AN INTRODUCTION TO THE GLOBAL POSITIONING SYSTEM AND SOME GEOLOGICAL APPLICATIONS

T. H. Dixon JetPropulsion Laboratory Divisionof Earthand PlanetarySciences Pasadena,California

Abstract.Receivers equipped to measuredual frequency data acquisitionand analysisare having a significantim- carrierphase signals from satellitesof the GlobalPosition- pact on studies of near-fault crustal deformation and ing System(GPS) havebeen capable, under special condi- earthquakeprocesses, until recently the province of con- tions, of determiningrelative horizontalpositions among ventional terrestrialgeodetic techniques.The enhanced stationsseparated by oneto a few hundredkilometers with satelliteconstellation, improved models, and establishment a precisionof one to severalmillimeters since the early of global tracking networks have extendedseveral mil- 1980s. The major obstaclesto making this capability limetershorizontal positioning capability to stationsepara- routine,extending it to all partsof theglobe, and extending tionsof 1000km or morein virtuallyall partsof the world. it to longerstation separations, have been equipment cost, This enablesstudy of new classesof tectonicproblems that limitations in the GPS satellite constellation,arduous data previouslywere difficult to attackwith any geodetictech- analysis,uncertainties in satelliteorbits, uncertainties in nique.Examples include a completekinematic description propagationdelays associated with variabletropospheric of ongoingcrustal deformation in broad, complexconti- watervapor, and difficultiesin resolvingcarrier phase cy- nentalplate boundaryzones, and measurementof relative cle ambiguities.Recent improvements have occurred in all plate motionat convergentboundaries where global mod- theseareas. The increasingease and reducedcost of GPS els maybe poorlyconstrained.

INTRODUCTION extendingthe range and accuracyof GPS measurements are emphasized.I have aimed for broadcoverage of most With the adventin the early 1980sof a satellite-based of therelevant topics and an intuitiverather than complete navigationsystem known as the Global Positioning System or rigorous treatment, giving more derailed references (GPS) operatedby the U.S. Departmentof Defense it whereappropriate. becamepossible for a user with the properreceiver to obtain almost instantaneousthree-dimensional position informationaccurate to severalmeters. With the comple- A BRIEF HISTORY OF THE GPS PROGRAM tion of the satelliteconstellation in the early 1990s this capabilitywill be extendedto virtually all parts of the The spacesegment of GPS is a constellationof satellites globe.This is a remarkableachievement and buildson a in highEarth orbit equipped with powerfulradio frequency number of technologicaladvances in the last several transmittersand highly stable atomic clocks. Easton [1978] decades. Even more remarkable is the fact that. with careful reviewsthe majordevelopments leading to thiscapability. attentiontoexperiment configuration anddat• analysis it is In 1967 an early prototypeof a GPS satelliteknown as possible to 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 enhancedperformance allows for measure- navigation.Weighing about 40 kg and consumingonly 6 ment of crustalstrain and fault motionrates in just a few W of power,it carrieda UHF transmitterslaved to a stable )rears. quartzclock, with a frequencydrift of several parts in 10• This paper reviews fundamentalprinciples of GPS, per day. Additional proof-of-conceptsatellites followed, discussessome geologicaland geophysicalapplications culminating10 yearslater in the NavigationTechnology and their accuracyrequirements, and considersimplica- Satellite (NTS) 2, very similar to subsequentGPS tions for GPS experimentdesign. Recent developments satellites.NTS-2 was launched into a 20,300-km-altitude

Copyright1991 by the AmericanGeophysical Union. Reviewsof Geophysics,29, 2 / May 1991 pages249-276 8755-1209/91/91RG-00152 $05.00 Papernumber 91RG00152 ß 249 ß 250 ß Dixon: THE GLOBAL POSITIONING SYSTEM 29, 2/REVIEWS OF GEOPHYSICS

orbit, weighed440 kg, consumed400 W of power, and oneanother in thesky provide correlated (redundant) range transmittedtwo L band (-1.2 and 1.5 GHz) timing and information,an effect known as geometricdilution of ranging signals based on a sophisticatedcesium clock, precision(GDOP). If observationsare limited to four witha frequencydrift of lessthan two parts in 10•3 per satellitesby receiverdesign, these geometric effects can be day. One year after NTS-2, the first "Block 1" GPS minimized by choosingsatellites which maximize the satellitewas launched,part of the operationaltest phase of volumeof a tetrahedron,defined by the pointsof intersec- the GPS program. By 1990 the Block 1 constellation tion on a unit spherecentered on the user, of vectors includedsix functioningsatellites launched between 1978 betweenthe satelliteand the groundreceiver. and 1985. It is alsopossible to obtaindistance information (strictly It hasbeen recognized for sometime thathigh-precision speaking,the changein distance)through phase measure- geodetic measurementscould be made by exploiting mentson the carrier signal itself, keepingtrack of the signalsfrom artificial satellites[e.g., Preston et al., 1972; number of cycles after signal acquisition.Assuming MacDoran, 1979; Counselmanand Shapiro, 1979]. The perfectclocks, and ignoring propagation effects, Block 1 constellationhas provensatisfactory for develop- ing andrefining experiment design and analytical concepts and for initiating a number of high-precisiongeodetic monitoringprograms. The first Block 2 satellite,with a = (v,/œ)(n + 0)) (2) number of improvementsrelative to its forebears,was wheren is the numberof integercarrier wavelengths 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 satellitesplus three sparesare plannedto be in operation cycles,)• isthe wavelength, f is the frequency, and v, isthe by the end of 1992. They will orbit at an altitudeof about phasevelocity (the importanceof distinguishingv, and 20,000 km in six orbital planes with 12-hour periods, groupvelocity, vg, will becomeapparent). Since the wavelengthof the carrieris considerablyshorter than that enablingsimultaneous observation of four or more GPS of the lower frequencycode modulations(Table 1), the satellitesin virtuallyall partsof the globe. resulting"distance" measurement, though ambiguous by the initial numberof wavelengths,is considerablymore POINT POSITIONING WITH GPS ,,,-ooiootho,, o pseudorangemeasurement and is one of the keys to high-precisionGPS measurements[Bossler et al., 1980; Counselmanand Gourevitch,1981; Remondi, 1985]. RangeMeasurement Carrier phase is not measureddirectly, as this would An observeron Earth can uniquelylocate his position requirevery high samplingrates. Rather,the signalis by determiningthe distancebetween himself and three mixed ("heterodyned")with a signal generatedby the satelliteswhose orbital positionsare alreadyknown. With receiver's internal clock (local oscillator) and, after GPS, distanceinformation is based on the travel time x of a band-passor low-pass filtering, the resulting lower satellite signal, obtained by measuringthe difference frequency"carrier beat phase"is sampled.Most current betweenthe transmit (t•) andreceive (tr) times at theGPS generationreceivers accomplish this "downconversion" receiverof a specialranging code, describedin the next with electronicsthat includeextensive analog circuitry. section.If we ignore transmissionmedia effects on the Some newer generationreceivers have largely digital speedof light c and any timing(clock) errors, then the true architecture,reducing production cost, size, and power rangep betweensatellite and receiver is just C(tr -- rs). consumptionand enablingdigital samplingof the carrier Errorsin receiveror satelliteclocks are present in therange phasesignal with only minimalpreprocessing [Melbourne, estimate,which for this reasonis referredto as pseudo- 1990]. rangeR, definedmore precisely as TABLE 1. Summaryof GPS SignalCharacteristics R = p + c(Atr - Ats+ Atp) (1) Carriers Code Modulations C/A whereAt r is thereceiver clock offset from "true" (GPS sys- L1 L2 P (L1 only) tem) time (we ignore any other receiver-inducederrors), At•is thesatellite clock offset, and At is thedelay associ- Frequency(carrier) 1.57542 1.2276 10.23 1.023 atedwith all other error sources, rn•ainly due to atmos- or chiprate GHz GHz MHz MHz (codemodulation) phericpropagation effects. Information from a fourthsatel- Wavelength 19.0cm 24.4 cm ~30 m ~300m liteallows a first-orderclock correction (Atr - Ats),and ap- proachesdiscussed below can be appliedto estimateand correct for At, enabling meter-levelpositioning under idealconditionS. For analysispurposes we considerthe pseudorangeor Observationgeometry affects the quality of theresulting phaseparameters in termsof what the receiveractually three-dimensionalposition. Satellites that appearclose to sees(the "observable"),explicitly accounting for major 29, 2 / REVIEWSOF GEOPHYSICS Dixon' THE GLOBALPOSITIONING SYSTEM ß 251 error sources.For example,in simplifiedform the phase wherem• andm 2 are the angularfrequencies (m = 2•rf) observable, sometimes called integrated Doppler or associatedwith the L1 andL2 carriersand A• andA c are accumulateddelta range, can be written(in unitsof cycles) the relative amplitudesof the P and C/A codes. The for a singlereceiver-satellite pair as amplitudeof the C/A signal is higher to facilitateinitial signalacquisition. One effect of the PRN code modula- •(t) =-f'c(t) + v(t) + a (3) tionson the carriersis to spreadthe energyof the P code signal+10.23 MHz aroundthe carrier centerfrequency; wherethe delay x(= p/c) is thegeometric delay due only to becauseof this wide bandwidththe GPS signal is often the satellite-stationgeometry, ignoring propagation effects, referred to as "spread spectrum."Figure 1 illustrates v representserrors ("noise") including propagation effects, schematicallyhow a carrier signal is biphasemodulated and a is a constant,representing a combinationof the with a PRN code.A receiverwith knowledgeof the code carrierphase cycle ambiguityand an initial phaseoffset structure and an internal clock can recover an estimate of between satellite and receiver oscillators.Equation (3) signaltransit time by cogeneratingthe codesequence and ignorestime-dependent clock errors in both the satellite performinga crosscorrelation between the receivedsignal andreceiver; also, the variouseffects represented by v and andits internalcode, determining the time delaynecessary a mustbe consideredin moredetail. King et al. [1985] and to matchthe two sequences. Leick [1990] give a completederivation of the phase P-CODE CHIP 1 2 3 4 5 6 7 8 9 1011121314-.. observableequations. SEQUENCE

C/A CODE 1 1 1 1 1 1 1 1 1 1 0 0 0 0... SignalStructure VALUE +1 The followingdiscussion is abbreviatedfrom Spilker C/A CODE [1978]. GPS satellites transmit two L-band carrier SCHEMATIC

frequencies,each modulatedby severallower frequency P-CODE 1 0 1 1 0 1 0 0 1 0 0 0 1 1"' signals(Table 1). The carriers(L1 at 1.57542GHz andL2 VALUE at 1.22760 GHz) are coherentmultiples of a 10.23-MHz P-CODE atomicclock, a stableoscillator that providesa frequency SCHEMATIC standardon each satellite(L1 = 154 x 10.23 MHz, L2 = 1 "CHIP"=9.77 x 10'8 sec - 54 L1 CYCLES 120 x 10.23 MHz). The clock frequencyis actually set -- i i i i slightlylower than thisnominal frequency to accountfor I I I _ I I I I I relativisticeffects [McCaskill and Buisson,1985] so that I I I I I I MODULATED an observeron the ground"sees" 10.23 MHz almost CARRIER(C/A) I, I I I I I exactly. The L1 carrier has two components.The "inphase" componentis modulatedby the precision(P) code. A lower frequencycoarse/acquisition (C/A) code is modu- lated in quadrature,i.e., on the same carrier frequency I SHh=Tr ß = P-CODE = C/ACODE shifted90 ø (Figure1). TheL2 carrieris normallymodu- TRANSITION TRANSITION lated only with the P code. All three carriersalso are modulatedwith a low bit rate (50 Hz) data streamtrans- Figure 1. Schematic illustration of biphase and quadrature mittingsatellite health, ephemeris, and other housekeeping modulationof 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) pseudorandomnoise (PRN) consideredas squarewaves with valuesof +1 and are codes. Small circles on modulated carriers denote phase termedpseudorandom noise (PRN) codesbecause they inversionpoints. Note 90 ø phaseshift between P andC/A havesufficiently long repeat times that they appear random modulations.In this illustrationthere is only one carriercycle per to a user without knowledge of code structure.For P codechip. In actualitythere are 154L1 cyclesper P codechip. example,the P coderepeat time is 37 weeks.Each satellite is assigneda unique1-week portion of thecode, and, since Receiverdesign affects the type of observablethat can the number of active satellites will not exceed 24, each be extractedfrom the spectrumof GPS signals.There are satellite can be uniquely identified by a PRN number currentlytwo basicreceiver architectures in commonuse correspondingto thecode portion transmitted. for high-precisiongeodesy, code correlating(as described We canrepresent L 1 andL2 signalsas above)and codeless,where only the carrierphase observ- able is available. Code-correlatingreceivers can recover Sl(t) =A•,t P(t)D(t) cos(tot t) ("reconstruct")the phaseobservable as a by-productof the (4) correlationprocess. Recovery of carrierphase without code + AcC(t)D(t) sin (tot t) knowledgerequires a nonlineardetection scheme such as signalsquaring. In effect, the signalis multipliedby itself, S2(t) = A•,2P(t)D(t) cos(0•2 t) (5) making the original phase inversions(equivalent to an 252 ß Dixon: THE GLOBAL POSITIONING SYSTEM 29, 2/REVIEWS OF GEOPHYSICS

amplitudechange of +1) unity,giving the secondharmonic 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 positioningand of the codelessapproach is that the high-precisionpart of associatedconcepts that enable high-accuracygeodetic the signal (the carrier phase) can be utilized without measurementswith GPS. Relative positioninginvolves knowledgeof the classifiedP code, which may not be simultaneousobservation of a group of satellitesby a availablein the future.Disadvantages include reduction of network of groundreceivers. Three-dimensional vectors, signal-to-noiseratio, possiblyimportant under marginal termed baselines, are def'med between all stations in the observingconditions such as periodsof high ionospheric network, relative to one or more fixed stations whose activity,and reducedeffective wavelength, making carrier positionsare known a priori. The combinationof simul- phasecycle ambiguityresolution more difficult. Also, P taneousnetwork observations and the analytical techniques code data, which are otherwise useful for clock designedto accommodatesuch data enables us to eliminate synchronization,data editing, and, dependingon data or greatlyreduce the errorslisted above, resulting in the quality,resolution of carrierphase cycle ambiguities,are millimeter-to centimeter-levelposition data we requirefor obviouslyunavailable. Some civilian receiversemploy a mostgeological and geophysical applications. hybridapproach, with code-correlatingcapability on L1 to recoverthe nonclassifiedC/A codefor clock synchroniza- FrequencyStandards, Time, and ReferenceFrames tion and navigationinformation, and a codelesschannel to One key to high-precisiongeodesy with GPS is recover the second harmonic of L2. simultaneoussatellite observations by a numberof ground Block 2 satelliteshave additionalsecurity features that receivers.Simultaneity in this case is defined quite will affect civilian users. "Selective availability" (SA) stringently;in 1 millisecond(ms) a stationat mid-latitudes reducespoint positioningaccuracy to about 100 m by moves more than 30 cm to the east as a result of Earth reducingthe accuracyof the broadcastephemeris, altering rotation,a GPS satellite,orbiting at about3 km/s,moves 3 the clock epoch,and ditheringthe clock frequency,thus m, and a pseudorangesignal propagates300 kin. Ul- affecting both code-correlatingand codelessreceivers. timately,we will relate observationsat widely separated "Antispoofing"(AS) will be activatedperiodically for test ground stationsto better than a microsecond(gs), al- purposesand consistsof encryptionof the P code; this though,as we shallsee, physical clock synchronization at would not affect codelessreceivers. Most aspectsof SA anythingnear this level is unnecessary.We nevertheless and AS will not causeserious impact on 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 abilityto relatewith great precision the positions of ground important aspect of SA (clock dithering) is discussed receiversanywhere on the Earth to satellitesin orbit. The below. following discussionis summarizedfrom King et al. [1985],Lambeck[1980, 1988], and Leick [1990]. BothGPS satellitesand receivers have precise "clocks," HIGH-PRECISION GEODESY: RELATIVE i.e., high-frequency,highly stableoscillators. A receiver POSITIONING might employa quartzoscillator, a mechanicalresonator that exploits the frequency-selectiveproperties of the Uncertaintiesin a GPS point positionmay be several piezoelectriceffect, with a fractionalfrequency stability meters to several tens of meters, althoughMalys and Af/fof about1 partin 10•ø per day. Most GPS satellites 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 based on figuredglobal experiment. One sourceof uncertaintyin a the energy difference between two states def'me the GPS point positionis the inherentimprecision of the P "clock."For example, the cesium clock [Essen and Parry, code group delay measurement,meter level for most 1956] is basedon the splittingof the groundelectronic receivers,although at least one recentmodel achievesa stateof cesium133, depending on whetherthe spinof the severalcentimeter precision with just severalminutes of unpairedvalence electron is parallelor antiparallelto the averaging[Melbourne, 1990]. For this reasonwe make nuclearspin. The transitionbetween these two hyperfme phase measurementson the carrier itself [Bossleret al., levelshas a frequencyof 9,192,631,770Hz andis thebasis 1980;Remondi, 1985], with an inherentprecision of 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 clockstability is theAllan (two sample) receiver, althoughground receivers are likely to have variance[Allan, 1966]. Following Thompson et al. [1986], larger biases),the atmosphere,including the frequency- if thefractional average frequency deviation jr(t) fromthe dispersiveionosphere and the nondispersivetroposphere, nominalfrequency f0 over a timeinterval x is both of whichaffect signalvelocity and thusour estimate of satellite-receiverdistance, uncertaintiesin the satellite [(t) = [{(t + x) - d•(t)]/2gfo x (6) 29, 2/REVIEWS OF GEOPHYSICS Dixon: THE GLOBAL POSITIONING SYSTEMß 253 thenthe Allan variance O'2Ais mentedby leap seconds.Thus the differencebetween GPS time andUTC increasesby integralseconds, roughly one O'2n(1;) = <[f'(t+ 1;)- f'(t)]2>/2 (7) per yearsince 1980 whenthey were setequal. We will need to relate the orbital positionsof the GPS where the angular brackets denote the infinite time satellites, computed in a celestial (--inertial) reference average.The Allan standard deviations (N/-•a2 ) of several frame, to the locationsof the groundstations, defined in an commonclocks are plottedin Figure2. Note the long-term Earth-fixed (terrestrial)reference frame. This requires stability of cesium clocks, making them attractivefor preciseknowledge of 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 exceptionalstability of hydrogenmasers, these effects in order to relate inertial and terrestrial makingthem the best choice for groundreference stations. referenceframes. In terms of causeswe can distinguish externaltorques acting on the equatorialbulge of the Earth (forcedmotion), associatedwith relative orbital motion of the Earth and moon about the Sun, and free motion, associatedwith the responseof the Earth to internalmass redistribution and correspondingangular momentum exchangesin the Earth system,including the hydrosphere and atmosphere.Effects include changesin spin axis directionwith respectto inertial space(precession and nutation),spin axis changeswith respectto the Earth's crest(polar motion), and changesin spinrate. Following Sovers and Border [1990], it is convenient to consider the transformation from a terrestrial to inertial reference frame as a series of rotation matrices correspondingto these majoreffects. -16 0 1 2 3 4 5 6 7 Precession(P) is a large-amplitude(-23.5 ø) slow circularmotion of the pole with a periodof 26,000 years, Log • (sec) causedby lunar-solarattraction on the equatorialbulge. Figure 2. Frequencystability of "best commercial"clocks, Accurate models are describedby Kaplan [1981] and measuredby thesquare root of theAllan variance (oa, equation Melbourneet al. [1983, 1985]. Nutations(N) are smaller (7)), asa functionof timeinterval x. Notethe short-term stability (maximumamplitude 9 s of arc), more rapid oscillations of quartzcrystal oscillators and the long-term stability of Cs and superimposedon thisprecession, with periodsfrom 9 days H Maser clocks. Rb, Cs, and H Maser data from Allan et al. to 18.6 years.They are excitedby externaltorques, but [1989].Quartz crystal data from Hellwig [1979]. their responseincludes a free component.Nutation series are available from the 1980 International Astronomical Now that we can measure time, let's define it more Union (IAU) theoryof nutafions[Kaplan, 1981; Seidel- precisely. Sidereal "time," the kind many people are man, 1982], basedon the theory of Wahr [1981]. Addi- familiar with, is based on the Earth's irregularrotation tional annual and shorter-periodterms are now known about its axis and is no longer used as a time standard. from very long baselineinterferometry (VLBI) observa- Ephemeristime (ET) is more regular and exploitsthe tions [Herring et al., 1986]. Correctionsto the standard periodicnature of the orbital motionsof the Earth, moon, precession-nutationmodel can be incorporatedin a and Sun. Although no longer in use, ET is related to perturbationmatrix, f•. Polar motionrepresents both free currenttime standardsbecause the atomic (SI) secondis and forced responses.In addition to contributionsfrom defined equal to the ET second,in mm defined as a externaltorques, possible internal excitationmechanisms fractionof the tropicalyear 1900.Universal time (UT1) is include large earthquakes [Slade and Yoder, 1989], a form of siderealtime, correctedfor someirregularities aseismicdeformation, motions of the atmosphereand [Aokiet al., 1982]. Coordinateduniversal time (UTC) is an hydrosphere,and core-mantlecoupling. The dominant internationallyaccepted standard based on atomic time, periodis 14 months(the Chandler wobble), with additional roughly synchronizedto UT1 by adding"leap seconds" annualand shorter-periodmotions, causing the pole to whenrequired. Over the lasttwo decadesthis additionhas wander in a circle less than a few tens of meters in averagedabout one secondper year,reflecting the change diameter.Pole positionis describedby two coordinates,x of the Earth's rotationrate aboutits axis with respectto and y (correspondingtranslation matrices X and Y), gravitationalorbital motion (the basisof the ET, andhence representingrotations about two orthogonalaxes lying in atomic,second). GPS satellitesbroadcast time signalsthat theequatorial plane (1 cm of polarmotion corresponds to a are synchronizedwithin 1 gs of UTC but are not incre- rotation of about 0.3 millarc seconds(mas). The time- 254 ß Dixon: THE GLOBAL POSITIONING SYSTEM 29, 2/REVIEWS OF GEOPHYSICS

varyingrotation rate 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-periodvariations could only 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 referenceframe with comparableprecision. representedby the differencebetween atomic time (UTC) The completetransformation from a positionvex:tor in a and UT1. Time seriesof LOD or UT1-UTC show strong terrestrialreference frame r r to aninertial reference frame (several millisecondsamplitude) annual and semiannual r• isgiven by periodsassociated with seasonalchanges in wind circula- tion. 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-mantle interactions whereot is an empiricalcoordinate frame scalingfactor. (seereviews by Wahr [1988], Lainbeck[1988], and Hide Note that for local GPS networks,tidal and atmospheric and Dickey [1991]). Values for both polar motion termscan be ignored,and the precisionrequirements for (precision~1 mas)and UT1-UTC (precision~0.1 mswhen the otherparameters may be lessstringent than for large averagedover a day) come from VLBI observationsof apertureor globalnetworks. quasars[Carter et al., 1985]. Satellitelaser ranging (SLR) [Cohenand Smith, 1985] providescorresponding preci- EliminatingClock Errors: Single and Double sionsof approximately2-4 masand 0.2 ms averagedover Differencing severaldays [Smith et al., 1985;Tapley et al., 1985].Lunar A GPSpoint position is limitedto meter-levelaccuracy laser ranging (LLR) providesroughly comparabledata by a combinationof uncertaintiesin clocks, orbits, and [Dickey et al., 1985]. Sensitivityanalyses indicate that atmosphericeffects. However, for a relativepositioning systematicerrors in GPS orbit and baseline estimates experimentinvolving a networkof groundstations within a causedby theselevels of uncertaintyin Earth orientation specificregion, these errors are common mode or nearlyso are negligible [Lichtenand Border, 1987; Dixon et al., and can be eliminatedor greatlyreduced in subsequent 1991c]. data processing[Bossler et al., 1980]. Consider two In summary,spin axis changeswith respectto inertial receiverssimultaneously observing two satellites.There space(precession and nutafion)are largelydue to external are four geometricrange equations and a maximumof 16 torques(forced motion), can be largein amplitude,and can observationequations, since each range could be estimated be modeledquite well. Polar motionis a spinaxis change with botha pseudorangeand carder phase equation, each with respectto the Earth's crest, consistsof both forced at two frequencies.For a single-frequencypseudorange and free motions,is much smallerin amplitudeand less measurement,R (equation(1)), denotingsatellites (id) by easily modeled.UT1-UTC likewise cannotbe modeledat superscriptsand receivers (1,2) by subscripts,we havethe presentwith sufficientaccuracy. Both polar motion and followingfor satellitei: UT1-UTC are measuredadequately for our purposesby VLBI and SLR. R1 =p• +c(Aq -At / +A •) (9) Tide-relatedelastic deformationsmay be important, i i particularlywhere high accuracyis desiredfor widely R2= P2+ c(At2- At / + Att;2) (10) separatedstations (the effectslargely cancelfor stations closeto eachother). There are threemajor effects: the solid Subtractiongives the singledifference observable R '.' Earth tide Asd (maximumamplitude -50 cm); ocean loadingeffects in coastalareas, Aoc (maximum amplitude Ri' =pi' + c(Atl - At2+ At}' ) (11) ~5cm); and the pole tide Apd, representing elastic response of the Earth to changesin spinaxis orientation(maximum wherethe primed variables denote the differential pseudo- amplitude~1 cm).Differential atmospheric loading Aat m rangeor true range and differentialpropagation delay, [Rabbeland Schuh,1986] may causeeffects at the several respectively.Satellite clock error (At •) is eliminated,and millimeter level for large station separationsbut is not differentialpropagation delay is considerablysmaller than believed to be significant for local or regional GPS theoriginal delay. Similarly, for satellitej, networks.This effect may become importantwith the adventof high-precisionglobal GPS networks. Rj'=p/'+c(Atl -At2+ An Earth-fixed reference frame based on VLBI observationsdoes not preciselydefine the locationof the Subtractionof equations(11) and (12) gives the double Earth's center of mass. However, GPS-determined difference observable R ": positionsare intimately related to GPS satellite orbits, which are sensitive to the center of mass. To account for R"= p"+ c (6t;,") (13) offsets between the VLBI-based reference frame and the actualcenter of mass,we definea vector,rc, •, to be whicheliminates receiver clock error and leavesonly incorporatedin the final transformation.SLR definesthe differentialpropagation delays as a significanterror 29, 2/REVIEWS OF GEOPHYSICS Dixon: THE GLOBAL POSITIONING SYSTEM ß 255 source,which can be calibrated,or solvedfor, independ- bending. There are two main regions to consider:the ently(see below). Note thatfor stationsreasonably close to frequency-dispersiveionosphere (--50-500 km altitude) one another(<water vapor not in hydrostaticequilibrium. The dry delay correctedphase, and the effective wavelength is •c = c/(f• is typically200-230 cm at zenith(elevation angle, 0 = + f2) -- 10.7cm (5.4 cmfor completelycodeless receivers). 90ø) at altitudes near sea level, while the zenith wet delay The shortereffective wavelength of •c impliesthat mightrange from 3 to 30 cm. The delayat otherelevation resolutionof carrierphase cycle ambiguities will be more anglesis larger,increasing approximately as 1/sin(0),but complicated.For this reason, and also becausenoise othereffects are incorporated in a "mappingfunction" for (particularlymultipath) is amplifiedby L•, it maybe greateraccuracy, including the finite height of the deskableto usesingle-frequency measurements for very atmosphere,the vertical distribution of components,Earth short(a few kilometersor less,depending on ionosphericcurvature, and my bending[e.g., Black and Eisner, 1984; activity)baselines where ionospheric effects are smallor Lanyi,1984; Davis et al., 1985].The total path delay can processthe two frequenciesseparately. be written as 29, 2/REVIEWS OF GEOPHYSICS Dixon: THE GLOBAL POSITIONING SYSTEMß 257

t,(0)= + (0) (22) have a small bias that dependson site, season,or data reductionalgorithm. wherep0 refers to thepath delay at 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, and M(0) is a mappingfunction, involving WVR calibration and stochasticestimation assumedto be azimuthallysymmetric. generally indicates no significant differences between The dry zenith delay is determinedby measurementof the two approaches(for example, Figures 3 and 4), surfacepressure. If suchmeasurements are not available, implying that at current precision levels WVR calibra- standardatmospheric conditions can be assumed,and an tion of the wet delay is not required. In fact, it is even initial estimatefor the dry zenithdelay approximated from possibleto lump the wet and dry delays togetherand the standardpressure P (in bars), based on the surface estimatethem jointly, avoiding all neutral atmosphere elevationof the station,h, anda scaleheight H, typically7 calibration(including surface pressure) entirely [Tralli km [Tralli et al., 1988]: and Lichten, 1990]. This works at current levels of GPS precisionbecause the mappingfunctions for wet anddry P = 1.013e -4'm (23) delaysare very similarabove 10 ø or 15ø, the typical cutoff angle usedwith GPS to avoid groundmultipath Subsequentanalysis of the GPS data can improve the (see below). initial estimate and reduce the effect of errors in this In spite of the success of stochastic estimation parameteron thebaseline estimates. techniques,for GPS experimentsin regionsof high wet The wet zenith delay can be estimatedin at least three path delay and high variability (for example, some ways:by measurementof surfacetemperature and relative conditionsin the tropics), troposphericcalibration is humiditycoupled with a simpleatmospheric model [e.g., probablythe dominanterror sourcefor baselinesin the Chao, 1974]; with a water vapor radiometer(WVR), an geologicallyimportant length range of several tens to instrumentthat measuresatmospheric blackbody radiation several hundred kilometers. Orbital effects dominate at in the microwaveregion, which is affectedby a rotational longer baseline lengths, and receiver effects and other moleculartransition of water vapor near 22.2 GHz [e.g., errors dominate at shorter baseline lengths. This is Janssen, 1985; Robinson, 1988]; and by stochastic discussedin more detail in the sectionon precisionand estimation techniques without a priori calibration, accuracy.Tropospheric calibration does not appearto be exploiting the data strength of GPS and the known a significanterror sourcein some other regions (for elevationangle dependence of the wet delay [Tralli et al., example, typical conditionsin the southwesternU.S.) 1988; Dixon and Kornreich Wolf, 1990; Dixon et al., when the wet path delay is low and not particularly 1991a]. Stochastic models make use of the fact that the variable.Future improvementsin the accuracyof GPS, wet path delay is likely, in a statisticalsense, to vary particularlyfor the vertical component,will dependin within a limited rangeover a shorttime interval.For SM, part on improvedtropospheric calibration and estimation WVR, or other a priori calibrations,residuals (wet delay techniques.By analogywith VLBI, direct line of sight corrections) can be estimated along with geodetic calibrationwith a WVR or othertechnique may improve parametersof interest.This improvesthe final resultbut GPS baseline estimates when calibration accuracy implies errorsin the calibration.Studies have shownthat exceeds~5-10 mm [Herring, 1986;Elgered et al., 1991; SM calibrationby itself can lead to unreliableGPS results Dixon and KornreichWolf, 1990]. It hasbeen difficult to [Tralli et al., 1988;Dixon et al., 1991a],confirming earlier deployhigh-accuracy WVRs to a majorityof sitesin a studiesindicating poor correlation between surface relative GPS networkbecause of the excessivecost, weight, and humidityand the wet pathdelay [Reberand Swope,1972; power consumptionof these sophisticatedinstruments. Elgered,1982]. K. Hurstet al. (Estimationof GPSbaseline Recentdevelopments in the areaof monolithicmicrowave errorsdue to imperfectretrieval of wet atmosphericdelays integratedcircuit (MMIC) technologyhave the potential to using surface meteorologymeasurements, submitted to improvethe accuracyand reducethe cost, weight, and Journalof GeophysicalResearch, 1991) compareSM and powerconsumption of WVRs, makingextensive deploy- WVR estimatesof wet zenith delay and concludethat ment of these instrumentsviable. Improved stochastic SM-basedestimates would have an rmserror proportional modelsare anotherpromising area of research.Current to the magnitudeof the delay, given by 1.2 cm + 0.08 modelsassume azimuthal symmetry in the wet delay. (SMT), where "SMT" is the SM-basedzenith delay in However,azimuthal asymmetry has been observed [Dixon centimeters.Tralli et al. [1988] and Tralli and Lichten and KornreichWolf, 1990;Rocken et al., 1991]. Estimat- [1990] suggestedthat the errorin SM calibrationcan also ing two stochastic,orthogonal spatial gradients in addition show considerabletime variation. In contrast,corrections to thestochastic wet zenith delay should improve precision to WVR calibrationare generallysmall (1-2 cm in zenith in the presenceof asymmetryand shouldbe feasiblewith wet delay) and constantor nearly so over severalhours, sufficientdata strength,presumably available with the implyingthat these instruments can give a goodindication enhancedBlock 2 constellationand receiverscapable of of temporalvariability in the wet zenith delay but may trackingmore than four satellites. 258 ß Dixon: THE GLOBAL POSITIONING SYSTEM 29, 2/REVIEWS OF GEOPHYSICS

6 _ a4o/o LIMON - LIBERIA(270 km) _

,,.:.:.:.:.:.:.:.:.:.:.:.:.:.:, :::::::::::::::::::::::::::::::::::::::::• STOCHASTICESTIMATION --

::::::::::::::::::::::::::::: • - iiiiii!iii!i!ili!iiiiiiiiiiil85%86% 87% 87%

86%

86% 87%• 86% ..• 86% 88% iii.dliiill • ilili 86%87% 88ø/* --

::::::::.,..__::i::i:::: STOCHASTICRESIDUAL

0.0 0.48 0.80 1.0 1.2 1.8 2.4 0.48 0.80 1.0 1.2 1.8 2.4 O• (cm/x/h-• O• (cm/•--• WVR CALIBRATION STOCHASTIC ESTIMATION, NO PRIOR CALIBRATION

Figure 3. Short-termrepeatability for a 270-km baselinein CostaRica with varioustreatments of wet path delay (see also Figure 4). Percentsymbols above each bar indicatenumber of biases resolved.With WVR calibration(left side), estimationof a residualdelay, either as constantor with a stochasticmodel specified by therandom walk parameter, a (cm/•), improvesthe calibration,but only to the level reachedin the casewhere no calibrationis used(right side) and the entiredelay is estimatedstochastically [from Dixon and KornreichWolf, 1990].

Multipath range observables with systematic, time-dependent Antennas for almost all GPS receivers are omnidirec- sinusoidal signals associatedwith variable receiver- tional, enabling signals from several satellitesto be satellitegeometry over a pass.The magnitudeof multipath received simultaneously.Depending on the immediate is roughlyproportional to wavelength,and the effectsare environmentand the gain patternat low-elevationangles, considerablylarger for P code pseudorangerelative to suchantennas can be susceptibleto interference("multi- carrier phase.Low-angle observationstend to be most path")from multiple arrivals of thesame signal because of affected,and for thisreason a cut-offangle of 10ø-20ø reflectionsfrom nearbyobjects. Reflections at the satellite above the horizon is usually employed.This has the alsooccur [Young et al., 1985]but canbe ignoredfor most fortunateeffect of minimizingerrors due to third- and applications.Multipath corrupts the phaseand/or pseudo- higher-orderterms in the ionosphericdelay and minimiz-

LIMON - LIBERIA (270 km)

I I I I i i --•'--- WVRCALIBRATION • [ i ..,.T' ' Figure 4. Baseline estimates -O- STOCHASTIC ESTIMATION with and without WVR cali- brationof the wet path delay for thedata shown in Figure3. Estimates with WVR calibra- tion include a stochastic resi- dualmodel (a = 1.2cm/x/-•). Estimates without WVR cali-

w brationemploy a similar sto- > i chastic model for the entire wet delay. Error bars and el- lipses are one sigma [from Dixon and Kornreich Wolf, 19901. -4 -4 -4 -2 0 2 -4 -2 0 2 EAST (cm) LENGTH (cm) 29, 2/REVIEWS OF GEOPHYSICS Dixon: THE GLOBAL POSITIONING SYSTEMß 259 ing troposphericcalibration errors, including those due to nature of the perturbingforces on the satellitesand the incorrectmapping functions. However, lack of low-angle requirementfor high-accuracymeans that this model is not observationsis one of the conlributingfactors to poor adequatefor our purposes.Rizos and Stolz [1985] sum- resolution of the vertical component with GPS, so marize major accelerationson the GPS satellites.Atmos- multipathcontrol, as well as improvementsin the other phericdrag is negligibleat thesealtitudes, and the Earth's areasjust mentioned,is highlydesirable. Multipath can be nonsymmetricgravity field, while significant, is ade- minimizedthrough use of antennabackplanes or RF (radio quatelydescribed with currentmodels; the more poorly frequency)absorbent material around the base of the determinedshort wavelength components, related mainly antenna, mounting antennasclose to the ground (to to lithosphericstructures in poorly surveyedparts of the minimizethe effectsof groundreflection), and carefulsite Earth,have little effect on the high-altitudeGPS satellites. selection,choosing sites well away from planar-reflecting Additionallarge perturbationsare the gravitationaleffects surfacessuch as buildingsor vehicles. of the moonand Sun,which can be accuratelydetermined, Since multipathis related mainly to the geometryof and solarradiation pressure. nearbyobjects, and sincethis geometry is usuallyconstant One approachto the orbit problemis to begin with an over several days or longer, the temporal signatureof estimateof the six orbitalcomponents at someinitial time multipathtends to repeatfrom day to day, retardedby an and numericallyintegrate the equationsof motionusing amountequal to the offsetrise time of the satellites(4 min accuratemodels for variouspertubing forces and resulting earlier each day). The repetitivenature of multipath in accelerations,predicting location and velocity of the principlecan be usedto estimateand correctits major satellitesat later epochs;ground tracking data can be used effects, though few such studieshave been reported.If in a least squaresadjustment to improve the initial uncorrected,multipath can be an important source of position/velocityestimates, the modelparameters, and the systematicerror, dependingon environment,antenna/ subsequentposition/velocity estimates. The length of the backplanedesign, and lengthof observingsession. Carrier orbitarc over whichthe equationsare integratedmay be phasemultipath tends to have periodsshorter than 10-20 only a few hours ("short arc") in which case the force min, thus observationsover severalhours or longer will modelscan be relativelysimple, or may extendto several averageout mostof theeffects. weeks("multiday" or "long arc"), requitingsophisticated forcemodels. Solar radiation pressure produces a largeand Orbits somewhatunpredictable perturbing force becauseof the GPS satellites orbit about three Earth radii above the large cross-sectionarea of the solar panels, complex surfacein six orbitalplanes. For high-precisiongeodesy it satellitegeometry, and variablesatellite albedo, resulting is necessaryto know precisely the positions of the inaccelerations oforder 10-7m/s 2and perturbing theorbits satellitesat the time of observation,with acceptableerror a manymeters in just a few hours.Although a solarradiation functionof desiredbaseline precision and stationsepara- pressuremodel for the GPS satellitesaccounts for albedo tion; for millimeter-levelperformance on baselineslonger andgeometry, large residuals have been observed [Fleigel than about 100 km, meter-levelprecision in the orbit et al., 1985]. It is therefore common to solve for at least estimatesis required. one additional accelerationparameter in the estimation Satellite orbits can be describedby six parametersat process,representing departures from the ideal model.In a someinitial epochand a forcemodel to definesubsequent spacecraft-centeredcoordinate system where z pointsto the time evolution. The epoch state parametersare three centerof the Earthand y is alongthe solarpanel support componentsof position (x,y,z) and three velocities in beam and normal to the spacecraft-Sundirection, un- Cartesian coordinates,or equivalently six Keplerian modeledaccelerations are often observed 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 misalignedsolar panels [Fleigel et al., 1985; Schutzet al., parameterdescribing the positionof the satellitewithin 1990]. Residualstend to be largestduring the 2-month that orbit (for example,M, the "mean anomaly"or f, the eclipseseason, when the spacecraftperiodically enter 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 pressurecoefficients in all three describingthe positionand orientationof the orbitalplane. componentsfor data arcs up to 1 week, usingconstant In the absenceof additionaldisturbing forces, only the correctionsto the nominalmodel. Lichten and Bertiger anomalyis time dependent.A simpleKeplerian description [1989] usedthree componentstochastic corrections to the of an orbit in a plane can be obtainedby integrating nominalmodel for dataarcs longer than 1 week. Newton'slaws of motion,giving Informationon satellitetrajectories during the period of theexperiment may comefrom the broadcastephemeris in r = a(1 - e2)/[1+ e cos(f)] the GPS signal, or additional GPS data taken simul- taneouslyfrom siteswhose positions are well knownfrom where r is radial distance from the center of the Earth independentmeasurements such as VLBI or SLR. The [Kaula, 1966]. For near-circularorbits like GPS, e--0. The resulting tracking data allow generation of accurate 260 ß Dixon- THE GLOBAL POSITIONING SYSTEM 29, 2/REVIEWS OF GEOPHYSICS

ephemerides(satellite locations as a functionof time) and vantageousto employa multidayarc analysis,instead of define the reference frame for the experiment. The the singleday, short arc approach.The advantagesof subnetworkof known groundstations is called a fiducial multidayarc analysisare describedby Lichtenand Border network. Uncertainties in fiducial site location, or in the [1987]. With observationsover more than one revolution, groundsurveys connecting the phasecenters of the large orbital periodsare more accuratelydetermined and the VLBI antennasto the ground mark used by the GPS antenna,or low-quality GPS data at one or more of these 12 sites during an experiment,are a major sourceof sys- tematicerror in GPS geodesy. VERYICALIO' 10 o Thatpart of theerror in a baselineestimate •t,t dueto orbiterror •oa, can be estimated from the rule of thumb -- •oa,L/h, whereL is baselinelength, h is satellite I o altitude,-20,000km, and • is a geometricfactor (4).2) [Lichten, 1990a]. Since the error in the broadcast Siope• ephemeriscan be 50-100 m with selectiveavailability, we =1x10 -9' w 0 Ilb•)Slope ,,. requirea robustfiducial networkfor any experimentwith o_ •o o baselineslonger than about 50-100 km if we desire w II o centimeterlevel or better accuracy.The nature of the requirednetwork depends on the experiment.In general,at • I I . I t leastthree stations are required,with adequatenorth-south and east-west extent. Stations too close to each other ß I ß provide redundantinformation and may not contribute NO'RTH'I U.S. only greatly to network strength.On the other hand, mutual Global network satellite visibility is deskable for cancellationof clock errors,one or two redundantstations protect against data outages,and some pmximal stationscan facilitate cycle

ambiguityresolution. The optimumnumber and geometry 4 of stations may be difficult to predict. A covariance analysis,which estimatesuncertainties in baselinevectors from certainassumptions about the dataand its errors,can 2 be useful in testingtrial fiducial networks[Dixon et al., 1985;Freymueller and Golombek,1988]. In some cases,covariance analyses may suggestthe o need for fiducial sites in areas where VLBI or SLR sites are not available,or where the necessaryground tie data EASTI I .... 0 , ß are not available.It may still be advantageousto deploya 0 receiverto sucha locationto provideadditional geometric strengthto the trackingdata. Kornreich Wolf et al. [1990] . a$\•/- used data from the "CASA" experimentto show that addingtwo trackingsites in the southwestPacific and two in Europeto a fiducialnetwork consisting of threestations .• - in the U.S. improvedthe repeatabilityof the horizontal componentsof long (>400 kin) baseline estimatesin o o o 1 CentralAmerica and northernSouth America (Figure 5), / ' even thoughthe actual locationof 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 introducingsystematic error. In spiteof our bestefforts to establisha robustfiducial BASELINE LENGTH (km) network we may find that logisticallimitations or data outageslimit the quality of the resultingobservations. An Figure5. Improvementin short-term (3-8 days)repeatability (equation(30)) for seventeenbaselines in Central and northern importantclue to the causeof low-quality data can be SouthAmerica as a functionof fiducialnetwork configuration. found by plotting repeatability(day-to-day scatter) of Single-dayorbital arcs are used."U.S. only" indicatesthree baselineestimates asa function ofbaseline length; ahigh VLBI sims inthe U.S. "Global" indicates thisnetwork supple- dependenceonlength (>2-3 parts in l0s for baselines mented bytwo European andtwo southwest Pacificstations. Best longerthan 100-200 kin) suggests thepossible influence fit straight lines are shown (modified from Kornreich Wolfet al., 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 nodesbetter defined, with cor- six-stationGPS global trackingnetwork, expectedto be respondingimprovement in the ability to resolvecarder operationalin 1991, for trackingthe U.S.-FrenchTOPEX- phasecycle ambiguitiesthroughout the networkand the Poseidonsatellite, a low Earth orbiterfor oceanographic precisionand accuracy of all componentsof the baseline researchto be launchedin 1992.Neilan et al. [1990] givea vector.Figure 6 showsthe repeatability of a setof baseline currentsummary of internationaltracking networks. vectors from the northern Caribbean before and after a multiday arc analysis [Dixon et al., 1991a]. Geodetic Reducingthe Data estimateswere improvedby factorsof 2 or morewith the Considera hypotheticalGPS experimentinvolving 5 longer arcs for baselineslonger than about 500 kin. daysof observationsat 10 groundstations, with the posi- Single-dayarcs and the existingthree-four station fiducial tionsof threefiducial sites well known.Each station might network were apparently adequate to minimize the observesix satellitesfor an averageof 5 hourseach day contribution of orbit error for shorter baselines. (the total time spanof observationsmight be 8 hoursor more, but not all satellitesare visible simultaneously). BIASES RESOLVED EAST Pseudorangeand/or phase data are collectedcontinuously,

E3 SINGLE DAY ARCS 5 but let's considerphase data averagedat 1-min intervals. []MULTI DAY ARCS The totalnumber of "datapoints" collected in 1 day is then 4 10 stationsx 6 satellitesx 5 hoursx 60 pointsper hour= 3 18,000. From this large data set we might estimatethree position componentsfor each unknown ground station (21), six components(three position, three velocity at the initial epoch)for eachsatellite (36), a y bias term for each $ • o satellite(6), perhapsone troposphereparameter for each

NORTH groundstation (10) (additionalterms are requiredfor a stochasticmodel), and clock terms every minute for all satellites(1800) and nine groundstations (2700), sinceone stationacts as a referenceclock, giving a total of 4573 un- knowns.The systemis overdetermined,with data points greatlyexceeding unknowns. Since the dataare noisy,we cannotformulate simultaneous equations to solvedirectly for the desiredparameters. The challengeis to find, in a least squaressense, the best estimateof the parameters

VERTICAL from availabledata, using where possible additional infor- mation suchas troposphericcalibration and orbit models. Thereare numerous approaches to GPS dataanalysis, and I havetried to generalizethe followingdiscussion. However, 6 somedetails are specificto analyticalapproaches familiar to me, and I apologizefor thisobvious bias. The timeevolution of pseudorangeor phaseobservables (e.g., equation(3)) is largely a function of observation 0 - , , , • 200 300 400 51•0 9;0 1100 1300 geometry.Given prior knowledge of the approximate BASELINE LENGTH (km) ground stationpositions in a terrestrialreference frame, satellite orbits (from the broadcastephemeris), Earth Figure 6. Histogramsof short-termrepeatability (equation (30)) orientation,and initial estimatesof atmosphericdelay, it is for 15 baselinesin the northernCaribbean for singleday and possible to generate fairly accurate models for the multiday orbital arc analysis.Carrier phasecycle ambiguities observables.With thesemodels, and knowledge of how the ("biases")are resolved for eachtechnique to the extentpossible. observableob would changegiven a small changein a Improvementin the longer (>500 km) baselineswith multiday parameter,x, (given by the partial derivative,•5(ob)/fx), arcsresults in part from the increasedability to resolvethe biases leastsquares or Kalmanfilter techniquescan be appliedto [from Dixon et al., 1991a]. the observablesto improveestimates of x. FollowingKing et al. [1985], givenan observablemodel c, an observation The burdenof establishingadequate fiducial networks l, andan observationerror v (all functionsof time), for regionalGPS experimentswill be considerablyreduced in the next few yearsas variousnational and international c = l + v (25) agenciesestablish global tracking networks. The Coopera- tive InternationalGPS Network has been in operationwith This canbe linearizedby expandingthe left-handside in a six or more stations since 1989, administered in the U.S. Taylorseries and dropping second- and higher-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., basedon a priorivalues of x, x•, plus incremental constant parameters (y), a vector of process noise corrections,[?ffob)/15x]dx, to be constrainedby the dataand parameters(p) and v, definedhere as a zero mean white its errors.The linearizedobservation equation in matrix noise vector: form is Z= AxX +At, P+AyY+v (28) c. + A dx= l + v (26) where z is the difference between an observation and its where A (the design matrix) is the matrix of partial modelvalue. Parametersmodeled as processnoise include derivatives,dx is a vectorof smallcorrections to x., andv clocks(modeled as white noise)and wet troposphericpath is now the vector of postfit residuals.The quantity delays (modeledas random walks). Measurementsare minimizedis theweighted sum of squaredresiduals, given groupedinto finite time intervalsor batches(typically one byvrWv, where W, theweight matfix, is theinverse of P, to severalminutes) and processedsequentially, leading to the covariance matfix. The solution is updatedparameter estimates and covariancesafter each batch.After filteringall dataa smoothingalgorithm can be • = x. + dx (27) applied that works backward in time to update the estimatesand covariances,for example, allowing inves- where• isthe best estimate ofx, dx= N-1 ArW(/- c.), tigationof the time-varyingbehavior of the troposphere andN isthe normal equation matrix given by ArWA. This [e.g.,Dixon andKornreich Wolf, 1990]. approachmay be appliedto both differencedand undif- We sawfrom equations(2) and (3) that the initial phase ferenceddata. It is alsopossible to transformundifferenced measurementupon acquisitionof the carder signal is data to derived parameterscontaining no clock terms biased by an unknown number of cycles. Assuminga through orthogonalizationmethods without forming receivermaintains lock on the signal, the range change normalequations [Lawson and Hanson,1974]. Additional between the receiver and satellite can be determined, and discussionof GPS data analysisapproaches can be found the initial range (the cycle ambiguity) can be estimated in worksby Goad [1985],Bock et al., [1985a,b], Lindler along with the geodeticparameters of interest.However, and Wells [1985], Bock et al. [1986], Beutler et al. [1986, this degradesthe accuracyof horizontalbaseline compo- 1987],Schaffrin and Graffarend [1986], and Leick [1990]. nents relative to the case where the cycle ambiguity is Most of the datadiscussed in thisreport were processed 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 processesundifferenced data 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, conventionalKalman filter updatesmeasurements from troposphericeffects (Figure 7), multipath,and other error one observationepoch to the next on the basisof the sourcescan corruptGPS signalssuch that errorsbecome covariance matrix P. Factorization of P into upper significantrelative to one half wavelength,affecting our triangular(U) anddiagonal (D) matrices(P = UDUr) abilityto fix ambiguitiesto thecorrect value. It is common improvesaccuracy and computational efficiency [Thornton practiceto resolvethe ambiguitiesfirst on shorter(<100 and Bierman, 1980] and is the basis for the parameter kin) baselineswhere the effect of most of theseerrors is estimation.The formulationof the linearizedleast squares reduced[Bock et al., 1985b; Abbot and Counselman,1987; observationequation differs slightly from (26), partitioning Dong andBock, 1989;Blewitt, 1989].The methodused for parametersinto a satellitestate vector (x), a vectorof mostof the data presentedhere also exploitsthe fact that

lOO

L!.I X ,'7' 95 Figure 7. Percentage of carder phase cycle ambiguities(biases) resolved as a functionof the stochastic(random walk) modelused to characterize C9 90 the wet zenithpath delayfor a GPS networkin the northernCaribbean. Optimum model has random walkparameter ct= 1.0cmP•-• (compare toFigure C.) 85 3) [from Dixon et al., 1991a].

80 0

RANDOMWALK PARAMETER (a.. cm/,J'•'r) 29, 2/REVIEWS OF GEOPHYSICS Dixon' THE GLOBALPOSITIONING SYSTEM ß 263 the ionosphericgroup delay of the P codemodulation is et al. [1991a, b], Blewitt [1990], and Beyis [1991]. From the samemagnitude (though opposite sign) as the phase ourdiscussion of theprecision of carderphase measure- delay,provided the correct number of cyclesis associated ments and ionosphericeffects it should be clear that with thephase measurement [Melbourne, 1985; Wubbena, phase-tracking,dual-frequency receivers are critical. P 1985].Blewitt [ 1989]gives a completediscussion. codepseudorange capability is usefulbut not critical for Another"bootstrap" technique that assistsbias resolu- editingcycle slips,high-precision clock synchronization, tion involvesthe "wide-lane"ambiguity. A linearcombi- andcarder phase cycle ambiguityresolution. If P codeis nationof observables,the wide-lane phase L,•, defined as not present,C/A code pseudorangecan provide clock synchronizationto the microsecondlevel or better. The Lw =- (•)1- 1•2)•Lw ability to track six or more satellitesis helpful.If possible, (29) commonreceivers and antennas should be usedthroughout = (f 1L1 -- f2 L2 )/fl -- f2 ) the network to ensure common time tagging in the presenceof SA and reducethe effect of antennaphase isuseful because the wide-lane wavelength, • = c/(f1 -f2) centervariation [Rockenand Meertens, 1989]. -• 86 cm, is more easilyresolved than the "narrow-lane" In addition to adequatelysampling the geophysical (•1, L2)ambiguities (compare equations (20) and (29) and phenomenonof interest,two other aspectsof network definitionsof • and•). Subsequenttechniques can be designare important.First, if baselineslonger than about appliediteratively to resolvethe narrow-laneambiguities 50-100 km are to be determinedwith high accuracy,an (X•-- 10.7cm) as discussed above. With P codereceivers adequate (three or more station) fiducial network is the wide-laneambiguity may be resolvableeven if the P important.A covarianceanalysis can be performedto codedata are relativelynoisy (P codeerror averaged over determine optimum configuration. Variations in the severalhours just needs to be less than •/2, i.e.,<43 cm). number,geometry, and quality of fiducialstations between Codelessreceivers have an effectivewide-lane wavelength experimentscan lead to large, spuriousvariations in of only 43 cm, thus the wide-laneambiguity is more geodeticposition estimates [Larson, 1990a]. Second,the difficult to resolve. regionof interestmust be coveredwith a sufficientlydense Most high-precisionrelative positioningexperiments networkof stationsto ensureresolution of carrierphase with GPS are "static,"in the sensethat a given receiver cycleambiguities. If a largeregion must be coveredwith a observesin oneposition for manyhours or daysto obtain sparsenetwork, at least 2-3 baselinesshould be shorter datastrength adequate to resolvethe cycle ambiguities and than 100 km. If necessary,the networkcan be coveredin reduce other errors. However, over short (<30 km) two or more observingsessions with a few common baselines,it is also possibleto operatein a "kinematic" stations[e.g., Kellogg et al., 1989]. mode, with a receivermaintaining lock on the satellite With the satelliteconstellation available up until 1990, signalsand movingfrom stationto station,continuously high-precisionresults have been achieved in several recordingdata. As long as lock is maintained,only one regionswith 3-5 daysor moreof observation,assuming at cycle ambiguityper receiver-satellitepair needsto be least 7-8 hours of observationper day [e.g., Tralli and resolved,regardless of the numberof stationscovered by Dixon, 1988; Dong and Bock, 1989; Dixon et al., 1990; theroving receiver. The techniquehas obvious advantages Kornreich Wolf et al., 1990; Kellogg et al., 1990]. for rapidsurvey of largenumbers of points.A variationof Observingfor two or more days allows a check of data this approach,"rapid static" surveying, does not require qualityby treatingeach day independentlyin the subse- continuoussignal lock, exploiting high-quality P codedata quentanalysis and comparing results and reduces the effect or priordata to resolvethe ambiguitiesdirectly [Blewitt et of random error by averaging.Another advantageof al., 1989]. severaldays of observationis that multiday orbital arc The assumptionof linearity (equations(26) and (28)) analysisis possible. means that the initial model for the observables must be The durationof observationsin a givenday dependson sufficientlyaccurate that second-and higher-orderterms experimentlocation and satellite constellation geometry, as canbe neglected.Accuracy can be estimatedafter the fact well as receiverdesign, multipath environment, and by inspectingthe solutionoffset from the assigned a priori logistical considerations.Experience with the Block 1 value. The solutions should be iterated unless the satellite constellationusing mainly four-channelreceivers allows epochstates are accurateto about1 km, andinitial ground somegeneralizations. King andBlewitt [1990] suggestthat positionsaccurate to severalhundred meters [Lichten, observingfor 6.5 hours or less per day may degrade 1990b]. results.Freymueller and Golombek[1988] compared7- and9-hour observationsfor a GPS experimentin northern ExperimentDesign SouthAmerica with covarianceanalyses and suggested On the basis of the preceding discussionwe can improvementwith the longerobserving time. Davis et al. summarizethe importantaspects of a successfulhigh- [1989] reportedthat carrierphase cycle ambiguitiescould precisionGPS geodetic experiment. Additional discussions not be resolved reliably even on short (30-50 km) canbe foundin worksby King and Blewitt [1990],Dixon baselineswith short(4.5 hour or less)observing sessions. 264 ß Dixon:THE GLOBAL POSITIONING SYSTEM 29, 2/REVIEWSOF GEOPHYSICS

Optimumobserving times may decreaseonce the Block 2 addressmainly "best achievable" accuracies; in somecases constellationapproaches operational capability, assuming an applicationmay have less stringent accuracy require- receiversare capableof trackingmore than four satellites ments,and considerable savings in fieldoperation and data anddepending on SA effects.High-quality P codedata, if analysiscosts may be possibleby acceptingslightly available,could also shorten observing times by providing reducedaccuracy. constraintson carrier phase cycle ambiguities. Bothprecision and accuracy of GPSbaseline estimates Finally,site selectionand siteintegrity are important. dependstrongly on whether or notthe carrier phase cycle Sitesshould have adequate sky visibilityand be located ambiguitieshave been resolved [Blewitt, 1989, Dang and away from buildingsor other reflectingsurfaces that Back, 1989]. Bias fixing improvesthe precisionand introducemultipath. Instability of the groundmonument accuracyof GPShorizontal baseline component estimates maybe a substantialsource of noisein a long-termGPS by factorsof 2-3; the verticalcomponent is usually experiment,introducing spurious position shifts unrelated unaffected(Figure 8). The abilityto resolvethe biases is a to tectonicsignals, and is an importantand oftenover- good indicationthat certainerrors have been reducedto lookedaspect of GPS experimentdesign. Monuments not somethreshold value; in general,it is difficultto resolve emplacedin bedrockare most susceptible and may require thebiases with lower quality data. For consistency, most of specialdesign [Wyatt, 1989]. the examplesdiscussed in this section will therefore involve bias-fixed data. As with mostphysical measurements, it is usefulto PRECISION AND ACCURACY distinguishrandom errors, the effects of which can be reducedby averagingmore data, and systematicerrors. The precisionand accuracy achievable in a givenGPS The major randomerror sourcein GPS is the noiseof the experimenthave improved greatly in the lastdecade with phaseobservable. Systematic errors are of twotypes. The betterexperiment design, receiver hardware, and analytical firstis due to errors in theinput (nonestimated) parameters, techniques,and furtherimprovements are likely. I will for example,fiducial site location, Earth orientation, and

6 6 SINGLEDAY ARCS EAST 5 [] BIAS ESTIMATED [] BIAS-FIXED 4 4

3 3

2

1 1

0 - i , 0 Figure8. Histogramsof short-term repeatability (equation(30)) for 15 baselinesin the northern 4 NORTH Caribbeanbefore and after resolutionof carrier phasecycle ambiguities(bias fixing). Note improvementin horizontal components forbaselines shorterthan about500 kin. Biasescould not be resolvedon thelonger baselines with single-day orbitalarcs (compare to Figure6) [fromDixon et al., 199la].

VERTICAL

i i ß 200 300 11 oo 1300 BASELINE LENGTH (km) 29, 2/REVIEWS OF GEOPHYSICS Dixon: THE GLOBAL POSITIONING SYSTEMß 265 geocentercoordinates. Sensitivity analyses suggest that repeatability,the scatterof a seriesof GPS measurements only fiducial site locationerror is significant[Lichten and takenover monthsor years,which will show the effectsof Border, 1987; Dixon et al., 1991c]. The secondtype of slowly varying errors.For stationsnot separatedby an systematicerror is associatedwith modelsconstraining the activefault, no motionis assumedon thebaseline, and (30) estimatedparameters. For example,errors in modelsof the can be used.For stationsseparated by an active fault we Earth's gravity field and the solarflux causeerrors in the canassume steady motion and look at the scatterof points subsequentorbit and baseline estimates. Sensitivity abouta best fit straightline on a plot of positionversus analysessuggest that theseorbit-related errors are negli- time [Davis et al., 1989; Larson, 1990b]. An important gible. Other systematicerrors of this type result from questionis the relation between short- and long-term assumptionsand modelsconcerning the atmosphere,for repeatability. example, ignoring third- and higher-orderterms of the The overall precisionof an individual GPS baseline ionosphericcorrection, use of incorrectmapping functions estimatewill dependon severalfactors, whose contribution (both dry and we0 especiallyat low elevationangles, and to the total will be the squareroot of the summedsquares the assumptionof azimuthalsymmetry in thisdynamic and (rss) of the individual error terms. Some of these errors nonhydrostaticmedium. Although it is difficult at present will be constant,while otherswill dependon baseline to quantify these systematicerrors, the data presented length.A simpleformalism first proposedby Savageand belowsuggest some bounds. Prescott [1973] to describethe precisionof Geodolite One measureof the uncertaintyof a GPS vectorbaseline distancemeasuring equipment is estimate is the formal error, based on propagationof randomdata noisethrough the estimationprocess. Formal • = (a2 + b2L2)1/2 (31) error basedonly on data noise,even if scaledto reflect postfit residuals,underestimates true error as it does not wherec• is thestandard deviation, L is baselinelength, and accountfor all randomerrors nor most systematicerrors. a and b are the constantand length-dependentsources of However,formal error doesallow us to propagateuncer- error,respectively, with a = 3 mm andb = 2 x 10-7. For taintiesand describecorrelations in a statisticallyrigorous comparison,the length componentof a VLBI position manner. vectorhas a = 5.4 mm and b = 2.4 x 10-9 [Clark et al., Assumingthat eachday of datais treatedindependently, 1989]. Equation (31) has also been used to describethe the scatter of daily solutions for a GPS experiment precisionof GPS measurementsas a functionof baseline spanningseveral days (the "short-term"repeatability) is length[Dixon et al., 1990;Hager et al., 1991]. As we shah anothermeasure of GPS performanceand is one indicator see,this simple error model has weaknesses when applied of precision.The repeatabilityR of a vector baseline to GPS,but is adequateto addresssome basic questions, in component(east, north, or vertical)is theroot-mean-square particular the relation between short- and long-term (rms)scatter about the weightedmean: repeatability.Best fitting curves (31) through some short-termrepeatability data for GPS experimentsin the southwesternU.S. [Dixon et al., 1990] and the northern i--1(ci -O•i Caribbean[Dixon et al., 1991a] give, for horizontal(north R: • (30) andeast) components, a = 3-8 mm, andb • 1-2 x 10-g (Figure9). The eastcomponent is determinedmore poorly i=1•/ than the north, becausethe satelliteground tracks were orientedmainly north-south,giving large range-change where n is the numberof days, c is the estimateof the "signals"in the north-southdirection, and smallersignals componenton dayi, (c) is theweighted average of the in the east-west direction. The difference is reduced with componentestimate over all days,and •/ is the formal bias-fixeddata. The verticalcomponent is the mostpoorly error. Short-termrepeatability has been used to assessthe determinedwith GPS, reflectingthe geometriclimitation relative merits of various experimentaland analytical that satellitesare observedin the upperhemisphere only, approaches[Tralli and Dixon, 1988; Dang and Back, andhigher sensitivity (about a factorof 3 relativehorizonø 1989;Blewitt, 1989;Dixon et al., 1991a].Note that if just tal components)to errors in troposphericcalibration a few days of data are availablefor a small numberof [Herring, 1986]. Typical valuesfor verticalrepeatability baselines,the statisticalsignificance of theobserved scatter are in the range 20-60 mm, with the higher values may be limited. Also, short-termrepeatability will not indicativeof highand variable wet troposphericpath delay reveala slowly varyingsystematic error, for example,due (Figure9). Verticalrepeatability is not a strongfunction of to propagationdelay, multipath, or fiducial network baselinelength and thus is notwell fit by equation(31). inconsistencies,which might be constantover several Thequestion of short-versus long-term repeatability can days.On the other hand, sincerandom errors are reduced be partiallyanswered by comparingthese data to datafrom with repeatedmeasurements, repeatability might actually laterexperiments. Figure 10 showsresults from sevenGPS be pessimistic.With moredata now availableit is possible experimentsconducted over a 3-year period between to use another indicator of precision,the "long-term" Mojave (on the North Americanplate) and Vandenberg 266 ß Dixon: THE GLOBAL POSITIONING SYSTEM 29, 2 / REVIEWSOF GEOPHYSICS

overa 2-yearperiod of 3-8 mm (north),3-6 mm (eas0,

+ Vertical and11-33 mm(vertical). Thus it appearsthat for carefully [] East configured experiments(for example, robust fiducial 6 ß Nort,h networks,observing sessions of adequatelength, resolved carderphase cycle ambiguities), the repeatability observed over severalyears is not significantlydifferent from that 4 + observedover several days. These are encouraging results, + + suggestingthat the magnitude of slowlyvarying systematic + -!- errors is similar to or less than the level of random errors + + + + •.• manifestedover several days. [] + The accuracyof any long baselinemeasurement is •.+• ..... difficult to determinerigorously, since completely 0 'i• ,-1'•,•"•11 '", ,e , I , I , 15 MOJAVE-VANDENBERG (350 km)

+ + Vert,cal + + [] East + 10 -- o•' 6 ' + + ß Nort,'-

4' + + • . ++ + o muJ _ + [] 21 ß ß o/ , , ,-, ,-,', ',1 0 200 400 600 800 1000 1200

LENGTH (km)

Figure 9. Short-term(3-8 days)repeatability (equation (30)) for experimentsconducted in 1986 in southernCalifornia (top) and the northernCaribbean (bottom). Curves are bestfit (equation (31)) throughhorizontal components only. Note the higher scatterin the vertical componentestimates for the northern Caribbean,in part reflectinghigher levels of tropospheric -5 calibrationerror [from Dixon et al., 1990, 1991a]. 30

(on the Pacificplate), a baselinewhere VLBI dataare also available.Straight-line fits throughboth data sets show 20 northwestmotion of Vandenbergwith respectto Mojave,at a rateand direction broadly consistent with whatwe expect 10 to seefrom a globalplate model [DeMets et al., 1990].The scatter of the GPS data about the best fit line is 6 mm

(north), 10 mm (east),and 34 mm (vertical),not greatly 0 differentfrom the short-termrepeatability noted in Figure9 for a baselineof thislength and in fact similarto the VLBI scatter[Dixon et al., 1990].This suggeststhat at leastin this -10 I i I • I location,for thisperiod, the contributionof slowlyvarying 1986 987 1988 1989 systematicerrors was low.Davis et al. [1989]investigated YEAR long-term(~2 year) repeatabilityfor six short(<50 km) baselinesand a singlelonger (223 km) baselineand found Figure 10. Comparisonof GPS and VLBI results for the rms scattersabout best f'lt lines of 2-12 mm (north),2-19 Mohave-Vandenbergbaseline in California between 1986 and mm (eas0, and 4-49 mm (vertical),where the larger 1988. Solid line is VLBI global solutionGLB223; data from values were for nonbias-fixed cases. Larson [1990b] individualVLBI experimentsomitted for clarityand have scatter presenteddata on long-termrepeatability for 10 baselines similarto GPS data (from Larson [1990a] reprintedin the work between91 and 464 km in lengthand foundrms scatters by Dixon et al. [1990]). 29, 2/REVIEWS OF GEOPHYSICS Dixon: THE GLOBAL POSITIONING SYSTEMß 267 independenttechniques of known accuracy are not are independentof baselinelength. Atmospheric (mainly available for comparison.However, if two independent tropospheric)errors depend on baselinelength only up to a techniquesprovide similar results, we can be more certain distance, after which atmosphericvariation is confident in each. For baselines <50 km in length, uncorrelatedbetween two sitesand the correspondingerror conventionalterrestrial geodetic techniques can be used. reachesan approximatelyconstant value, dependingon Davis et al. [1989] reportcomparisons with creepmeters calibrationor estimationaccuracy. The correlationlength andalignment arrays on lines~10 km in length,suggesting for troposphere-relatederrors at the level of a few mil- GPS rate accuraciesof 1-2 mm/yr. The same study limeters,i.e., at a level exceedingreceiver noise, will vary describedGPS comparisonswith Geodolitetrilateration dependingon site,season, and local weatherbut is likely to data on lines in the range 10-50 km and found no be on the orderof troposphericthickness, perhaps ranging detectable differences within the standard deviations of the from a few kilometers to a few tens of kilometers. For measurements.For longer station separations,VLBI example,Ware et al. [1985] noticedsignificant differences providesimportant data for comparison.However, VLBI in the wet delay for stationsseparated by 22 km. Orbit- and GPS are not completelyindependent, sharing, for related errors are expected to exhibit an almost linear example,a similarsensitivity to troposphericeffects. Also, dependenceon baseline length up to several thousand GPS relieson Earth orientationdata provided by VLBI, as kilometers,ata levelof 1-2 partsin 108of baseline length well as fiducial site location data. Nevertheless, VLBI- GPS comparisonsallow the best available standardfor

GPS accuracyassessment for baselineslonger than about + Vertical

50 km, and I will use the term accuracyto describethe [] East -!- -!- results of GPS-VLBI comparisons,recognizing these ß North limitations.Figure 11 [Dixon et al., 1990] showsdiffer- encesbetween VLBI and GPS baselineestimates plotted as a functionof baselinelength for one experiment,with best fit curves (equation (31) and Figure 9) summarizing short-termrepeatability in the same experiment.The comparisonsuggests that accuracyis very similar to the short-termrepeatability, again limiting the magnitudeof + + • ß systematicerror. Figure 10 suggeststhat it is possibleto achievea similar level of performanceover longertime spans.Other GPS-VLBI comparisonsin Californiahave . _' • , ,O ,O • , I , been reported: the 223-km Palos Verdes-Vandenberg 200 400 600 800 1000 1200 baseline(six experimentsbetween 1986 and 1988) [Davis LENGTH (km) et al., 1989]; the 245-km Owens Valley-Mojave baseline (a singleexperiment in 1987) [Dongand Bock, 1989], and Figure 11. GPS resultsfrom an experimentin Californiain 1986, the 224-km Palos Verdes-Mojavebaseline (six experi- plotted as differencesfrom VLBI solutions.Curves are best fit mentsbetween 1986 and 1988) [Larson, 1990b]. All these throughshort-term repeatability data from the same experiment comparisonsindicate no significantsource of systematic (Figure 9) and are not based on the GPS-VLBI data. Note errorof a type that wouldcause GPS-determined rates to similarity between "precision"(based on repeatabilitycurves) differ from VLBI-determinedrates. It is interestingto note, and "accuracy"(based on VLBI comparison)[from Dixon et al., however,that comparisonsinvolving GPS and fixed-site 1990]. VLBI oftenexhibit near-constant offsets which fortunately do not affect the rate estimates.Figure 10 is a good or betterassuming good fiducial control. Note that for the example (see also Larson [1990b, Figure 4]. The best rangeof baselinelengths typical of mostGPS applications explanationfor theseoffsets is thatthe siteties relating the the projectionof local verticalonto the horizontalbaseline phasecenters of the large, fixed VLBI antennasto the componentsdue to Earth curvature is not critical to the nearby ground marks used by GPS may be in error. error analysis. With these constraintsin mind we can However,this hypothesisremains to be verified.Future modify(31) to give a simpleerror model more applicable work on GPS accuracy should also include GPS-SLR to GPS: comparisons,since VLBI and GPS have some common error sources. o= [a2 + b2(1- e-œA)• + c•L•]•/• (32) One problem with the two-parametererror model describedby equation(31) is that •dual-frequencyGPS wherea is thereceiver and set up error,b is the asymptotic measurementshave at leastthree significant error sources, troposphericcontribution to baseline error for station each with a different dependenceon baseline length. separationsexceeding several troposphericcorrelation Receivernoise and setup errorsare typicallyat the level of lengths,), is the troposphericcorrelation length, and c is 1-2 mm, determinedby "zero length"baseline tests, and orbit-relatederror. An examplemodel is shownin Figure 268 ß Dixon: THE GLOBAL POSITIONING SYSTEM 29, 2/REVIEWS OF GEOPHYSICS

12a.Note the crossoverpoint at about400 km, whereorbit 8 errorsstart to dominate.Tropospheric error dominatesfor stationseparations longer than about 10 km and shorter Total• than400 km in this example. Figure 12 also shows some long-termrepeatability data for a seriesof GPS experi- tr ments in California for comparisonto the model. The m , 4 posphere captionlists the criteriaused to minimizeinterexperiment differences,hopefully ensuring that data with similarerror z characteristicshavebeen compared. However, by using ¸N 2 Receiver long-termrepeatability, we have undoubtedlyaveraged O overa rangeof troposphericconditions. -,- Orbit Mostof theGPS-VLBI comparisons performed todate 0 are restrictedto a fairly limitedregion in the southwestern

resultingarguments on GPS precisionand accuracyto U.S.,otherandregions.anote Sensitivityofcaution studiesis in orderindicate when thatapplying fiducialthe 8• I ' ' ' ' ' ' ' ©'' ' systematicorbit and baseline error, the exact amount dependingon experimentlocation and baseline length and 4 networkorientation.uncertaintiesGeodetic networksare theinlargestthe southwestern contributor U.S.to areless susceptible tothis effect because ofthe proximal <• • © [-'] ß North locationof several high-quality fiducial sites, i.e., sites mn 2E3 [3 East with a long historyof VLBI occupationsand well- rr • Model (Total) establishedlocal ties between the VLBI antenna phase 0 , • ., • , • , ' , center and the GPS mark. Even so, uncertainties in VLBI-GPS ties at the 1-2 cm level are common [Tralli andLichten, 1990; Dixon et al., 199lc]. Theeffect of these 40

will not causeincreased scatter over severaldays, as the orbitgeometry is essentiallyconstant over this period. The errorseffectsgrows may evenwith bedistance smallfrom over the1-2 fiducialyear periodsnetwork if butthe • 30• O O O satellite constellationhappens to be similar for several consecutiveexperiments, and if the same or similar 2o fiducial networksare used.However, over the 5-10 year or longer periods likely to characterizemanyGPS mm 10 Ve'•'•cal experiments (seebelow), such fidelity is unlikely, mm Model implying the need for care in the establishmentand 0 I • I , i • I • I • maintenance of fiducial networks. 0 1 O0 200 300 400 500 When we perform a seriesof GPS experiments,we LENGTH (km) generallywish to obtain estimatesof strainrate or fault motion rate. Many geologic problemsrequire rate ac- curaciesin the range 1-5 mm/yr [Jordan and Minster, 1988].In orderto plan experimentsit is importantto know Figure12. (a) Individualerror components and total (rss) error how long it will take to acquirea damset with a givenrate fromequation (32) for thehorizontal component of a hypotheti- accuracy,or how often an experimentmust be repeated calGPS experiment, with a = 2 mm ("receiver"),b = 4 ram,• = within a fixed period(for example,5-10 years)to achieve 10km (b and• togetherdefine "troposphere"), andc = 1 x 10-8 thataccuracy. The uncertaintyin a rateestimate •r is a ("orbit").(b) Long-termrepeatability (weighted rms scatter about functionof thesingle measurement position accuracy a bestfit straightline) for GPShorizontal component measure- the intervalbetween experiments At (assumedconstant in ments in central and southernCalifornia between 1986 and 1988 therelation below), and the totaltime spanof observations, comparedto an errormodel (equation (32)). All dataplotted are T, and can be estimatedfrom [Coateset al., 1985]: fromTI-4100 receivers, represent four or moreexperiments, and have all or most carrierphase biases resolved. Data are from Davis et al. [1989] andLarson [1990b]. "Model" curve is same as "total"in Figure12a. (c) Similarto Figure12b, for vertical component.Model curvefrom equation(32)), with a = 5 ram, b C•r=-•- c•,• [ (1 + T/At)(2 12T/At+T/At) (33) = 20mm,k= 10km, andc = 1 x 10-8. 29,2/REVIEWS OF GEOPHYSICS Dixon:THE GLOBAL POSITIONING SYSTEM ß 269

Figure 13 plots the rate uncertainty(1(•) for single In the limit of shortlength and time scales(implying measurementuncertainties of 1 and 2 cm (1(•), for several densenetworks and frequentobservations), high-accuracy valuesof At. While the singlemeasurement uncertainty is GPSis currentlylimited mainly by economicand logistical clearlyimportant, the mostefficient way to reducethe constraints,receiver limitations, and data processing uncertaintyin a rateestimate is to extendthe totaltime efficiency.In the nearfuture, with receiverimprovements spanof observations.Comparison of modern GPS or other and the enhanced constellation, ~5-mm horizontal geodeticdata and lower quality historical geodetic data accuracyshould be achievableover a widerange of station taken20 or more yearsago is a goodexample of this separationswith kinematicor rapid static techniques, principle.Significant compressional deformation along enablingdense spatial sampling. As receivercosts fall and partsof thesouthern San Andreas fault system in Caldor- the efficiency of data analysisimproves, continuous nia hasbeen documented with this approach[Feigl et al., monitoringof densenetworks with GPS also becomes 1990b; Webb, 1990]. feasible,allowing greatly improvedtemporal sampling. This capabilitywill complementseismic networks and 10 1 •.• I ' ' I ' strainobservatories (D. M. Tralli, Spectralcomparison of q,,, =- 1•m,1 yr' continuousGlobal Positioning System and strainmeter • %, '...... O...... 2cm/lyr measurements of crustal deformation, submitted to GeophysicalResearch Letters, 1991), enabling monitoring • %,, ....I--- 1cm/3yr of phenomenaat periodsjust beyondthe limit of long- periodseismometers, for example,creep and strain events •O,.....'""O.."%,"1-1, ----13-- 2cm/3 yr (Figure 14) and other possibleeffects related to stress migration.Since 1-mm horizontalprecision in theoryis achievablewith severalminute samplingof carrier phase andprecise P codepseudorange data over short (<10 km)

12 10 STRAIN = 10 plate boundary 0 , I • I • I , I , zone 0 2 4 6 8 10 tectonics 103 yr 10 post TIME AFTER FIRST EXPERIMENT (yrs) 10 volcanic post-seismic events rebound, glacial rebound coupling Figure 13. Uncertaintyof a geodeticallydetermined rate estimate •' asa functionof time,T (equation(33)) for two values of single •• 108 SLR measurementuncertainty, o• (1 and2 cm,listed first in legend) LU 1 yr and two valuesof experimentrepeat interval, At (1 and 3 years, < listedsecond). Note that rate uncertaintyis betterthan 3 mm/yr ca VLBI after10 years for all cases shown. • 10 GPS-1990

1 day

GEOLOGICAL APPLICATIONS AND ACCURACY lO4 GPS-1995 ? REQUIREMENTS 1 hr

In this sectionwe review severalapplications of GPS 2 eadhquakes 10 and their correspondingaccuracy requirements. The examplesencompass a wide range of stationseparations, 10¸ 101 102 103 104 an importantfactor in determiningboth the accuracy SPATIAL SCALE (km) achievablewith GPS and critical aspectsof experiment design. Figure14 (modifiedfrom Minster et al. [1990]shows Figure 14. Cartoonshowing order of magnitudeestimates for thetemporal and spatial scales of a varietyof geologicalspatial and temporal scales of somegeological phenomena (lower phenomenawhere geodetic monitoring can provide useful caseletters, stipple pattern) comparedto detectioncapability of information.The regions of applicabilityfor several some geodetic techniques(upper case letters, diagonal line geodetictechniques are also shown. It is apparentthat even pattern),assuming 10-7 strainor higherand typical operating at currentlevels of performance,GPS can contribute to the conditions.OB, observatoryinstruments (for example,continu- studyof a wide varietyof phenomena.This figurealso ously recording strainmeters,tiltmeters); VLBI, very long indicatesa roughestimate of GPS performance5 years baselineinterferometry; SLR, satellitelaser ranging;EDM, two hence, based on the expectedimpact of some recent color laser electronicdistance measuring equipment, assuming daily measurements.Modified fromMinster et al., [ 1990]. developments. 270 ß Dixon: THE GLOBAL POSITIONING SYSTEM 29, 2 / REVIEWSOF GEOPHYSICS

baselines,it is temptingto speculateon thedetectability of zonethrust faults are givenby Savage[1983] and Thatcher seismic precursorsproposed by Lorenzetti and Tullis and Rundle[1984]). The fault-parallelsurface displace- [1989] with a densearray of continuouslyoperating GPS mentrate v duringan earthquakecycle for a simpleelastic stations. half-spacemodel is [Savageand Burford, 1973]: In the limit of long length scales,GPS accuracywill improveas global trackingnetworks evolve to includea v = (Vo/•)tan -• (x/D) (34) uniform set of well-distributed,high-accuracy ground systemswith strict configurationcontrol, and as force where D is the fault or "locking" depth related to the models for solar radiation pressure and other orbit brittle-ductiletransition depth, x is distanceperpendicular perturbationsare ref'med.This will enablea numberof to thefault, and v 0 is the totalslip rate at largedistances platemotion and regional plate boundary zone problems to from the fault or beneaththe locking depth. If T is the be attacked.Even with currentperformance there are many earthquakerecurrence interval, then the coseismicslip that moreapplications of GPS thanwe canaddress here, but to relievesaccumulated elastic strain (and stress) is justvoT. give someflavor for the diversity,I will considertwo types Geodeticdata may be invertedfor D and v. Constraintson of applications,divided arbitrarily by lengthscale. D are also availablefrom seismicity(large earthquakes tend to nucleate near the base of the fault, at the brittle- Short (0-100 km) Baselines:The Near Field ductiletransition), and v 0 maybe determinedfrom far-field Geodetic measurements within about 50-100 km of a spacegeodetic measurements (VLBI, SLR, or GPS) or, for fault systemmay provideconstraints on crustand mantle a simpleplate boundary, from a globalplate motion model. rheology,faulting mechanics, and the earthquake cycle. This For strike-slipfaults in continentalcrust, D is typically lengthscale is alsoappropriate for monitoringupper crustal about10-20 kin, on the basisof inversionof geodeticdata deformation associated with inflation/deflation of shallow [e.g.,King et al., 1987],seismic depth data, and rheologi- magmachambers. Until recently,most suchstudies were cal propertiesof crustalmaterials from laboratoryexperi- performedby conventionalterrestrial geodetic techniques ments [Sibson, 1983]. Figure 15 plots differencesin (triangulation,tri!aterafion distance measurement, and level- fault-paralleldisplacement rates for elastic half-space ing). The accuracy,site flexibility, ease of dataacquisition, modelswith D = 10, 15, and 20 kin. The largestrate andincreasingly straightforward dam analysis allows GPS to differences are observed at horizontal distances from the complement,and in somecases, replace these approaches. fault roughlyequal to fault depth,giving someguidance as The accurate three-dimensional vector nature of GPS to stationplacement. Distinguishing among the models positioningalso allowsconstraints to be placedon block would require resolutionof fault-paralleldisplacement rotations,something that is not possiblewith conventional ratesat the 1-3 mm/yr level. techniquesalone [e.g.,Davis et al., 1989].Minster et al. , [ [1990], Bilham [1991], and Hager et al. [1991] review ap- 41 I ' I I plicationof geodeticstrain determination to near-fieldand •, 10-15 earthq•e studies.In thissection I will focusmainly on one • 15-20 aspect,namely the accuracy required for GPSdata to distin- • 10-20 guishamong some plausible models of faultingand crustal rheology. A seriesof GPS experimentsallows determination of sur- face strainand strainrate tensors,reflecting stresses and rheologicalproperties in thecrust and mantle. Simple elastic half-spacemodels and more complicatedrheological dis- tributionshave been used to explainobserved surface defor- -4 , , i • mation [e.g., Melosh, 1976, 1977; Savageand Prescott, -40 -20 0 20 40 1978;Spence and Turcotte,1979; Turcotteet al., 1984;Li DISTANCE (km) and Rice, 1987; Li and Lira, 1988]. The manner in which deformationchanges in space(away from a fault) and time (beforeand after a largeearthquake) may reveal infomarion Figure15. Differencein faultparallel surface displacement rate aboutthe rheologyof the lower layersbut may alsorequire forseveral models assuming a locked, vertical strike-slip fault in anelastic half space (equation (34)) for faultdepths, D = 10, 15, largenumbers of accuratemeasurements on any givenfault and20 km.A farfield displacement rate of 30 mm/yris assumed. segmentto describeadequately the spatial and temporal dis- tributionof strain.Important questions include whether GPS dataallow discriminationamong these models, what accu- A more realistic model for deformation near a fault racy is required,and whether we nee• to incorporateany ex- accountsfor the layeredrheology of the crustand upper traordinaryfeatures into the GPS experimentdesign? mantleand the coupling that occurs between layers. Recent Considerthe surfacestrain pattern for a locked,vertical models assumethat lower layer or layers behave like strike-slipfault (strainmodels appropriate to convergence Maxwellviscoelastic solids. Like its mechanicalanalogue, 29, 2/REVIEWS OF GEOPHYSICS Dixon: THE GLOBAL POSITIONING SYSTEM ß 271 a springand dashpotin series,a viscoelasticsubstance degreesor betterin azimuth [DeMetset al., 1990]. These behaveselastically over shorttime scalesand viscously models are based on spreadingrates from mid-ocean over long time scales.Viscous relaxation in the viscoelas- ridges,determined from magneticanomalies averaged over tic "asthenosphere"damps short-term perturbations in the severalmillion years, azimuthsof transformfaults that overlyingelastic "lithosphere." The rate of stressrelaxa- averageover a similartime interval,and earthquakeslip tion is related to a time constant2•l/!.t, where •1 is the vectors representing essentially instantaneouslocal viscosityof the lower layer and !x is the averagemodulus azimuthinformation. There is no reasonto suspectthat of rigidity of the two layers.Expressions for the surface plate motionaveraged over severalmillion yearsdiffers strainspredicted by couplingmodels are given by Savage over shorter(for example, geodetic) time scaleswhen and Prescott [1978], Turcotte et al. [1984], and Li and measuredbetween plate interiors,away from the transient Rice [1987]. Inspectionof these models suggeststhat effects of plate boundarydeformation, unless a plate differencesin fault-parallelsurface displacement rates of boundaryis undergoingrapid evolution.This hypothesis 1-2 mm/yr or better are required for discrimination. hasbeen largely confirmed by NASA's CrustalDynamics However,for mostparts of the earthquakecycle, predicted Project,which has made velocitydeterminations between surfacedeformation with the elastichalf-space model and severalmajor plates with VLBI [e.g.,Argus and Gordon, even the simplest(two layer) viscoelasticcoupling model 1990] and SLR [e.g., Harrison and Douglas, 1990]. may be surprisinglysimilar and thus difficult to dis- Nevertheless,there are some plate boundarieswhere criminatewith GPS or any othergeodetic technique alone relativeplate motion is poorlyconsmined, and where GPS [Savage and Prescott, 1978; Thatcher, 1983; Savage, (as well as VLBI and SLR) measurementscan make 1990]. On the otherhand, time-varying signals over part of uniquecontributions. Plates lacking a spreadingcenter the earthquakecycle are a potentiallyuseful observable boundary(for example,the Philippineplate) or with only [Thatcher, 1983; Li and Rice, 1987], and we should not small or poorly definedspreading center boundaries (for rely on long(> 10-15 years)observation times to achievea example,the Caribbeanplate) will have poorly defined given rate accuracy.Davis et al. [1989] obtained rate relativemotion rates with respectto adjacentplates. Plate uncertaintiesof 1 mm/yr in horizontalcomponents and boundarieswhere transform fault trends or earthquakeslip 3-4 mm/yr in the verticalcomponent for baselinesup to azimuths may be biased due to the influence of 12 km in lengthin 3 yearsusing GPS data taken several heterogeneouscontinental crust (for example,the Pacific- timesper year. North Americaplate boundarynorth of the southernGulf GPS is also well suited to define the coseismic strain of California)will havepoorly determined relative motion field associatedwith earthquakes.These strains may azimuths.Predicted motion at subductionzones may be exhibit great spatial variation within a few tens of biasedbecause of systematicbiases in trenchearthquake kilometersof the fault, particularlyfor complex fault slip vectors[DeMets et al., 1990]. In caseswhere local geometry,implying the needfor densesurveys. It may be data are lackingor biased,relative plate motion is deter- useful to separatecoseismic rupture from postseismic minedmainly from closure conditions on otherplates, and relaxationphenomena, implying a requirementfor rapid GPS or otherspace geodetic technique can provideuseful responseand frequentresurvey after an event.If required complementaryinformation. GPS stands out as the accuraciesare not high (severalcentimeters are sufficient appropriatetechnique if economicor logisticalconsidera- for most studiesof coseismicoffset), kinematic or rapid tionspreclude the useof SLR or VLBI, bothof whichhave static techniquesenabling survey of large numbersof larger, more cumbersomeground systems,and if the points may be useful. Larsen [1990] used kinematic problemcan be solvedwith stationseparations less than techniquesto measure coseismicoffsets for the 1987 1000-2000km (bothVLBI andSLR havebetter perform- Superstition Hills earthquake sequence in southern anceat longerstation separations). California. Figure 16 showsthe uncertaintyin relativevelocities of someplate pairs with convergentboundaries. If at leastone LongBaselines (100 > 1000 km) of theplates is oceanic,nearby measurement sites spanning Long baselinemeasurements with GPS are challenging the trenchplate boundary may not exist.One of the goals becausethe influenceof orbit errors is directly propor- of the CASA (Centraland SouthAmerica) GPS experi- tional to baselinelength, and becausecarrier phase cycle ments[Kellogg et al., 1989; Kellogg and Dixon, 1990] is ambiguity resolutionbecomes more difficult at station to determine the rate and direction of subduction of the separationsexceeding 100-200 km. Special care is Cocos and Nazca plates beneath Central and northern requiredin networkdesign, and covarianceanalyses prior SouthAmerica, among the mostpoorly determined plate to the actualexperiment may be important.The geologic pairsin the NUVEL 1 plate motionmodel [DeMets et al., rationalefor pursuingsuch difficult GPS experiments, 1990](Figure 16). Thisrequires measurements from Cocos someexamples, and preliminary results are givenbelow. Island(the only point of land on the Cocosplate) and Relativevelocities of mostof the majorplates are now MalpeloIsland (the only point of landon theNazca plate well known from globalgeologic models, with precisions within 500 km of mainlandSouth America) to pointsin of a few millimetersper year or betterin rate and a few Central and South America, defining numerouslong 272 ß Dixon: THE GLOBAL POSITIONING SYSTEM 29, 2 / REVIEWSOF GEOPHYSICS

INCREASING UNCERTAINTY has at least three major faults that are importantin

! ! accommodatingthe tangential components of relativeplate CO-NA •////•.///•.////////•J motionbetween the Pacific and North Americanplates. ! I Rotations are predicted theoretically [Ivins, 1989] and NZ-SA //////////•///////////////////• Neogenerotations in partsof theplate boundary zone have CO-CA /•///////////////////• ! ! beenobserved with paleomagneticand otherdata [Carter CA-SA et al., 1987; Luyendykand Hornafius, 1987; Nur et el.,

0 8 16 24 1989]. Kinematicmodels of the region [e.g., Bird and F Rosenstock,1984; Weldonand Humprheys, 1986; Rundle, FO.05 FO.01 1986]are being tested with GPS, employinga networkthat includes islands of the southern California borderland and station separationsmainly in the range 100-500 km Figure 16. Uncertaintyin relativeplate motion vectors (NUVEL [Agnewet al., 1988; Dixon et al., 1990]. Precisionof the 1 model)[DeMets et al., 1990] for plate'pairs where local motion GPS-based rate estimates should be commensurate with is constrainedmainly by trenchslip vectors;most other plate the models (a few millimeters per year) [Bird and pairsin themodel are better determined. Uncertainty is basedon Rosenstock,1984; Jordan and Minster, 1988], achievable theF ratio testfor plateclosure [Gordon et al., 1987]."F.01" and in 5 yearswith yearly GPS measurements(Figure 13). "F.05" indicate1% and5% risk levels,respectively (99% and Preliminaryresults are reviewed by Hager et al. [1991]. 95% confidencelevels). CO, Cocos;NA, North America;NZ, Total relative motion between the Pacific and North Nazca; SA, South America; CA, Caribbean. Modified from Americaplates can be measuredby spanningthe relatively DeMets et al. [1990]. simpleridge-transform boundary in the southernGulf of California,avoiding tectonic complexities associated with baselines. It was known in advance that a robust fiducial the TransverseRanges and Basin and Range extension. networkwould be requiredto performsuch measurements, The relevantbaselines are 300-500 km in length.GPS and covariance analyseswere performed to determine measurementswere made in 1985 and 1989, allowinga optimal configurations [Freyrnueller and Golombek, preliminarydetermination of the relative motion vector 1988]. These analysessuggested that a North American (Figure 17). Using a sensitivityanalysis to quantify fiducial network by itself was inadequate.A global randomand systematicerror, Dixon et al. [1991c] showed network of tracking stations was instituted for this that the Pacific-North America relative motion vector experiment,the first civilian globalGPS network[Neilan determinedby GPS in the southernGulf of Californiais et al., 1989].Preliminary results using this network tend to consistent,within lc• totalerror (+_7 mm/yr in rate,_+6 ø in confirm the predictionsof the covarianceanalyses, with azimuth),to the NUVEL 1 global model [DeMetset al., significantimprovements in long baselineestimates when 1990]. the expandedtracking network is used[Kornreich Wolf et al., 1990;Freyrnueller and Kellogg,1990; Figure 5). The secondCASA experimentis scheduledfor 1991. _•1 I I Relativemotion between plates is occasionallyaccom- 3.0 modated within a continent. Continental crust is both weaker and older on average than oceanic crust, and inherited weaknessesof varying orientation may be reactivatedunder the newer stressregime of the plate boundary[Ivins et al., 1990], complicatingthe boundary's geometryand evolution,and often leadingto broadzones of distributeddeformation. Examples of complexcontinen- tal transform boundaries include the San Andreas fault zone in California, the Motogua-Polochicfault zone in Central America, and the Alpine fault zone in New Zealand.High-precision GPS measurementscan be useful I I I I 4.0 3.0 2.0 1.0 in characterizingboth the integratedmotion acrossthe WEST VELOCITY(cm/yr) entire deformation zone and details of crustal behavior near individual faults within the plate boundaryzone, includingrotations of quasi-rigidblocks betweenfaults. Figure 17. Vector diagram summarizingrelative motion of Knowledgeof individualfault ratesand the totalplate rate Pacificplate (at CaboSan Lucas) with respectto North America, can lead to developmentof internallyconsistent kinematic based on GPS measurement,and NUVEL 1 global vector, block models.In effect, a completemass balance can be closure fatting vector (CFV), and best fitting vector (BFV) obtainedbecause adequate spatial samplingis possible [DeMets et al., 1990]. Error ellipsesare lo from Dixon et al. with GPS. For example,the southernSan Andreas system [1991]. 29, 2/REVIEWS OF GEOPHYSICS Dixon: THE GLOBAL POSITIONING SYSTEMß 273

CONCLUSIONS Bilharn,R., Earthquakesand sea level: Spaceand terres•al met- rologyon a changingplanet, Rev. Geophys.,29(1), 1-29, 1991. Bird, P. and R. W. Rosenstock,Kinematics of present•ust and I have coveredin a generalway the major principles manfie flow in SouthernCalifornia, Geol. Soc. Am. Bull., 95, behindGPS, particularlythose relevant to high-precision 946-957, 1984. geological and geophysicalapplications. GPS is an Black,H. D., andA. Eisner,Correcting satellite Doppler data for impressivenew measurementtool; the rate of improve- troposphericeffects, J. Geophys.Res., 89, 2616-2626, 1984. mentin the qualityand quantityof geodeticmeasurements Blewitt, G., Carrier-phaseambiguity resolution for the Global PositioningSystem applied to geo&tic baselinesup to 2000 with GPS over the last decade is astonishing.We can km, J. Geophys.Res., 94, 10,187-10,203, 1989. expectto seecontinued improvements in receiveraccuracy Blewitt, G., GPS techniquesfor monitoringgeodynamics at and portability,reduced receiver cost, decreased time and regional scales, paper presented at Second International costfor data processing,improved models and calibration Symposiumon PrecisePositioning with Global Positioning of major error sources,corresponding increases in the System,Int. Assoc.of Geod.,Ottawa, 1990. Blewitt, G., U. J. Lindqwister,and K. W. Hudnut,Densification numberof diversityof scientificapplications, and some of continuouslyoperating GPS arraysusing a rapid static very interestingscientific results. surveytechnique (abstract), Eos Trans.AGU, 70, 1054, 1989. Bock,Y., R. I. Abbot,C. C. CounselmanIll, R. W. King, andS. A. Gourevitch,Three-dimensional geo&tic control by ACKNOWLEDGMENTS. Comments on a preliminary interferometrywith GPS: Processingof GPS phaseobserv- versionof themanuscript by Eric Ivins, SteveLichten, and Larry ables,paper presented at First InternationalSymposium on PrecisePositioning With Global PositioningSystem, Int. Young were very helpful. I am indebtedto Jim Davis and Bob Assoc. of Geod., Rockville, Md., 1985a. King for thorough reviews, and editor-in-chief Marcia Bock, Y., R. I. Abbot, C. C. CounselmanIll, S. A. Gourevitch, Neugebauerfor patience and many helpful suggestions.Ken andR. W. King, Establishmentof three-dimensionalgeodetic Hurst,Chris Rocken, Dave Tralli, andBob King kindlyprovided controlby interferometrywith theGlobal Positioning System, manuscriptsin advanceof publication.This work was conducted J. Geophys.Res., 90, 7689-7703, 1985b. at the JetPropulsion Laboratory, California Institute of Technol- Bock,Y., S. A. Gourevitch,C. C. CounselmanHI, R. W. King, ogy,under contract to NASA. and R. I. 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