精密单点定位最原始的论文,J. F. Zumberge,1997

JOURNAL OF GEOPHYSICAL

RESEARCH, VOL. 102, NO. B3, PAGES 5005-5017, MARCH 10, 1997

Precise point positioning for the efficient and robust analysis of GPS data from large networksJ. F. Zumberge, M. B. Heftin, D.C. Jefferson,M. M. Watkins,and F. H. WebbJet Propulsion Laboratory, California Institute of Technology,Pasadena

Abstract.

Networks of dozens to hundreds of permanently operating precision

Global Positioning System(GPS) receivers emerging spatialscales are at that range

from100to 10 km. To keep computational the burden associated the analysis withof such data economicallyfeasible, one approach is to first determine preciseGPS satellite positionsand clock correctionsfrom a globally distributed network of GPS receivers. Then, data from the local network are analyzed by estimating receiverspecific parameters with receiver-specificdata; satellite parameters are held fixed at their values determined in the global solution. This"precisepoint positioning" allows analysis of data from hundreds to thousandsof sites every day with 40Mfiop computers,with results comparable in quality to the simultaneousanalysis of all data. The referenceframesfor the global and network solutionscan be free of distortion imposed by erroneousfiducial constraintson any sites.Introduction

The Global PositioningSystem(GPS) has emergedin the 1990s as the space geodetic technique with the

The volume of GPS data is growing rapidly, and a means to analyze this volume in a consistent, robust,and economical manner is essential. In this article we

Heftin, 1995] plate-boundary deformation[Feiglet al., 1993], motion associated with earthquakes[Blewitt et al., 1993; Bock et al., 1993], and changingEarth orientation[Herring et al., 1991;Lindqwister al., 1992] etand rotation rate. More recent uses of GPS include

first discussthe computational burden associatedwith simultaneous virtues of accuracyand economy[e.g., processing GPS data in the 'contextof data volume and Yunck, 1995]. It has been applied to a variety of number of parameters estimated. We show that the geophysicalphenomena, including the motion of tec- computationalburden associated with the rigorousleast tonic plates[Larsonand Freymueller, 1995; Argusand squaresanalysisof data simultaneously from R receivers

scales with R3. To the extentthat globalparameters, that is, orbits of GPS satellites(expressed an Earthin fixed referenceframe) and satellite clocks,can be esti-

mated with a subset of the R receivers, then data from the others can be analyzed one at a time. Various analvolcanomonitoring[Webbet al., 1995], ground-based ysesof data from R= 49 receiversare used to quantify measurements atmospheric for[Busingeret al., 1996] the approximations involved in this technique, which and ionospheric[e.g., Wilsonet al., 1995]applications, we call"precisepoint positioning." The validity of the as well as atmosphere and ionosphere sounding[Mel- technique is also demonstrated based on data from over

bourne,1995] usinglow-Earth orbiters equippedwithreceivers.

GPS

102receivers 104stationdays. and

In heavily populated regionsof significantseismicac- Computational Burden tivity, deployment of dense networks of precisionGPS The two GPS data types,carrierphase(L) and pseureceiversis in progress. Already hundreds of receivers dorange(P), measurethe receiver-to-transmitter disare in operation in Japan and dozens in southern California. Networks suchas these allow the on-goingmea- tance with and without, respectively,an unknown bias. L is a much lessnoisy measurementthan P, which offsurement of the surface deformation field and are exsets the fact that it requires estimation of a bias term. pected to be valuable both in understandingthe complex systemof faults in the region and also in hazard Both data types exist at each of two frequencies,apmitigation. Additionally, space-basedarrays of GPS- proximately 1.2 and 1.6 GHz; this allows the formalinear combinationfor each, equipped receiversare expected over the next few years tion of the ionosphere-free which to first order is independentof the ionospheric to exploit the potential applications of GPS to climate, electron density. We assume in what follows that we weather, and the ionosphere.have formed this combinationand transmitter

for L and P.

A phase measurement at time t between receiver rCopyright 1997 by the American GeophysicalUnion.Paper number 96JB03860.x is modeled as

0148-0227/97/96 JB-03860509.005005

Lrxt+ Prxtb t+c t+ - , t+ C, t - z tm(O t) (1)+

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where Prxt is the true range, brxt is the phase bias or

ambiguity,Zrt is the zenith tropospheric delay,m(Orxt) is the mappingfunction for elevationangleOrxtbetweenreceiver and transmitter, and OJrx is the phase windup t term to account for changesin the relative orientation

ters. It is approximately a factor of 2 slower than least squaresestimation using normal equations. The least squaresestimate of n parameters with ra measurementsrequires a number of arithmetic operations

of the receiving and transmittingantennas[Wu et al., 1993]. Receiverand transmitterclockcorrections areCrt and cxt, respectively. The term lYrx accounts for t

B (xn2m,

(5)

data noisein the measurement phaseto make (1) of an equality. A pseudorange measurementis similarlymodeled as

where the constant of proportionality dependson the choice of estimation algorithm but is of order unity

[Bierman,1977]. We neglect termsinvolving 3 and n nm because they are small comparedto n2m in this application.Thus (3), (4), and (5) give

Prxt - Prxt-}-Zrtm(Orxt) Crt - Cxtq-T]rxt, q-

(2)

-

+...,

(6)

where rxt accountsfor pseudorangedata noise. Implicit in Prxt are receiver coordinates and all nonclocktransmitter parameters.Undifferenced Data

where the ellipses denoteterms of order R 2 and lower.

The valueof k is easilyshown equal(5 q-X)2ml, towhere ral is the number of

measurementsper receiver

Considerdata from R receiversspanninga period A; typically we might have A= 24 hours. Let be the

time betweenmeasurements and let f /47r be the average probability that a satellite is in view above an

((3) with R - 1). In (6) we assume that the constant of proportionality (5) is 1. in Total processing times includethree passes through the filter module. Identification of phase-biasbreaks is performed after the first, and the third represents aniteration, required becauseof nonlinearities in our mod-

satellites' yaw attitude[Bar-Sever et assumed elevationcutoff ( 0.25 for a 15ø cutoff). If eling of eclipsing timesalsoincludeauxiliary there are X transmitters(X= 24 for the operational al., 1996].Total processing GPS constellation), then the numberof measurements modulesfor input/output, data cleaning,calculationsis given by

m= RX(f /4 r)(A/6)d,

(3)

where d is the number of data types. Normally, d= 2, corresponding to ionosphere-free phase and pseudorange.

of predicted measuredvalues and their derivatives with respect to estimated parameters, and so on. Shown in Figure I are actual processing times as a function of

The number of parameters to be estimated can bewritten

R, as well as the theoretical filter timesusing(5), with X - 24, f /47r - 0.25, A - 30 hours, 6 - 5 min, d- 2, a- 29, b- 10, and c- 5. From Figure 1it is clear that somewhere the region 50 _< R _< 100, insimultaneousanalysisof data from R receiversstarts to

n= aR+ bX+ c.

(4)

For example, station-specific parametersmight include three Cartesian coordinates, a zenith tropospheric delay parameter, a clock parameter, and one phase bias parameter for each transmitter in view by the receiver

becomecomputationallyinfeasiblewith 40-Mfiop computing devices.PartitioningOne method to reduce the burden is to divide the

over the period of interest. Thus a - 5+ X in (4). data into J groups. For simplicity we will assumethat Transmitter-specificparameters might include epoch- thereare J= 2 groups with ra/2 measurements each. in state position and velocity, two solar radiation parame- Let n be the number of parameters common to both ters, a Y bias parameter, and a clock parameter, giving groupsdivided by the total number of parametersn. b= 10. In the case that transmitter parameters are Then (1- n)n/2 parameters apply to the first group fixed, we have b= 0. Polar motion values and rates are only,and another(1 -n)n/2 apply to the second group estimablewith GPS, as is length of day, giving c= 5. only. In the contextof the GPS problemeachdata parOver the period A it is essential to allow stochastic tition couldcorrespond a distinct groupof receivers; to variation of clocks because most receiver and transmiteachreceiverbelongs one and only one group. The toter clocks look like white noise on timescales of minutes.

common parameters are Earth orientation and satellite

Temporal variation of phase biasesis also necessary to accoun

tfor abrupt changeswhen the receiverloseslock or slips a cycle. Stochasticvariation for zenith tropospheric delays and solar radiation parameters has also been found to be effective. Becausewe use square root

parameters,while the group-specific parametersare the parametersof receivers the group. in The ral= ra/2 measurements group I are usedto in

estimate= n q-(1 - )n/2= (1q- )n/2 parameters, nl

requiring n ml -- n2m(1 )2/8 operations. soluqThe

groupwill require an equal number information filter (SRIF) sequential estimation[Bier- tion for the second man, 1977;Lichten,1990],suchparameters contribute of operations,so that together the two solutionscost only onceto the parameter count; the aboveexpressions n2m(1 n)2/4 operations.+for a and b reflect this. SRIF sequential estimation is used because it is numerically stable and allows considerableflexibility for stochasticvariation of parameNext, the two solutions must be combined. For

the combinationone can think of n parametersbe-

ing determined from nl+ n2 measurements, costing

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r-

o

E o

E o

0.1

0.01

10

number

of receivers

100

Figure 1. Computational burden B as a function of the number of receiversR. The open circles are actual total processing times on a 40-Mfiop computer and include three passes through the parameter estimation module, as well as auxiliary programs. The dotted line is the theoretical time requiredfor three passes through the parameter estimationmodule only, assumingB=

2.68n2m;the 2.68is chosen that Equations (4), and(5) givetheobserved execution so (3), filtertime for the R= 48 and 49 cases.The units of B are the number of days required by a 40-Mfiop computer to perform the analysis.

of all data simultaneously,is thus

n2(nl q-n2) --n3(1 q-n) operations. total cost)/, Double Differencing The relativeto the n2m operations required the analysis For receiversq and r in view of satellites w and x at fortime t, (1) can be usedto showthat the quantity)/(1+ n)

dLqrwxt Lqwt- Lrwt - Lqxtq-Lrxt -chronization of all measurements

(8)

This can be generalizedto

is independentof all clock parameters. In practice, synto GPS time is made

x-[lq-n(J-1)] q--[1 j2 q-n(J-1) nm

(7)

prior to the formation of the double difference. A similar double difference can be formed for the pseudorange. Least squaresestimation of nonclockparameterscan be made based on double-difference matrix measurements. in double-

when the m measurements are divided equally into J

groups. Note that (7) approaches unity as n - 1, assumingn/m is small. Thus, if all parameters comaremon, partitioning is ineffective.

It is straightforwardto form a measurementcovariancedifferenced

that accounts for known correlations data.

With X - 24, f /4 r= 0.25,A - 24 hours,= 5 min, (5d - 2, a - 29, b - 10, c - 5, and R - 40 receivers, we have n - 1405 parameters of whichnnbX+ c245

The technique ha

s the advantage of a reduction in both the measurement and parameter counts. Oneneed not.include the entire set of double-difference measurements because it contains redundant information. For R receivers in view of X transmitters at time t

there are (R - 1)(X - 1) independent double differare common(n 0.174). There are m= 138240mea- ences that form a complete measurement set. One surements,and the ratio of parametersto measurements suchset is dLqrwxt for q - w - 1, 2 _<r _<R, and is n/m 0.01. Equation(7) then shows 64% reduc- 2< x< X. Thus the measurement count m is reduced a tion in burden for J= 2 and an 88% reduction for by RX - (R- 1)(X - 1) - R+ X - 1 compared the to J= 40. Clearly, partitioning is one effectivemeansof undifferenced case. An identical reduction in the pareducing CPU when n is small and has been demon- rametercountn occurs because (R- 1)(X- 1) is alsothe number of phase bias parameters to be estimated, comparedwith RX in the undifferencedcase. Finally, When many parameters are treated as stochastic,rig- receiverand transmitter clocksare no longer parameorouspartitioning is difficult. We feel that the benefits ters, so that the parameter count is further reducedby R+X. of using stochastics outweigh this disadvantage.strated by some analysis centers in the International

GPS Service Geodynamics for (IGS).

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For X= 24, then, the value of n2m is reducedby about 34% for R= 10 and by 19% for R= 100. From the standpoint of computational burden the doubledifference approach has approximately this advantage over the undifferencedapproach when data from R receivers are simultaneously analyzed. To simplify the calculation, we have ignored the extra computational burden needed to form the doubledifferences and a measurement accounts for known correlations covariance matrix that in data noise. We haveat all times.

orbitsor a few tensof metersfor GPS clocks[Zumberge and Bertiger 1996]. Our useof precise point positioning differsfrom the classical of the term in two ways. useFirst, both carrier phase and pseudorange are used as data. Second,the quality of transmitter parameters is a few centimetersor better. When applied to singlereceiver data, the resulting accuraciesare comparableto whattational

is obtainedcost.

when

data

from

all receivers

are

simultaneouslyreduced and come at far lower compu-

also ignored issuesof common visibility by assumingthat all R receivers view all X transmitters

AlthoughBlewitt[1993]givesan exampleof a caseinwhich it is not possible to form a set of double differences which contains all of the information content

Method

available for estimation of nonclockparameters, performanceof IGS analysiscentersthat usedoubledifferencing indicates that this is not a practical limitation. The double-differencetechnique views receiver and transmitter clocks as nuisance parameters that can be eliminated up front by

the appropriate linear combi-

Supposethe analysis of data from R receiversis divided into two parts. First, a subset of S receivers is usedto estimate satellite parameters,Earth orientation, and S setsof receiverparameters. Then, data from each

of the remaining R-

S receivers analyzed,one reare

ceiver at a time. The computational burden B can be

written as the sumof two terms,onefor the globalsolution (Bg) andonefor the point-positioned sites.For the we to nation of data[Wu, 1984],exactly analogous elim- globalsolution haveng- aS q-bX q-c parameters to inating ionosphericeffects by forming the ionosphere- estimatefrom mg= SX(f /4 r)(A/6)d measurements.free combinations described earlier. In the method de-

Thecorresponding burden Ba= kSa+..-, just asin is(6) with R replacedby S.The computational burden to determine nl ----5 q- X

scribed here, however, transmitter clock parameters, to-

gether with GPS orbits, are the key for the efficient analysisof large networksof GPS receivers. (From the point of view of information content as opposedto

receiver-specific parameters givenrrt ((3) with R= 1)

computational burden,consultKuang et al.[1997]forcomparisonsof the differenced and undifferencedapproaches.)

measurements - n m ; thisapplies R-S sites. is B to The total burdenis thusB= Ba+ (R- S)B1 or

B= (5 q-X)2ml(S q-R- S) q- ..-. 3

(9)

Precise Point PositioningSupposedata from a globally distributed network ofR receivers are used to estimate receiver and transmit-

The Sa termtotallydominates, that the receiver so parametersfrom the additional R- S sitesare essentially free. Note that the computational burden includes the parameter-estimation processes only. The total pointposition computational burden is of the order of 1 min per site on a 40-Mfiop computer. Aside from computationalefficiencyan additional advantageof the one-receiver-at-a-time processing that is it allows easy diagnosisof receiver-specific problems. For geodeticquality receiversthe rms value of the postfit residuals, that is, the rms difference between the

ter parameters. Denote the estimatesof transmitter parameters by P. Consider an additional receiver and the analysisof data from Rq- 1 receiversto yield transmitter parameters Q. In the approximation that P= Q, that is, data from the additional receiver has negligible effect on the values of estimated transmitter parameters, a more efficient way to estimate the parameters spe-

cific to the additional receiveris to analyze data from it in seclusionand fix the transmitter parameters in the analysis to be P. When GPS orbits in an Earthfixed frame and GPS clock corrections are not estimated

modeled valuesin (1) or (2) and the measured values,is 5 to 10 mm for phaseand 50 to 100 cm for pseudorange. If measurements from a particular receiverare significantly noisier, it may be an indication of a hardware fault, a poor multipath environment,an interfering RFsource, or other unusual circumstance.

Reference Frame

but, instead, fixed at some predetermined values and the model has no other spatially correlated parameters

(e.g., tropospheric delay), then there is no needto analyze data from all receiverssimultaneously. The term"point positioning" in GPS analysishas referred historically to the estimation of receiver-specific parameters by the analysisof receiverpseudorange data, fixing transmitter parametersto valuesbroadcastin the navigation message.The quality of results is limited in

One way to impose a referenceframe in the analysis of GPS data is to assume that coordinates of certain

sitesare sufficientlywell known that they can be fixed or

tightly (< 1 cm) constrained the analysis data from in of

a global network[Blewitt, 1993,andreferences therein].For example, the analysis centers of the IGS fix the locationsof 13 globally distributed sites in their deter-

this case because pseudorange inherentlynoisy, (1) is nobetter than about 40 cm, based on our experience with postfit residualsof geodetic-quality GPS receiversand

minations precise of GPS ephemerides[Kouba, 1995].A disadvantageto the fiducial approach is that errors in assumed locations of the fiducial sites distort the

(2) the transmitter parametersbroadcastin the navigation messageare accurate to a few meters for GPS

精密单点定位最原始的论文,J. F. Zumberge,1997

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GPS orbits in a way that is not easily undone. (This disadvantagedoes not apply to site coordinates produced by IGS analysis centers;such coordinatesdo not reflectstrongfiducialconstraints.)In our experience at

data will be required if the mistakes are significant enough. To avoid the consequences fiducial errors, a priof ori uncertaintiesof coordinatesare set fairly large at

Park, Canada). JPL's (Jet Propulsion Laboratory)IGS analysis center 1 km (10 m for the site at Algonquin[Zumberge al., 1995],such"errors"can arisein two Although the resulting"free-network"solution will be etways. First, the"standard" referenceframe, the Interin a poorly defined referenceframe, all estimated pa-

nationalTerrestrialReference Frame(ITRF)[Boucher rameters will be immune to fiducial errors, and exof et al., 1994]in the caseof the IGS, itself evolves with pensivereprocessing global network data is avoided.time, so that a discontinuity in assumedlocations of Furthermore, a particular reference frame can be refiducial sites will occur, typically at midnight between alized by transforming estimated receiver parameters two calendar years. These discontinuities are not re- into that frame. A differentframe (an updatedITRF, lated to actual motion of sitesbut indicate only a change for example)can be realizedsimplyby applyinga new in assumed locations. transformation. Estimated parametersof sites subject Fiducial orbits on December 31, 1994, that are based to precise point positioning with free-network transon ITRF 92, for example, will exhibit a discontinuity mitter parameters will be in the same r

eference frame comparedwith orbits on January 1, 1995, that are based as the original free-networksolution. A single sevenon ITRF 93. These discontinuitiesare small, typically a parameter transformation can be applied to all sites for few centimetersand can be reducedby applying a seven- alignment with the ITRF. The appropriate transformaparameter Helmert transformation but only to the ex- tion shouldbe availableto usersfrom the analysiscenter tent that the two referenceframes are related by such that produced the free-network orbits and clocks. The a transformation. Without reprocessing discontinu- free-networkapproachhas been demonstratedfor Earth the[e.g.,Herringet al., 1991],baselines[He in ities cannot be entirely eliminated, and a time seriesof orientation[Blewittet al., 1992],and veresultsderived from fixed-orbit analyseswill inherit the et al., 1992],coordinatesdiscontinuities. The effect can be several millimeters in

locities[Feiglet al., 1993;Argus andHeftin,1995].Single-Day Tests

positions.

Second,mistakesin assumed antenna heightsof fiducial sites, or other similar blunders, cannot be completely eliminated. Expensive reprocessing global of

Transmitter

parameters.

Shownin Figure 2 areare constructed to

sites from which various networks

.................. .... ...... ...:!ii . := i :!:? :.i .?;:::: :::; ::; ": ............. ..... ......... . '................ i :.,.%::* % ...... '................................. ...........:;::; ........ .....;:i :. . ......... . ....... ............... i o .:. o

........ :..: :::::. .: :-.'i -..43 ........................ '..:' '" ':::::I: ............ ':' .... .........:::'"::::i:11'": .: ..... : ::. ::ii':: i ........ ..... 43 ii:' ':.... '......

........ .?::i. ......... 0ß:"...........

?_ o o - .. ::' ...............'.- ..:.::. o

:..o?!....? n:; i:;!:. i: :. ..........::: ß ..... i ....:::: '............ !.. ...... .................... 11 :'ii : .:"'i

.::::"'

...........

'"::

...... 0

: i!:

...... ':':'"

'"::::..... i; !:=

: ::.:iii:' :::' ../

. ii!i:=

ß

..................... ..::::::'"'" i

' .. ..

'............... i!

":::: ::'.::. '::::":: :.. '" ":::L....:.....:...

?

'"':::::' '

.:::?: .... ...:.-'::'""::'" .:::::?...:,..:.-':"

"::....!;:.

............................................................ ...:. f:? ....

(3

.

......................................................................ß ........................................................................................................ : :::::' : =( .......................?:::i: ......................................................................................

. ....................................

.......................

Figure 2. Sites used for tests of point positioning. The nine large solid circl

es correspondto receivers from left to right, Usuda, Japan; Tidbinbilla, Australia; Kokee Park, Hawaii; Casey, at, Antarctica; Fairbanks,Alaska;Pietown, New Mexico; AlgonquinPark, Canada; Fortaleza, Brazil; and Metsahovi, Finland. The 12 smaller solid circles correspondto receiversat Lhasa, China; Perth, Australia; Guam, Pacific Ocean; Irkutsk, Russia; Chatham Island, New Zealand; Thule, Greenland; Arequipa, Peru; St. Croix, U.S. Virgin Islands; La Plata, Argentina; Maspalomas, Canary Islands; Noto, Italy; and Kerguelen Islands, Indian Ocean. The dotted open circles correspond three sitesin Australia and New Zealand, eight sites in North America, and seven to sites in Eurasia. Finally, the small open circlescorrespondto additional sites in North America (eight) and Europe (two). Althoughthere are permanently operatingreceivers the southern in Pacific Ocean and Africa, their data on November I were either not available or of poor quality.

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Table 1. Average Formal Errors of Transmitter Parameters Estimated From Different Networks of Figure 2S9

Orbit 3-D, cm19.4

Var, cm5.9

Coverage,%90.5

Clock, cm9.7

Var, cm3.5

10

16.7

4.4

90.9

8.9

3.1

2122

6.15.9

1.01.0

98.998.9

4.74.6

0.40.4

3839 48 49

4.64.6 4.4 4.4

0.70.7 0.7 0.7

98.998.9 98.9 98.9

3.83.8 3.7 3.7

0.30.3 0.3 0.3

The nine-station network is shown as the nine large solid circles in Figure 2. The 21station network includes, in addition, the 12 smaller solid circles. The 39-station network includes the 21 plus an additional 18 indicated by the dotted open circles. Finally, the 49-station network includes all sites in Figure 2. Networks with S - 10, 22, 38, 48 consist of the network with 5' -F 1 receivers, with the addition or subtraction of the receiver in

Belgium (a dotted open circle in Figure 2). Thirty hoursof data centeredon GPS noon,November 1, 1995, were analyzed. The coverage column indicates that for the 9- and 10-

station networksthe geometrywas insufficientto determineclock parametersabout 9%of the time. The var columns indicate the 1-standard-deviation variability in the formal errors among satellites and over time for the 24-hour period of November 1, 1995. To remove contributions from reference-frame uncertainties, the locations of the nine large solid circles in Figure 2 are assumed known for purposesof Table 1 only.

determine transmitter parameters, in the Earth-fixed frame, based on November 1, 1995, data. The solid circles compose a network of 21 stations with good

pected 1/v/ , because thecontribution to from (1) duereference-frame uncertainty has been removed, as dis-

cussed above,and (2) the additionalsitesare clustered

global distribution. This network consists a sparse in already populated regions. of To what extent do the data from additional receivers nine-stationnetwork (large solid circles)with reasonaffect the estimated transmitter parameters? Shown abl

e globalcoverage exceptnear 90øE longitudeand an additional12 sites (small solidcircles)chosen give in Figure 3 is a comparison of transmitter parameters to improved global distribution.Given the distribution of receiversshownin Figure 2,

the uniformity of the global distributioncannot be improvedmuchbeyondthis 21-stationnetwork. ShowninTable 1 are the formal errors of estimated transmitter

35 30

1

3d rms clock

parametersas a function of the network used as data, assuming that the locationsof the nine large solid circlesin Figure 2 are preciselyknown. This assumption is similar to one made by IGS analysiscentersin producing fiducial orbits. In the presentarticle it is invoked only so that the formal errors of the transmitter parameters will not reflect reference-frame uncertainties. Free-

25

20cm

15

10

network estimated transmitter parameters are used in all analyses. Table 1 showsthat$ - 9 provides estimates of orbits and clocks that are, on the average, determined with formal errors that are factors of 4.4 and 2.6, re-

0 0

IlO

I30

I ......:: 40 50

20

number of receivers in global solution

spectively,larger than those determinedwith$ _ 38. Furthermore,the$ - 9 and$ - 10 networkshavelarge variations in the formal errors, as well as geometry thatis insu cient to determine clock parameters for about

Figure 3. Transmitter parameters determined with fewer than 49 receivers, compared with those determined from the 49-station solution. The solid circles

give the three-dimensional rms differencein satellite po10% of the time and/or satellites. sitions, while the open circles give the rms difference On the other hand, Table 1 also indicates that for in clock solutions. The gradual improvementbeyond S _ 21 the reduction in formal error that accompanies 20 receivers the global solution is costly in terms of in the increase in$ is somewhat less than would be excomputational burden.

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Table

2. Formal

Errors in Coordinates

of Point-Positioned

Sites

North, mmAverageVariation

East, mm2.00.5

Vertical, mm4.80.7

3-D, mm5.20.8

0.80.2

These reflect only assumeddata noise (1 m for P and 1 cm for L), the sampling interval, and a priori uncertainties in estimated receiver parameters. The data noise contribution dominates. By definition of point positioning, orbit and clock errors do not contribute to the formal errors. Thus formal errors are essentiallyindependent of the network from which global parameters were derived. (However, becauseof incomplete coverageas indicated in Table I, formal errors of point-positioned sites based on orbits and clocks derivedfrom the 9- and 10-stationnetworksare slightly,about 8%, larger than for the other networks.) The average row indicates the average value over all sites in Figure 2, and the variation row indicates i-standard-deviation variability.

estimated

from

different

subsets of the sites shown in

Figure 2. To avoid contributions to the uncertainties that arise from imperfect knowledgeof receiver coordinates only weak, 10-m to 1-km, a priori constraints are imposed on these, as describedearlier. Becausethe resulting transmitter parameters are in a poorly definedEarth-fixed reference frame, the rms differences shown

are after a seven-parameter (three translations,three rotations, and a scalefactor) transformationto align,orient, and scale the referenceËames to each other.

solution, the rms differencesin receiver coordinates due to this changein A are 0.7 mm in the north, 2.6 mm in the east, and 5.4 mm in the vertical. Shown in Figure 4 are the differencesin estimated receiver parameters as a function of$, with$ - 49 as the basis for comparison. To account for the reference frame uncertainty of the free-network transmitter parameters, all solutions are aligned to the 39-station results, using a seven-parameter Helmert transformation to minimize the rms coordinate difference of the

From Figure 3 it can be seen that the 9- and 10station networks provide transmitter parameters that differ by tens of centimetersfrom those estimated with$ _>21. On the other hand, the differencesamong the$>__

21 networks

is of the same order

as the formal

nine-station network. Clearly, the$ - 9 and$ - 10 networks give rise to several tens of millimeters differencesin estimated receiverparameters. Beginningwith$ - 21, however,receiverparameters are lessand less affected, and differencesare comparableto the formalerrors.

errors. Such differencesmay not be worth the significant increase in computational burden from$- 21 toS-49.

Receiver parameters. Another measure of the technique's validity is the degree to which estimated receiver parameters depend on the division of the total R- 49 sites into$ used in the global solution and the R-$ sites that are subsequentlypoint positioned.For reference we show in Table 2 the formal errors ofmm

50

I

I

I

I

ß40

vertical horizontal -

o

30

point-positioned sites. By definition, uncertainties in estimated transmitter parameters do not contribute to formal errors in point-positioned sites and so are essen-

20

tially independent$. (The exceptionhas to do with of$ - 9 and 10, for which insufficient coveragemeansthat transmitter clocks were not estimable about 10%

10

of the time.) The formal errorsare about 1 mm in thenorth component, 2 mm in the east, and 5 mm in thevertical.

0 0

I

i

lO

2O

3O

4O

5O

To improve estimatesof transmitter parameters near the midnight boundaries, the global solution uses 30 hours of data centered on GPS noon of November 1, 1995. Receiver parametersdetermined in the global solution correspondto the same 30 hours, whereaspointpositioned solutions correspondto only 24 hours of receiver data. To force all receiver parameters to correspond to the same interval, point positioningis applied to 24 hours of data from each of the 49 sites, and all receiver para

meters are reestimated. For the$ - 49

numberof receiversin globalsolution

Figure 4. Point positioningrms differences over all sitesin Figure 2, showingthe effectof different divisions

of R - 49 sitesinto$ used determine to transmitterparameters. All solutions alignedto the 39-stationrearesults, using a seven-parameterHelmert transformationto minimize the rms coordinate difference of the nine-

station network. Above 20 receivers the global soluin tion, receiverparametersdo not dependvery strongly on the division. The horizontal points refer to the sum in quadratureof the north and east components.

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We havealsocomputed rmscoordinate the differences areas and Africa, with no receivers. That is, in Figfor the 11 stationsthat participatedin the$= 49 global ure 2 the distancesfrom open circles, both dotted and solution but not that for$= 38. That is, if one uses small, to nearby solid circlesare small comparedto the distancesfrom isolated regionsto solid circles. the$= 38 network to determine transmitter parameFigures 3 and 4 implicitly assumethat the$ - 49 ters, then usesthoseand precise point positioning for 11 additionalstations,how do the resultscomparewith solution is the"truth" case to which$< 49 results the inclusionof the 11 stationsin the globalsolution? shouldbe compared. From Figure 3 one might conclude The answer is 2.5-mm rms in the horizontal and 9.1that there is a slight but steady improvementin the mm rms in the vertical. (If the site with the largest quality of transmitter parameters for S> 21. To a might be drawn from vertical difference is excluded, the rms vertical is re- lesserdegreea similar conclusion ducedto 6.8 mm.) For the verticalthis valueis slightly Figure 4 in the caseof receiverparameters.For several includingmismodeling, is not necessarily it the higherthan that shownin Figure 4 for$= 38. On reasons, the scale of the formal coordinate errors of the 11 sites, casethat$ - 49 representstruth more than, say,$ 2.6 mm horizontal and 4.5 mm vertical, the differences38.

Even with no mismodeling, Table 1 indicates only are marginally significant. marginalimprovement the ability to determinetransin (Sincethe$= 48 and$= 49 transmitterparameters differby only about 1 cm or less(Figure3), the 5-mm mitter parameters with increasing$ beyond$ - 21, rms difference the vertical component the$= 48 given the distribution of sites as shown in Figure 2. for for solidcirclein Figure4 may seemsomewhat high. How- In fact, for the site distributions and values of$ used ever, this is most likely explained by the reestimation here, formal errors of transmitter parameters are aplinearlyrelatedto , with drr /d 9.4 cm of receiver parameters once transmitter parameters are proximately per 103km anddrr2/d 3.8 cm per 103km, where determined,sothat all receiversolutions correspond to rrl denotes the three-dimensional rms orbit formal er24 hou

rsinsteadof 30 hours,as described above.) Geographical distribution and truth case. To ror and rr2 denotes the clock formal error. Thus an evaluate the uniformity of$ sites distributed on a increase in$ which is not accompanied by a decrease in is of little value in improving the overall, global sphere,we first definethe function quality of transmitter parameters, even in the absenceof mismodeling. Furthermore, known aspects of mismodeling include where0 is colatitude,½ is longitude,and r is the great a number of simplifying assumptionsrelated to tropovariations in receiver and transmitter phase centers, to name a few. Becauseof the geographicaldistribution of receivers,the$ - 49 solution will suppressmanifestations of mismodeling, or systematic errors, more in 47r Europe and North America than in other regions. In the presenceof systematic errors the least squaressoShown Figure5 is a plot of ( versus usingSimpin$, son's rule in two dimensions to numerically evaluate lution for$ - 49 may not be nearer to truth than that (10). Clearly,( decreases negligibly aboveS . 21, for$- 21 or$- 38. In addition to mismodeling,unwhich is simply a consequence large regions,oceanic even data quality from common or additional stations of could cause the$ - 49 solution to be better or poorer, respectively,than that for$- 38.The rms isolation is then

R(O,ok) min[r , r2,''- rs], --

circledistance between½) and siten. ThusR(O,ok) (0, sphericdelay,oceanloading,satellite forcemodels,and is the distance from (0,½) to the nearestof the S sites.

_ 2' dO<b)(10) fo f sin do) 0 R2(0,

4OOO

Weekly reportsby the IGS analysiscoordinator(see http://igscb.jpl.nasa.gov/IGSREPORT.html) evaluatethe quality of estimated transmitter parameters from all IGS analysis centers. The number of receiversused varieswith center and rangesfrom fewer than 30 to over 100. There is almost no correlation, however, between the quality of transmitter parameters and the number of sites. Our interpretation is that beyond$ 30 with the global distribution as in Figure 2, the value of data from additional receiversin determining transmitter param-

3000

2OOO

eters is marginal at best and may even be outweighed by systematicerrors. As new receiversare installedin currently isolated regionsthereby moving the curve i i i i downward, we of course expect that data from these o lO 20 30 40 50 new receiverswill be valuable in determining transmitnumber of receivers ter parameters. As of November1996, the globaldistriFigure 5. The rmsisolation (Equation asa func- bution is such that analysisof data from approximately 10)tion of the number of receivers.

lOOO

35-40 well-distributed sites is appropriate.

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Multi-Day

Tests

daily deviationsare alsoshown. From Figure 6 it is clearthat there is little or no discernible difference betweenthe two classes.

The operational GPS analysis JPL[Zumb

erge at et al., 1995]has beenin placesince1992. Shownin Figure 6 is the detrended time series of coordinates for

Point positiontime series from 151 sitescan be viewed

the receiver at JPL spanning 2.5 years following the Northridge earthquakeof January 17, 1994. The 109 circlescorrespond days when data from this receiver to were included in the global solution, while the 426 shaded points correspondto coordinatesdetermined from precise point positioning.The distributions the of

at http://sideshow.jpl.nasa.gov/mbh/series.html. Histogramsof daily repeatabilities based theseare shown onin the left of Figure 7. Repeatability is about the estimated linear trend. Sites with known and uncorrected

changes antennaheight are not included,leaving139 in sites. The number of days for a given site rangesfroma few to a few hundred; the median value is 106 days.

o-point= 4.0 mm o-global= 4.4 mm

latitude

(cm)-2 -4

I

I

I

I

I

I

o-point= 6.7 mm o-global= 6.0 mm

longitude0(cm)-2 -4I I I

o

o-point= 11.0 mm o-global= 9.9 mm

height 0(cm)-2 -4I I

I

I

I

1994

1995

1996

0

20

40

60

80

lOO

days

Figure 6. Time series geocentric of coordinates (detrended) the receiver the Jet Propulsion for at Laboratory (JPL), beginning January1994after the Northridge late earthquake. The opencircles indicate 109 days when JPL data were includedin the daily global solution. The shadedpoints indicate 426 days when JPL data were analyzed with precisepoint positioning. Shown also are the distributions deviations the global (dotted line) and point-positioned of for (solidline) solutions; histograms the havea 2-mmbin width. The differences the means in and widths(as)of the distributions are both small and statistically insignificant.

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_lO

,.;::,

latitude,,

_,

,,

__

sites

:

!

:.

::__

i

..... i

-

_

lO

--i

i- ,

longitude:....:__

--

'-'1

I

_

_

sites_;..' ',

__

!

t .......:,

-

-!"i

': F

'lq

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:

' ....

-'!

_..

vertical

-

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-

:

-

_.7.,,

.

:..!i

:-

i

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0

li ........5 10

115

- 'Fimm

F--0

i5

,i10

" .... ....15 20

Figure 7. Distributions daily repeatabilities of (dottedlines)for 139 sites(left) that havebeen analyzed with precise point positioning and 59 sites(right) that participatedin manydaily global solutions.The solidlinesindicatesitesin the southern hemisphere on the left and 15 on the (23 right). The medianvaluesare 3.9 mm in latitude (north), 6.3 mm in longitude(east), and 11.1mm in the vertical for the point-positionedsites and 4.4 mm in latitude, 5.6 mm in longitude, and 10.3 mm in vertical for global sites.

Except for a few outliers, someof which are due to receiverswith a history of poor performance,the repeatabilities do not vary too much from site to site and, in particular, do not depend strongly on northern versus southern hemisphere. The median values are 3.9 m

m for the north component, 6.3 mm for the east, and 11.1mm for the vertical.

It is well known that detailsof an estimationstrategy (the choice relativeweightbetween of pseudorange andphase measurements, the a priori uncertainties in parameters, and the model of tropospheretemporal vari-

ability) can changethe estimatedvaluesof parameters with little changein the quality of the fit, reflecting some degree of degeneracyamong parameters. Nevertheless, the concept of satellite clock is no less valid in principle than that of satellite orbit. For GOA-II theformal errors of orbits and clocks from Table I indicate that to the few-centimeter level or better there is

For comparison,the corresponding valuesfor 59 sites included in global analysesare 4.4 mm, 5.6 mm, and10.3 mm; their distributions are indicated on the right of Figure 7. The differences insignificant.What was are demonstrated detail for the site at JPL (Figure 6) is in thus true in general.

Mismodeling There are severalhigh-precisionsoftwaresystemscurrently in useto analyze GPS data, of which Gipsy-Oasis

strength in the data to determine them separately from other parameters. Comparisonsamong resultsfrom sevendifferent IGS

analysiscentersusingsix distinct softwaresystems have shownthat estimated parametersfrom different analy-

II (GOA-II) usedin this work is one. Can satelliteparameters determined with one system be used in other? To the extent that there is consistencyamong software systemsin how GPS observablesare modeled as a function of these parameters, the answer is yes.

sesagreeto (1) about 20 cm, three-dimensional (3-D) rms, for daily orbits; (2) a few millimetersfor horizontal stationcoordinates, averaged over7 days;(3) abouti cm for vertical station coordinates,averagedover 7 days;and (4) about 5 mm for daily estimatesof total zenith troposphericdelay. Theselevelsof agreementin-

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dicate that the modeling among the different systemsisvery similar.

to the single-site coordinaterepeatabilities, depending on component, baseline for lengthsup to about 200 km.That this is not reflected in the formal errors for base-

(By examining the covariancematrix of estimated site coordinates amongsitesdeterminedin our regular ent sets of satellite parametersused: (1) JPL clocks global solutions,we may be able to determine approxideterminedevery 5 mid and JPL orbits and (2) Geo- mately correctsite-to-sitecorrelations sitessubject for ForschungsZentrum (GFZ)[Gendt et al., 1995]clocks to precise pointpositioning. example,= e-xd'J For pij determined every 30 mid and GFZ orbits. Parameters might modelthe correlationbetweensitesi and j sepafrom set 2 were retrieved from the NASA Crustal Dy- rated by distance dij, with/k an empiricalparameter.) A consequence point 2 is that formal errors of of namics Data InformationCenter(CDDIS)[Noll, 1995].The 3-D rms variation in satellite positions over thepoint-positioned results fr

om isolated sites will be nodifferent from those in dense areas. We use"isolated"

To further quantify the level of mismodeling,or different modeling, among the different systems,we have applied precise point positioning to 94 sites, globally distributed, using data from March 4, 1996. GOA-II was used for all analyses, but there were two differ-

lines is the result of a compromisebetween mathematical correctness and computationalfeasibility.

day (March 4, 1996) betweenJPL and GFZ is 33 cm. (For the presentpurpose,fiducial orbits are used. The improvementin orbit agreements negligibleif a sevenis parameter transformation is allowed, indicating good reference frame agreement betweenJPL and GFZ.) Therms agreementbetween JPL clocksand GFZ clocksis 23 cm. Because precise GFZ clocks are available from the CDDIS only at 30-mid intervals, the data rate available for analysis is 6 times less frequent than for the JPL

here in the sensethat data from no nearby receivers were used in the global solution that produced thetransmitter parameters. Transmitter parameters are probably not as well determined in such areas. This

effect,alreadysmallas evidenced the lack of strong by northern/southern hemisphere dependence Figure7 in (left), will become negligible the globalnetworkconastinues its expansion.

A secondlimitation has to do with the availability of precisetransmitter clockestimates. The technique The rms differences in estimated site coordinates dewill yield resultsof the quality shownhere only when termined in set 2 relative to set I are reasonablygood: theseare accurate the few-centimeter to (100 ps) level 9 mm in the north, 16 mm in the east, and 28 mm in at each data epoch. Becauseof selectiveavailability, the vertical. (We have had lesssuccess mixing or- GPS clocks in vary by the orderof 103 cm (101 to l02bits from one analysis center with clocksfrom another, possibly becauseof correlations in orbit and clock er-

solutions (seediscussion selective on availabilityin the sectionon limitations).

ns)overtimescales minutes. of Thusprecise knowledgeof their valuesevery, say, 5 min is of no use in predicting their valuesmore frequently, at least not to the few-centimeter level. As mentioned earlier,navigationmessage clockcorrections, thoughavailablein real time, are essentiallylinear approximationsover 102 rain

rors.) The dominant contributor to these valuesmay be the decreasedamount of data analyzed using GFZtransmitter parameters. To the question posed at the beginning of this section, we can thus say, yes, at thecentimeter level. To establish level awaits further work. this at the few-millimeter

andareaccurate onlyto the 103-cm level.Some participantsin the IGS[Kouba, 1995]nowroutinelyproduceGPS clockestimateswith subnanosecond precision at least as frequently as every 15 min and every 5 min from somecenters. However,until preciseGPS clocks are producedmore frequentlythan onceevery minute and preferably onceevery10 s, data to be an

alyzed withprecise point positioning will have to be decimated to

Limitations

One limitation to the precisepoint positioningtechnique is that the covariancematrix of estimated receiver parameters will not necessarilybe indicative of the ac-

tual quality of resultsbecause(1) it will showno cor- coincidewith epochsfor which precise GPS clock solurelation betweensitesand (2) its nonzeroelements will tions exist. be independentof receiverlocation. Consequently, the To achievesubcentimeterresults requires dual frecovariancematrix may be inappropriate for someappli- quency, geodetic-quality receiversso that ionospheric cations. variationscan be differenced away and data cleaning Because of point 1, if the location of one site from can be done on a single-receiver basis. Whenever the point positioning is estimated as with covarianceV results from a givenreceiver limited by the accuracy are and the location of another site is estimated as b with of orbits and clocks the broadcast in message, however, covariance Vb, then the baseline estimate between these useof precise clocks and orbitswill providean improvesites is - xq with covariance V + Vb. Common error ment, even for inexpensivereceivers. sources, from transmitter parameters, for example, do A detaileddiscussion the integerresolution phase of ofnot contribute to the individual covariances and cannot

bias parameters[Blewitt, 1989] is beyondthe scopeof this article. Because of receiver- and transmitter-

get removedin the formation of the vector difference. Real measuresof performance, like daily repeatability, will show an improvement if common error sources are differencedaway. For example, repeatabilities of baselinesformed from point-positionedresultsof southern California sites are smaller by 20%-60% compared

specificphase delays, which are generally not known and may vary with time, it is only the double-difference phase biasesthat will be integers. Thus resolution of these parameterscannot be performed on pointpositionedresultsfrom a single receiver. Results from

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multiple point-positioned sites in a local network can be combinedto resolvedouble differencephase ambiguity parameters, with reduction in baselinedaily repeatabilities, especiallyfor the east and vertical components. For those applications that do not require accuraciesbetter than a few millimeters in the horizontal dimen-

Acknowledgments.and referees for their

We thank the Associate Editoruseful comments and criticisms. The

sion and approximately 1 cm in the vertical dimension, ambiguity resolutionis not necessary, providedthat the observationtime is of the order of 1 day; for shorter observingtimes, ambiguity resolutionis increasinglyvaluable. For applicationsinvolving long baselines,for example, global networks,ambiguity resolutionis difficult at best. For denseregional networksthe rigoroussolution of resolving double-difference phas

e biasesinvolving hundreds of sites is not computationally feasible. Most approachesuse clusters of stations with overlapping subnetworks; these probably suffer somedegradation when compared to a rigorous solution. Additional work is necessary quantify the trade-off betweencomto putational feasibilty and mathematical rigor as it relates to ambiguity resolution.ConclusionsWe have shown that a robust and economical means

global network of precision GPS receiversis the result of international collaboration among many groups. The authors are grateful for the rich data set that this network provides. This work was performed at the Jet Propulsion Laboratory, California Institute of Technology,under contract with the National Aeronautics and Space Administration. References Argus, D. F., and M. B. Heftin, Plate motion and crustal deformation estimated with geodeticdata from the Global

PositioningSystem, Geophys.Res. Lett., 22 (15), 19731976, 1995.

Bar-Sever, Y. E., W. I. Bertiger, E. S. Davis, and J. A. Anselmi, Fixing the GPS Bad Attitude: Modeling GPS Satellite Yaw During EclipseSeasons, Navigation, 3 (1),1996.

Bierman, G. J., Factorization Methods for Discrete Sequential Estimation, Academic, San Diego, Calif., 1977. Blewitt, G., Carrier phase ambiguity resolution for the Global Positioning System applied to geodetic baselines up to 2000 km, J. Geophys. Res., 9 10,187-10,283, 1989. Blewitt, G., M. B. Heftin, F. Ho Webb, U J. Lindqwister, and R. P. Malla, Global coordinates with centimeter accuracy in the international terrestrial referenceframe using

GPS, Geophys.Res. Lett., 19 (9), 853-856, 1992.

Blewitt, G., M. B. Heftin, K. J. Hurst, D.C. Jefferson, of analyzing data from hundredsor more GPS receivers F. H. Webb, and J. F. Zumberge, Absolute far-field disinvolves(1) analyzinga globally distributed subsetto placements from the 28 June 1992 Landers earthquake determinetransmitter parameters (and receiverparamsequence, Nature, 361, 340-342, 1993. eters for the subset)and (2) analyzingthe remaining Blewitt, G., Advances in Global Positioning System Tech-

nology for Geodynamics Investigations: 1978-1992, in data one receiver at a time. The quality of results for Contributions of Space Geodesyto GeodynamicsTechnolstation parameters is the same whether the receiver is ogy, Geodyn. Set., vol. 25, edited by D. E. Smith and in the first or secondcategory. Beyond a certain numD. L. Turcotte, pp. 195-213, AGU, Washington, D.C., ber the quality of transmitter parameters does not im1993. prove by including more receivers the first category. Bock, Y., et al., Detection of crustal deformation from the in

toring, Bull. Am. Meterol. Soc., 77 (1), 1996. Feigl, Kurt L., et al., Spacegeodeticmeasurementof crustal deformation in central and southern California, 19841992, J. Geophys. Res., 98, 21,677-21,712, 1993. Gendt, G., G. Dick, and Ch. Reigber, IGS Analysis Center at G FZ Potsdam, in International GPS Service for mately10 sites beprocessed a si

ngle can with 40-Mfiop Geodynamics 1994 Annual Report, edited by J. F. Zumcomputer every day All solutionsderived from a set of berge, R. Liu, and R. E. Neilan, JPL Publication 95-18, Pasadena, Calif., 1995. fixed transmitter parameters are in the same reference usingGPS without fiducial frame, which is the referenceframe of the global solu- Heftin, M., et al., Global geodesy sites, Geophys.Res. Lett., 19 (2), 131-.134,1992. tion from which the transmitter parameters were determined. We expect that the efficacyof the techniquewill Herring, T. A., D. Dong and R. W. King, Sub-milliarcsecond determination of pole position using Global Positioning be demonstrated in the denseGPS arrays in Japan and System data, Geophys.Res. Lett., 18 (10), 1893-1896,southern California.1991.

The number of receivers that belongin category(1) dependson the global distribution. Although it is currently about 35-40, that number may increase as permanently operating receiverscome on-line in currently isolated regions. Current interest in GPSõeodesyis the densification of the terrestrial reference frame[Zumberge and Liu, 1994]. A commonview, in our opinionerroneous, is that the ability to processGPS data from hundredsor thousandsof receiversevery day, usingeconomicalcomputers, is beyond the capability of any single analysis center. With the technique described here, approxi-

Landers earthquake sequenceusing continuousgeodeticmeasurements, Nature, 361, 337-340, 1993. Boucher, C., Z. Altamimi, and L. Duhem, Results and analysis of the ITRF93, IERS Tech. Note 18, Obs. de Paris, Paris, Oct., 1994. Businger, S., S. R. Chiswell, M. Bevis, J. Duan, R. A. Anthes, C. Rocken, R. H. Ware, M. Exner, T. VanHove, and F. S. Solheim, The promise of GPS in atmospheric moni-

It has long been recognizedthat preciseGPS orbits, such as those producedby analysiscentersin the IGS, improve the quality of GPS analysis. However, the value of precise GPS clocks has not been widely recognized. As we have shown, together with preciseorbits, they allow analysis of data from one receiver at a time with few-millimeter daily precisionin horizontal componentsand centimeter precisionin the vertical.

Kouba, J., Analysis coordinator report, in International GPS Servicefor Geodynamics 199 Annual Report, edited by J. F. Zumberge, R. Liu, and R. E. Neilan, JPL Publication 95-18, Pasadena, Calif., 1995. Kuang, D., B. E. Schutz, and M. M. Watkins, On the structure of geometric positioning information in GPS mea-

surements,J. Geodesy,71 (1), 35-43, 1997.Larson, K. M., and J. Freymueller, Relative motions of the Australian, Pacific and Antarctic plates estimated by the

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Yunck, T. P., GPS data, acquisition,environmentaleffects, U.S. Natl. Rep. Int. Union Geod. Geophys.,1991-199 ,Rev. Geophys., 33, 349-352, 1995.

Lichten, S. M., Estimation and filtering for h

igh-precision GPS positioning applications, Manuscr. Geod., 15, 159176, 1990.

Lindqwister, U. J., A. P. Freedman, and G. Blewitt, Daily estimates of the Earth's pole position with the Global Positioning System, Geophys.Res. Lett., 19 (9), 845-848,1992.

Melbourne, W. G., Sounding the Earth's atmosphere and ionosphere with GPS, Eos AGU Trans., 76 (,16), 465-466,1995.

Zumberge, F., and R. Liu (Eds), Densification the IERS J. of Terrestrial ReferenceFrame ThroughRegional GPS Networks,JPL Publication 95-11, Pasadena,Calif., 1994. Zumberge, J. F., M. B. Heftin, D.C. Jefferson, M. M. Watkins, and F. H. Webb, Jet Propulsion Laboratory IGS AnalysisCenter 1994 annual report, in International GPS Service for Geodynamics 199,1Annual Report,edited by J. F. Zumberge, R. Liu, and R. E. Neilan, JPL Publication 95-18, Pasadena, Calif., 1995.

Noll, C. E., CDDIS Global Data Center report, in International GPS Servicefor Geodynamics199 Annual Report, edited by J. F. Zumberge, R. Liu, and R. E. Neilan, JPL Publication 95-18, Pasadena, Calif., 1995. Webb, F. H., M. Bursik, T. Dixon, F. Farina, G. Marshall, and R. S. Stein, Inflation of Long Valley Caldera from one year of continuousGPS observations, Geophys.Res. Lett., 22 (3), 195-198, 1995. Wilson, B. D., A. J. Mannucci, C. D. Edwards, Subdaily northern hemisphereionosphericmaps using an extensive network of GPS receivers, Radio$ci., 30, 639-648, 1995.Wu, J. T., Estimation of clock errors in a GPS based tracking system, paper presented at the Am. Inst. of Aero-

Zumberge,J. F., and W. I. Bettiger, Ephemerisand Clock NavigationMessage Accuracy, GlobalPositioningSysin tem - Theory and Applications,vol. I, edited by B. W. Parkinson and J. J. Spilker, 585-599, Am. Inst. of Aeronautics and Astronautics,Inc., Washington,D.C., 1996.

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naut. and Astronaut./Am. Astronaut. Soc. Astrodynamics Conference, AIAA, Seattle, Wash., 1984. Wu, J. T., S. Co Wu, G. A. Hajj, Wo I. Bertiger, and S. M. Lichten, Effects of antenna orientation on G PS carrier phase, Manuscr. Geod., 18, 91-98, 1993.

4800 Oak Grove Drive, Pasadena,CA 91109-8099. (email: mbh@cobra.jpl.nasa.gov, djeff@cobra.jpl.nasa.gov, mmw@cobra.j pl.nasa.gov, fhw@cobra.jpl.nasa.gov,jfz@cobra.jpl.nasa.gov)

(Received May 3, 1996; revised December 1996; 2,acceptedDecember9, 1996.)