers(1,2) by subscripts, we havethe presentwith sufficientaccuracy.Both polar motionand followingfor satellitei: UT1-UTC are measuredadequatelyfor our purposesby VLBI and SLR. (9) R1 =p• +c(Aq -At / +A •) Tide-relatedelastic deformationsmay be important, i i particularlywhere high accuracyis desiredfor widely (10) R2=P2+c(At2 - At/ +Att;2) separatedstations(the effectslargely cancelfor stations closeto eachother).Therearethreemajoreffects:the solid Subtraction givesthe singledifferenceobservable R '.'

Earth tide Asd (maximumamplitude-50 cm); ocean loading effectsin coastal areas,Aoc(maximum amplitude

Ri' =pi'+c(Atl- At2+At}' )

(11)

~5cm);andthepoletideApd, representing elastic response

of the Earthto changesin spinaxisorientation(maximum wheretheprimedvariables denotethedifferential pseudodelay, amplitude ~1 cm).Differential atmospheric loadingAat m rangeor true rangeand differentialpropagation [Rabbeland Schuh,1986] may causeeffectsat the several respectively. Satellite clockerror(At•) is eliminated, and millimeter level for large stationseparationsbut is not differential propagation delayis considerably smallerthan believed to be significant for local or regional GPS theoriginaldelay.Similarly,for satellitej, networks.This effect may become importantwith the adventof high-precision globalGPSnetworks. Rj'=p/'+c(Atl-At2+ An

Earth-fixed

reference

frame

based

on

VLBI

observations doesnot preciselydefinethe locationof the

Subtractionof equations(11) and (12) gives the double

Earth's center of

difference observable R ":

mass. However, GPS-determined

positionsare intimately related to GPS satelliteorbits, which are sensitive to the center of mass. To account for

offsets between the VLBI-based reference frame and the

R"= p"+ c (6t;,")

(13)

actualcenterof mass,we definea vector,rc,•, to be whicheliminatesreceiverclockerror and leavesonly incorporatedin the final transformation. SLR definesthe

differentialpropagationdelays as a significanterror

29, 2/REVIEWS OF GEOPHYSICS

Dixon: THE GLOBAL POSITIONING SYSTEM ß 255

bending.There are two main regions to consider:the frequency-dispersive ionosphere(--50-500 km altitude) and the nondispersive neutralatmosphere, especiallythe troposphere (4}-10 km altitude).Thompsonet al. [1986] give a comprehensive review. Free electronsexist in the ionospherebecausesolar ultravioletradiationis absorbedby gaseousmoleculesin the upperatmosphere, liberatingoutershellelectrons.The electrons interactstronglywith any electromagnetic signal contribute to more than one differenced observable without in the frequencyrange of GPS. Ionosphericeffects are adding new information;the observables are correlated, proportional to the integratedelectroncontentalongthe complicating error calculation.While thereare methodsto signalpathandthusdependon solaractivity,the elevation deal with thesecorrelations[Beutler et al., 1986; Goad and angleof theobservation, time of day,andlatitude. Mueller, 1988], it may be desirableto operatedirectlyon The traveltime• for a groupdelaymeasurement canbe undifferenceddata (sometimescalled one-way range or represented by phase),considering clocktermsasnuisance parameters to be •=p+A B eliminatedby otherapproaches [e.g.,Goad,1985]. • • +f• +... (14) The simplepictureof clock error cancellationthrough simultaneous observations is complicatedby $A and re- where thetermA/f represents mostof theionospheric ceiverclockepochand samplingoffsets.King et al. [1985] delayexperiencedat frequencyf, A is constant,and B is providea completetreatment, but we canillustratetheprob- proportional to the averagemagneticfield strength.If we lemby considering observation in thepresence of $A, where ignore the third- and higher-orderterms, the differential the effectsare particularlyimportant. $A affectssatellite groupdelayA• betweenobservations at f2 (the L2 clocksin two ways.First, the satelliteclockcorrectionrela- frequency) andfl (theL1 frequency) is [Spilker,1978]: tive to "true"GPS time,normallybroadcast in thedatamessage,is corrupted. Fortunately, thishaslittleimpacton nonreal time relative positioningapplications.Second,clock "dithering"changesthebasicP codefrequency, resultingin commensurate changesin thederivedL1 andL2 signals.Irregularvariations of severalhertzaroundthenominalcarrier centerfrequencyover periodsof a few minuteshavebeen (15) observedwith the Block2 satellites.Sincegroundmeasurementsare taggedby receivetime,but SA effectsare com= '1;1 [(fl If2 )2 _ 1] mononly at the sametransmittime,doubledifferencing or analogous schemes to eliminateclockerrorscanresultin incompleteerrorcancellation. The maximumnonsimultaneity where• is thegroupdelayatL1. Similarequations canbe due to $A for receivers at different locations on the Earth is constructedfor the carrier phase observable.Dual freabout 20 ms, which, at currentlevels of $A, inducesmilliquencyobservations thereforeallow eliminationof major meter-levelerrorsonly on the longestbaselines. The effects ionospheric effects(but not third-andhigher-order terms) of nonsimultaneity dueto receiverclocksare usuallynegli- for bothpseudoronge andcarrierphasemeasurements. gible with the double-differencing approach,sincethese The magnitudeof the ionospheric effectcan be large. clockscanbe periodicallysynchronized to GPStime,for ex- Recallingthatn = c/v, ample,with the C/A code. More seriousnonsimultaneity (severaltenthsof seconds)canresultsimplyfrom different (16) nion = 1- Nq•2/2e0 m•o)2 receivertypessamplingthe signalat arbitrary(anddifferen0 times.While thereare techniques for dealingwith thisnon- wherenio n is theionospheric indexof refraction approprisimultaneity[Feigl et al., 1990a],thesimplestapproachis to ateto thephasevelocityof a particularfrequency, N is the ensurethatall receiversare in a networksampleat common number density ofelectrons (-10• - 10•2/m3), q•andm• GPS time. Recent resultssuggestthat with this simple aretheelectron charge andmass, respectively, ande0isthe precautionthe impactof $A on high-precision geodesyis permittivity. Equivalently, the index of refraction at a negligible[RockenandMeertens,1990;Feigl et al., 1990a; given frequencycan be estimatedfrom the plasma source,which can be calibrated,or solvedfor, independently(seebelow).Note thatfor stationsreasonably closeto one another(

f•,,explaining thefrequency selection forGPScarriers.

applications, but if highprecision is required,theycan

integrated electron density N/(electrons/m2):

onlybe usedon theveryshortbaselines mentioned above, wheredifferentialionospheric effectsaresmall. The impact of ignoring higher-orderterms in the

We are more interestedin the integratedeffect of the index of refractionalong the ray path and requirethe

N•=l•th Ndz

ionospheric delaydepends on theelectrondensityin the

(19)ionosphereand the dot productof the ray pathandthe

Earth'smagnetic field.Severalmillimeters of rangeerror at low elevationcanresult.Thiscanbe ignoredfor most butmaybecomemoreimportantin thefuture equivalent length units, L, isgiven byL -- 10-17N/.For applications zenith observations the magnitudeof the time delay for improved vertical component estimates and A usefulrule of thumb is that the delay at zenith in

associated with theP codeobservable maybe 10 nsor less troposphericstudies,both of which benefit from lowelevation observations.Dual frequencyobservations ns or morefor daytimeobservations nearthe geomagnetic enableestimation of theintegrated electron densityandin

atnight atmid-latitudes (N/~1017 electrons/m 2)and100

equator andpolarregions (N/ ~ 1028electrons/m•), principlecouldbe usedto derivemodelsof ionospheric

equivalentto path length variationsof severalmetersto gradients. severaltensof meters.The delaycanalsovaryas a result of magneticstormsandis sensitive to the 11-yearsunspot TheNeutralAtmosphere cycle. In thissectionwe consider nondispersive atmospheric Sincetheionosphere is frequency-dispersive, thegroup effectson the GPS signal.These effectsare due to the

troposphere, tropopause, and mesosphere, but roughly velocity vg (associated with theP codepseudorange observable) differsfromthephasevelocityv• (associatedthreequarters of thetotalnondispersive atmospheric delay with thecarrierphaseobservable); theeffectsareequalin magnitude butopposite in signandarereferredto asgroup delayand phaseadvance.This is suggested by equations (16) and(17), wherenio n is lessthan1 for a givenfrequency, implyingthatthephaseof a wavepassing through theionosphere is advanced relativeto a wavetravelingin

and most of the variability is associatedwith the troposphere.

A GPSsignalis bentandslowedin itspassage through

theloweratmosphere. Thedelayxat mis thedifference in

travel time betweenactualsignalpropagation and the theoretical transittimein vacuum. It is usuallyexpressed vacuum.Fundamental principlesare not violated,because asequivalent pathlengthby multiplying by thespeedof light and can be defined by the difference of two path it is vg(thevelocity of thecodemodulation) thatdeter-

minesthe speedat whichinformationis carried. For many applicationsit is deskableto considerlinear combinationsof data, to isolate or reduce errors, or to reducecomputation timeby compressing observations to a

integrals[Daviset al., 1985]:

X,tm =l,tm n(s)ds - l•,½ ds

(21)

singledatatype.A goodexampleis the ionosphere-free wheren(s)is theindexof refraction at thepoints alongthe

phase observable Lcgivenby [e.g.,Blewitt,1989]:

pathandn,a½ = 1 is omittedfromthe second integral. Evaluation of the first integralrequiresan atmospheric

If•2L1 -f22L2) (20)

model.All components of theatmosphere contribute to the delay,but it is convenient to consider two components separately. The "dry" delay is associated with molecular whereL1 andL2 hererepresent carrierphaseranges at the constituents of theatmosphere in hydrostatic equilibrium two frequencies for a particular receiver-satellite pair (i.e., (including H20), andthe"wet"delayis associated with L1 = -cO)l/f1 andsimilarly forL2),Lc is theionosphere-watervapornotin hydrostatic equilibrium. Thedrydelay corrected phase, andtheeffective wavelength is•c = c/(f• is typically200-230 cm at zenith(elevation angle,0 = +f2) -- 10.7cm(5.4cmforcompletely codeless receivers). 90ø)ataltitudes nearsealevel,whilethezenith wetdelay The shortereffectivewavelength of •c impliesthat mightrangefrom3 to 30 cm.Thedelayat otherelevation resolution of carrierphasecycleambiguities will be more anglesis larger,increasing approximately as 1/sin(0),but complicated.For this reason, and also becausenoise othereffectsareincorporated in a "mapping function" for (particularly multipath) is amplified by L•, it maybe greateraccuracy,includingthe finite height of the deskableto usesingle-frequency measurements for very atmosphere, theverticaldistribution of components, Earth short(a few kilometers or less,depending on ionosphericcurvature,andmy bending[e.g.,Blackand Eisner,1984; activity)baselineswhereionospheric effectsare smallor Lanyi,1984;Daviset al., 1985].Thetotalpathdelaycan be written as process thetwofrequencies separately.

29, 2/REVIEWS OF GEOPHYSICS

t,(0)=

+

Dixon: THE GLOBAL POSITIONING SYSTEMß 257

(0)

(22)

have a small bias that dependson site, season,or data reductionalgorithm. where p0refers to thepathdelayat zenith (hereafter the Unfortunately,most WVRs are big and expensive. zenith delay), subscriptsd and w refer to dry and wet Fortunately, comparison of GPS baseline estimates components, respectively,andM(0) is a mappingfunction, involving WVR calibration and stochasticestimation assumedto be azimuthallysymmetric. generally indicates no significant differences between The dry zenithdelay is determinedby measurement of the two approaches(for example, Figures 3 and 4), surfacepressure.If suchmeasurements are not available, implying that at current precision levels WVR calibrastandardatmosphericconditionscan be assumed,and an tion of the wet delay is not required. In fact, it is even initial estimatefor the dry zenithdelayapproximated from possibleto lump the wet and dry delays togetherand the standardpressureP (in bars), basedon the surface estimatethem jointly, avoidingall neutral atmosphere elevationof the station,h, anda scaleheightH, typically7 calibration(includingsurfacepressure)entirely [Tralli and Lichten, 1990]. This works at current levels of GPS

km [Tralli et al., 1988]:

P = 1.013e-4'm

precisionbecausethe mappingfunctionsfor wet anddry (23) delaysare very similarabove10ø or 15ø, the typical cutoff angle usedwith GPS to avoid groundmultipath

Subsequentanalysisof the GPS data can improve the initial

estimate and reduce the effect of errors in this

parameteron thebaselineestimates. The wet zenithdelay can be estimatedin at leastthree ways:by measurement of surfacetemperature andrelative humiditycoupledwith a simpleatmospheric model [e.g., Chao, 1974]; with a water vapor radiometer(WVR), an instrumentthat measuresatmospheric blackbodyradiation in the microwaveregion,which is affectedby a rotational moleculartransitionof water vapor near 22.2 GHz [e.g., Janssen, 1985; Robinson, 1988]; and by stochastic estimation techniques without a priori calibration, exploiting the data strength of GPS and the known elevationangledependence of the wet delay [Tralli et al., 1988; Dixon and Kornreich Wolf, 1990; Dixon et al., 1991a]. Stochasticmodels make use of the fact that the

wet path delay is likely, in a statisticalsense,to vary within a limitedrangeover a shorttime interval.For SM, WVR, or other a priori calibrations,residuals(wet delay corrections) can be estimated along with geodetic parametersof interest.This improvesthe final resultbut implies errorsin the calibration.Studieshave shownthat SM calibrationby itself can lead to unreliableGPS results [Tralli et al., 1988;Dixon et al., 1991a],confirmingearlier studiesindicatingpoorcorrelationbetweensurfacerelative humidityandthe wet pathdelay[Reberand Swope,1972; Elgered,1982].K. Hurstet al. (Estimationof GPSbaseline errorsdueto imperfectretrievalof wet atmospheric delays using surface meteorologymeasurements,submittedto Journalof Geophysical Research,1991)compareSM and WVR estimatesof wet zenith delay and concludethat SM-basedestimates wouldhavean rmserrorproportional to the magnitudeof the delay, given by 1.2 cm + 0.08 (SMT), where "SMT" is the SM-basedzenith delay in centimeters.Tralli et al. [1988] and Tralli and Lichten [1990] suggested that the errorin SM calibrationcan also show considerabletime variation. In contrast,corrections

(see below). In spite of the success of stochastic estimation

techniques, for GPS experimentsin regionsof high wet path delay and high variability (for example, some conditionsin the tropics), troposphericcalibration is probablythe dominanterror sourcefor baselinesin the geologicallyimportantlength range of several tens to several hundred kilometers. Orbital effects dominate at

longer baseline lengths, and receiver effects and other

errors dominate at shorter baseline lengths. This is discussed in more detail in the sectionon precisionand accuracy.Troposphericcalibrationdoesnot appearto be a significanterror sourcein some other regions(for example, typical conditionsin the southwesternU.S.) when the wet path delay is low and not particularly variable.Future improvementsin the accuracyof GPS, particularlyfor the vertical component,will dependin part on improvedtroposphericcalibrationand estimation techniques. By analogywith VLBI, direct line of sight calibrationwith a WVR or othertechniquemay improve GPS baseline estimateswhen calibration accuracy exceeds~5-10 mm [Herring, 1986;Elgeredet al., 1991; Dixon and KornreichWolf, 1990]. It hasbeendifficult to deployhigh-accuracy WVRs to a majorityof sitesin a GPS networkbecauseof the excessive cost,weight,and power consumptionof these sophisticatedinstruments. Recentdevelopments in the areaof monolithicmicrowave integrated circuit(MMIC) technology havethepotentialto improvethe accuracyand reducethe cost,weight,and powerconsumption of WVRs, makingextensivedeployment of these instrumentsviable. Improved stochastic modelsare anotherpromisingarea of research.Current modelsassumeazimuthalsymmetryin the wet delay. However,azimuthalasymmetryhasbeenobserved[Dixon and KornreichWolf, 1990;Rockenet al., 1991]. Estimating two stochastic, orthogonalspatialgradientsin addition to thestochastic wetzenithdelayshouldimproveprecision in the presenceof asymmetryand shouldbe feasiblewith sufficientdata strength,presumablyavailable with the enhanced Block 2 constellation and receiverscapableof

to WVR calibrationare generallysmall(1-2 cm in zenith wet delay) and constantor nearlyso over severalhours, implyingthattheseinstruments cangive a goodindication of temporalvariabilityin the wet zenith delay but may trackingmorethanfour satellites.

258 ß Dixon: THE GLOBAL POSITIONING SYSTEM

29, 2/REVIEWS OF GEOPHYSICS

6

_ a4o/o LIMON - LIBERIA(270 km)

_

,,.:.:.:.:.:.:.:.:.:.:.:.:.:.:,

::::::::::::::::::::::::::::::::::::::::: • STOCHASTIC ESTIMATION

--

:::::::::::::::::::::::::::::

•

- iiiiii!iii!i!ili!iiiiiiiiiiil 85%86%

87%

87%

86%

86%

• .. iii.dliiill • ilili

::::::::.,..__ ::i::i:::: 0.0

0.48

87% •

86% 88%

86%

86% 87% 88ø/*

--

STOCHASTIC RESIDUAL 0.80

1.0

1.2

1.8

2.4

0.48

0.80

1.0

O• (cm/x/h-•

1.2

1.8

2.4

O• (cm/•--•

WVR CALIBRATION

STOCHASTIC ESTIMATION, NO PRIOR CALIBRATION

Figure 3. Short-termrepeatabilityfor a 270-km baselinein CostaRica with varioustreatmentsof wet path delay (see also Figure 4). Percentsymbolsaboveeachbar indicatenumberof biases resolved.With WVR calibration(left side), estimationof a residualdelay, either as constantor

witha stochastic modelspecified by therandom walkparameter, a (cm/•), improves the calibration,but only to the level reachedin the casewhereno calibrationis used(right side) and theentiredelayis estimatedstochastically [fromDixonandKornreichWolf, 1990].

range observables with systematic, time-dependent sinusoidal signals associatedwith variable receivertional, enabling signals from several satellitesto be satellitegeometryovera pass.The magnitudeof multipath receivedsimultaneously. Dependingon the immediate is roughlyproportionalto wavelength,and the effectsare environment and the gainpatternat low-elevationangles, considerablylarger for P code pseudorangerelative to suchantennas can be susceptible to interference ("multi- carrier phase.Low-angle observationstend to be most

Multipath

Antennas for almost all GPS receivers are omnidirec-

path")frommultiplearrivalsof thesamesignalbecause of reflections fromnearbyobjects.Reflections at thesatellite alsooccur[Younget al., 1985]butcanbe ignoredfor most applications. Multipathcorruptsthephaseand/orpseudo-

affected, andfor thisreason a cut-offangleof 10ø-20 ø above the horizon is usually employed.This has the fortunateeffect of minimizingerrorsdue to third- and higher-ordertermsin the ionospheric delay and minimiz-

LIMON - LIBERIA (270 km) I

I

I

I

i

i

•

--•'--- WVRCALIBRATION -O-

[

i ..,.T '

'

Figure 4. Baseline estimates with and without WVR

STOCHASTIC ESTIMATION

cali-

brationof the wet path delay for thedatashownin Figure3. Estimates with WVR calibration include a stochastic resi-

dualmodel(a = 1.2cm/x/-•). Estimates without WVR i

>

cali-

brationemploya similar sto-

w

chastic model for the entire

wet delay. Error bars and ellipses are one sigma [from

Dixon and Kornreich Wolf, 19901.

-4

-4 -4

-2

0

EAST (cm)

2

-4

-2

0

LENGTH (cm)

2

29, 2/REVIEWS OF GEOPHYSICS

Dixon: THE GLOBAL POSITIONING SYSTEMß 259

ing tropospheric calibrationerrors,includingthosedue to natureof the perturbingforces on the satellitesand the for high-accuracy meansthatthismodelis not incorrectmappingfunctions.However,lack of low-angle requirement observationsis one of the conlributingfactors to poor adequatefor our purposes.Rizos and Stolz [1985] sumresolution of the vertical component with GPS, so marize major accelerationson the GPS satellites.Atmosmultipathcontrol,as well as improvements in the other phericdragis negligibleat thesealtitudes,and the Earth's areasjust mentioned, is highlydesirable. Multipathcanbe nonsymmetricgravity field, while significant, is ademinimizedthroughuseof antennabackplanesor RF (radio quatelydescribedwith currentmodels;the more poorly frequency)absorbentmaterial around the base of the determinedshortwavelengthcomponents, relatedmainly antenna, mounting antennasclose to the ground (to to lithosphericstructuresin poorly surveyedpartsof the minimizetheeffectsof groundreflection),andcarefulsite Earth,havelittle effect on the high-altitudeGPS satellites. are the gravitationaleffects selection,choosingsiteswell away from planar-reflecting Additionallargeperturbations surfacessuchasbuildingsor vehicles. of the moonandSun,whichcanbe accuratelydetermined, Since multipathis related mainly to the geometryof andsolarradiationpressure. One approachto the orbit problemis to begin with an nearbyobjects,andsincethisgeometryis usuallyconstant at someinitial time over several days or longer, the temporalsignatureof estimateof the six orbitalcomponents multipathtendsto repeatfrom day to day, retardedby an and numericallyintegratethe equationsof motionusing amountequalto theoffsetrisetimeof the satellites(4 min accurate modelsfor variouspertubingforcesandresulting earlier each day). The repetitivenature of multipathin accelerations,predictinglocation and velocity of the principlecan be usedto estimateand correctits major satellitesat later epochs;groundtrackingdatacanbe used effects,thoughfew such studieshave been reported.If in a least squaresadjustmentto improve the initial estimates,the modelparameters,and the uncorrected,multipath can be an important sourceof position/velocity systematicerror, dependingon environment,antenna/ subsequent position/velocity estimates.The lengthof the backplanedesign,and lengthof observingsession.Carrier orbitarc over whichthe equations are integratedmay be phasemultipathtendsto haveperiodsshorterthan 10-20 only a few hours ("short arc") in which case the force min, thus observationsover severalhoursor longer will modelscanbe relativelysimple,or may extendto several averageoutmostof theeffects. weeks("multiday"or "long arc"), requitingsophisticated forcemodels.Solarradiationpressure produces a largeand Orbits somewhatunpredictableperturbingforce becauseof the GPS satellites orbit about three Earth radii above the large cross-section area of the solar panels, complex surfacein six orbitalplanes.For high-precision geodesyit satellitegeometry,and variablesatellitealbedo,resulting is necessaryto know precisely the positions of the inaccelerations oforder 10-7m/s 2andperturbing theorbits satellitesat the time of observation, with acceptableerrora manymetersin just a few hours.Althougha solarradiation functionof desiredbaselineprecisionand stationsepara- pressuremodel for the GPS satellitesaccountsfor albedo tion; for millimeter-levelperformanceon baselineslonger andgeometry,largeresidualshavebeenobserved[Fleigel than about 100 km, meter-levelprecisionin the orbit et al., 1985]. It is therefore common to solve for at least one additionalaccelerationparameterin the estimation estimatesis required. Satellite orbits can be describedby six parametersat process,representing departuresfrom the ideal model.In a someinitial epochanda forcemodelto definesubsequent spacecraft-centered coordinatesystemwherez pointsto the time evolution. The epoch state parametersare three centerof the Earthandy is alongthe solarpanelsupport componentsof position (x,y,z) and three velocities in beam and normal to the spacecraft-Sundirection, unCartesian coordinates,or equivalently six Keplerian modeledaccelerations areoftenobserved in they direction elements:the semimajoraxis (a) and eccentricity(e), ("y-bias"), perhaps related to thermal radiation or describingthe size and shapeof the elliptical orbit; a misalignedsolarpanels[Fleigelet al., 1985; Schutzet al., parameterdescribingthe positionof the satellitewithin 1990]. Residualstend to be largestduring the 2-month that orbit (for example,M, the "meananomaly"or f, the eclipseseason,when the spacecraftperiodicallyenter the "true anomaly"); and the inclination (i), argument of Earth's shadow [Schutzet al., 1990]. Lichten and Border perigee(co),and fight ascensionof ascendingnode (f•), [1987] adjustedsolar pressurecoefficientsin all three describingthe positionand orientationof the orbitalplane. components for data arcs up to 1 week, usingconstant In the absenceof additionaldisturbingforces,only the corrections to the nominalmodel.Lichtenand Bertiger anomalyis timedependent. A simpleKepleriandescription [1989] usedthreecomponentstochasticcorrectionsto the of an orbit in a plane can be obtainedby integrating nominalmodelfor dataarcslongerthan 1 week. Newton'slaws of motion,giving Informationon satellitetrajectories duringtheperiodof theexperiment maycomefromthebroadcast ephemeris in r = a(1 - e2)/[1+ e cos(f)] the GPS signal, or additional GPS data taken simultaneouslyfrom siteswhosepositionsare well knownfrom where r is radial distance from the center of the Earth independentmeasurementssuch as VLBI or SLR. The [Kaula, 1966]. For near-circularorbitslike GPS, e--0. The resulting tracking data allow generation of accurate

260 ß Dixon- THE GLOBAL POSITIONING SYSTEM

29, 2/REVIEWS OF GEOPHYSICS

ephemerides (satellitelocationsas a functionof time) and vantageous to employa multidayarc analysis,insteadof define the reference frame for the experiment. The the singleday, shortarc approach.The advantages of subnetworkof known groundstationsis calleda fiducial multidayarc analysisare described by LichtenandBorder network. Uncertainties in fiducial site location, or in the [1987]. With observations over more than one revolution, groundsurveysconnectingthe phasecentersof the large orbital periodsare more accuratelydeterminedand the VLBI antennasto the ground mark used by the GPS antenna,or low-qualityGPS data at one or more of these 12 sites during an experiment,are a major sourceof sysVERYICALI O' tematicerror in GPS geodesy. o

10

Thatpartof theerrorin a baseline estimate •t,t dueto

orbiterror•oa,canbeestimated fromtheruleof thumb -- •oa,L/h, whereL is baselinelength,h is satellite

o

I

altitude,-20,000km, and • is a geometricfactor(4).2) [Lichten, 1990a]. Since the error in the broadcast

ephemeris canbe 50-100 m with selectiveavailability,we requirea robustfiducialnetworkfor any experimentwith baselineslonger than about 50-100 km if we desire centimeterlevel or better accuracy.The nature of the requirednetworkdependson the experiment.In general,at leastthreestationsare required,with adequatenorth-south

w o_ w

•o II •

and east-west extent. Stations too close to each other

provide redundantinformationand may not contribute greatly to network strength.On the other hand, mutual satellite visibility is deskablefor cancellationof clock errors,one or two redundantstationsprotectagainstdata outages,and some pmximal stationscan facilitate cycle ambiguityresolution.The optimumnumberand geometry of stations may be difficult to predict. A covariance analysis,which estimatesuncertainties in baselinevectors from certainassumptions aboutthe dataand its errors,can be useful in testingtrial fiducial networks[Dixon et al., 1985;Freymuellerand Golombek,1988]. In some cases,covarianceanalysesmay suggestthe need for fiducial sites in areas where VLBI

NO'RTH' I

0 I

Siope•

=1x10 -9' Ilb•)Slope

o I

,,.

o .

I

t

ß

I

ß

U.S. only Global network

4

2

o

or SLR sites

are not available,or where the necessarygroundtie data EAST I I .... 0 , 0ß are not available.It may still be advantageous to deploya receiverto sucha locationto provideadditionalgeometric strengthto the trackingdata.KornreichWolfet al. [1990] used data from the "CASA" experimentto show that addingtwo trackingsitesin the southwest Pacificand two .• in Europeto a fiducialnetworkconsisting of threestations in the U.S. improvedthe repeatabilityof the horizontal o o o 1 componentsof long (>400 kin) baseline estimatesin / ' CentralAmerica and northernSouthAmerica(Figure 5), even thoughthe actuallocationof the new stationswas uncertainat the 10-20 cm level. The positionof such tracking sites is estimatedin the data analysisto avoid 200 400 600 800 1000 introducingsystematicerror. BASELINE LENGTH (km) In spiteof our besteffortsto establisha robustfiducial networkwe may find that logisticallimitationsor data in short-term (3-8 days)repeatability outageslimit the qualityof the resultingobservations. An Figure5. Improvement (equation(30)) for seventeenbaselinesin Central and northern importantclue to the causeof low-quality data can be SouthAmericaasa function of fiducialnetwork configuration. found by plotting repeatability(day-to-dayscatter)of Single-day orbitalarcsare used."U.S. only"indicates three

. a$\•/-

baseline estimates asafunction ofbaseline length; ahighVLBI sims inthe U.S. "Global" indicates this network suppledependence onlength (>2-3parts inl0sforbaselines mented bytwo European and two southwest Pacific stations. Best longer than100-200 kin)suggests thepossible influence fitstraight lines areshown (modified from Kornreich Wolf etal.,

of orbit errors (Figures5 and 6). It may then be at-

[1990].

29, 2/REVIEWS OF GEOPHYSICS

Dixon: THE GLOBAL POSITIONING SYSTEM ß 261

positionsof the orbital nodesbetterdefined,with correspondingimprovementin the ability to resolvecarder phasecycle ambiguitiesthroughoutthe networkand the precisionandaccuracyof all components of the baseline vector.Figure6 showstherepeatability of a setof baseline

six-stationGPS global trackingnetwork, expectedto be operationalin 1991,for trackingthe U.S.-FrenchTOPEXPoseidonsatellite,a low Earth orbiterfor oceanographic research to be launchedin 1992.Neilan et al. [1990]givea currentsummaryof internationaltrackingnetworks.

vectors from the northern Caribbean before and after a

multiday arc analysis [Dixon et al., 1991a]. Geodetic estimateswere improvedby factorsof 2 or morewith the longer arcs for baselineslonger than about 500 kin. Single-dayarcsandthe existingthree-fourstationfiducial network were apparently adequate to minimize the contribution of orbit error for shorter baselines.

BIASES RESOLVED E3 SINGLE

[]MULTI

EAST 5

DAY ARCS

DAY ARCS 4

3

$

•

o

NORTH

VERTICAL

6

0

-

,

200

,

300

,

400

51•0

•

9;0

1100

1300

BASELINE LENGTH (km)

Figure 6. Histograms of short-termrepeatability(equation(30)) for 15 baselinesin the northernCaribbeanfor singleday and multiday orbital arc analysis.Carrier phasecycle ambiguities ("biases")areresolvedfor eachtechniqueto the extentpossible. Improvementin the longer(>500 km) baselineswith multiday arcsresultsin part from the increasedabilityto resolvethebiases [from Dixon et al., 1991a].

Reducing the Data Considera hypotheticalGPS experimentinvolving5 daysof observations at 10 groundstations,with the positionsof threefiducialsiteswell known.Eachstationmight observesix satellitesfor an averageof 5 hourseachday (the total time spanof observations might be 8 hoursor more, but not all satellitesare visible simultaneously). Pseudorange and/orphasedataarecollectedcontinuously, but let's considerphasedataaveragedat 1-min intervals. The totalnumberof "datapoints"collectedin 1 day is then 10 stationsx 6 satellitesx 5 hoursx 60 pointsper hour= 18,000. From this large data set we might estimatethree positioncomponentsfor each unknown ground station (21), six components (threeposition,threevelocityat the initial epoch)for eachsatellite(36), a y bias term for each satellite(6), perhapsone troposphere parameterfor each groundstation(10) (additionalterms are requiredfor a stochasticmodel), and clock terms every minute for all satellites(1800) and ninegroundstations(2700), sinceone stationactsas a referenceclock,giving a total of 4573 unknowns.The systemis overdetermined, with data points greatlyexceedingunknowns.Sincethe dataare noisy,we cannotformulatesimultaneous equationsto solvedirectly for the desiredparameters.The challengeis to find, in a least squaressense,the best estimateof the parameters from availabledata,usingwherepossibleadditionalinformation suchas troposphericcalibrationand orbit models. Therearenumerous approaches to GPSdataanalysis,andI havetriedto generalizethefollowingdiscussion. However, somedetailsare specificto analyticalapproaches familiar to me, andI apologizefor thisobviousbias. The timeevolutionof pseudorange or phaseobservables (e.g., equation(3)) is largely a function of observation geometry.Given prior knowledgeof the approximate groundstationpositionsin a terrestrialreferenceframe, satellite orbits (from the broadcastephemeris),Earth orientation, andinitialestimates of atmospheric delay,it is possible to generate fairly accurate models for the observables. With thesemodels,andknowledgeof how the observableob would changegiven a small changein a parameter,x, (given by the partial derivative,•5(ob)/fx), leastsquares or Kalmanfilter techniques canbe appliedto theobservables to improveestimates of x. FollowingKing et al. [1985], givenan observablemodelc, an observation l, andan observation errorv (all functionsof time),

The burdenof establishing adequatefiducialnetworks for regionalGPS experiments will be considerably reduced in the nextfew yearsas variousnationalandinternational c= l + v (25) agenciesestablishglobaltrackingnetworks.The Cooperative International GPSNetworkhasbeenin operationwith Thiscanbe linearizedby expandingtheleft-handsidein a six or more stations since 1989, administered in the U.S. Taylorseriesanddroppingsecond-andhigher-order terms, by the National GeodeticSurvey.NASA will sponsora expressingc as a vectorof initial estimatesof the observ-

262 ß Dixon: THE GLOBAL POSITIONING SYSTEM

29, 2/REVIEWS OF GEOPHYSICS

ables, c.,based onapriorivalues ofx,x•,plusincrementalconstant parameters (y), a vector of process noise corrections, [?ffob)/15x]dx, to be constrained by thedataand parameters(p) and v, definedhere as a zero meanwhite its errors.The linearizedobservationequationin matrix noise vector: form is

Z= AxX +At, P+AyY+v

c. + A dx= l + v

(28)

(26) where z is the difference between an observation and its

where A (the design matrix) is the matrix of partial modelvalue.Parametersmodeledas processnoiseinclude path derivatives, dxis a vectorof smallcorrections tox., andv clocks(modeledas whitenoise)andwet tropospheric is now the vector of postfit residuals.The quantity delays (modeledas random walks). Measurementsare minimizedis theweightedsumof squared residuals, given groupedinto finite time intervalsor batches(typicallyone byvrWv,where W, theweight matfix, istheinverse of P, to severalminutes)and processedsequentially,leadingto the covariance matfix. The solution is updatedparameterestimatesand covariancesafter each batch.After filteringall dataa smoothing algorithmcanbe • = x. + dx (27) applied that works backward in time to update the estimatesand covariances,for example, allowing inveswhere • isthebestestimate ofx, dx= N-1ArW(/- c.), tigationof the time-varyingbehaviorof the troposphere andN isthenormal equation matrix given byArWA.This [e.g.,Dixon andKornreichWolf, 1990]. We sawfrom equations(2) and(3) that the initial phase approachmay be appliedto bothdifferenced and undifferenceddata.It is alsopossibleto transformundifferenced measurementupon acquisitionof the carder signal is data to derived parameterscontainingno clock terms biasedby an unknown number of cycles. Assuminga through orthogonalizationmethods without forming receivermaintainslock on the signal,the range change normalequations[Lawsonand Hanson,1974]. Additional between the receiver and satellite can be determined, and discussion of GPS dataanalysisapproaches canbe found the initial range (the cycle ambiguity) can be estimated in worksby Goad [1985],Bocket al., [1985a,b], Lindler along with the geodeticparametersof interest.However, and Wells [1985], Bock et al. [1986], Beutler et al. [1986, this degradesthe accuracyof horizontalbaselinecompo1987],SchaffrinandGraffarend[1986],andLeick[1990]. nentsrelative to the case where the cycle ambiguity is Most of the datadiscussed in thisreportwereprocessed knownor can be fixed to the correctvalue.Techniquesto with the GPS Inferred Positioning System (GIPSY) resolvethe ambiguityrely on the fact that, given enough software developedat the Jet PropulsionLaboratory. data,the rangebias can be estimatedto betterthan half a GIPSY processesundifferenceddata using a modified carderwavelength(cycle),after which the bias is fixed to Kalman filter, describedby Lichten [1990b]. Briefly, a the nearest integer value. Orbits, ionosphericeffects, effects(Figure 7), multipath,and othererror conventionalKalman filter updatesmeasurements from tropospheric one observationepochto the next on the basisof the sourcescan corruptGPS signalssuchthat errorsbecome covariance matrix P. Factorization of P into upper significantrelative to one half wavelength,affectingour triangular (U) anddiagonal (D) matrices (P = UDUr) abilityto fix ambiguitiesto thecorrectvalue.It is common improvesaccuracyandcomputational efficiency[Thornton practiceto resolvethe ambiguitiesfirst on shorter(