GEOPHYSICAL
RF-.•EARCH
LETTERS,
VOL.18,NO.6,PAGF_.•
1135-1138,
JUNE1991
FEW MILLIMETER
PRECISION FOR
BASELINESIN THE CALIFORNIAPERMANENT GPSGEODETIC ARRAY
Ulf J.Lindqwister,
James
F.Zumberge,
FrankH. Webb,andGeoffrey
Blewitt
JetPropulsion
Laboratory,
California
Institute
of Technology
Abstract.Geodeticmeasurements
with RogueGlobal
Positioning System (GPS) receivers from sites in the
CaliforniaPermanentGPS GeodeticArray (PGGA) have
been analyzed using the GIPSY orbit-determinationand
baseline-estimationsoftware. Based on an unbiasedselection
of 23 daily measurementsspanning8 months,the low-
frequency(•> weeks) contributionsto the long-term
repeatabilities
of baseline
measurements
areapproximately
5,
3, and 8 mm for the east,north, and verticalcomponents.
Short-term
contributions
to thelong-termrepeatabilities
were
evaluatedby examiningdata from the week of October21,
1990, which showed the lowest short-termscatter.For this
week,daily repeatabilities
of 2-3 mm in the horizontaland4
mm in the vertical have been achieved for the 172-kin JPL-
Pinyonbaseline,consistentwith carrierphasedatanoiseof
~6 min. High quality (%5 mm) repeatabilities
havebeen
achieved
for all components
of theotherbaselines
aswell.
Methods
The PGGA currentlyconsistsof GPS receiverslocatedat
JPL (Pasadena),ScrippsInstitutionof Oceanography(La
Jolla),andPinyonFlat Observatory,approximately172 km
ESEof JPL(Figure1). Eachstationis equippedwitha Rogue
SNR-8
receiver,
which
measures dual
band P-code
pseudorange
andcarrierphaseobservables,
and a choke-ting
backplane
antennato minimizemultipath.The monuments
at
JPL andPinyonFlat are collocatedwith Very Long Baseline
Interferometry(VLBI), and hence GPS estimatesfor this
baseline can be compared against an independent
measurement
techniquefor assessing
accuracy.
For referenceframecontroland improvedGPS orbits,data
from three Cooperative International GPS Network
(CIGNET) sitesin North America were combinedwith the
PGGA
data. The CIGNET
sites are located at Richmond
(FL), Mojave (CA), and Westford (MA). Each station
employsa MiniMac 2816 AT receiver,providingdual band
carrierphaseobservablesonly. For both receiversall data
typesweredecimatedto 6-minuteintervals.The coordinates
Introduction
A continuouslymonitoredarray of three GPS Rogue
receivers
wasdeployed
in southern
California
in thespring
of of the CIGNET stations and the JPL site were held fixed at a
1990.The arrayhassincebeenoperated
jointly by theJet
priori valuesbasedon the VLBI GLB223 solution(J. Ryan,
Propulsion
Laboratory(JPL) and ScrippsInstitution
of et al., GoddardSpaceFlight CenterVLBI group,unpublished
Oceanography.
GPSnetworks
of thisnaturearecapable
of
results,1987).The ScrippsandPinyonsiteswereestimated
yielding
highprecision
(fewmmoverseveral
hours)
orhigh asconstantsaswere all satellitestateparametersandcarrier
temporal
resolution
(~cmoverfew minutes)displacementphasebiases.Clockswere estimatedas white processnoise
measurements
over baselinelengthsup to severalhundred
and all the a priori sigmas were left unconstrained
[Lindqwister
et al., 1990].
numerous
applications
including
mapping
thestrainpattern The tropospheric
pathdelaysmay be subdivided
into a dry
in faultzones;
pre-,co-,andpost-seismic
faultmonitoring;(hydrostatic)anda wet delay. The dry delay is typicallyvery
kilometers.Such networks will facilitate the use of GPS for
long-termtectonicdeformation
studies;
monitoring
of stable and a nominal offset of 2 m was removed. The total
volcanic
uplift;studying
sealevelchanges;
andsupportingwet delayandthe residualdry delay were estimatedusinga
preciseorbit determination
of remotesensing
satellites zenithbiasparameterand the Lanyi wet mappingfunction
carryingGPS receivers. In addition, permanentarrays
[Lanyi, 1984]. For the October data set low-angle
providean opportunityto studyGPS systematic
error mismodelingwas studiedby varyingthe low elevationcutoff
sources,such as antennamultipath,tropospheric
delay angieasdescribedbelow.The wet delayvariesconsiderably
mismodeling,
andsystemnoise.
In thisletter,PGGAdatafromselected
weeksin 1990
havebeenanalyzed
to obtain
preliminary
estimates
of longtermbaseline
repeatabilities.
Theselection
criteria,
believed
tobestatistically
unbiased,
wereessentially
dataavailability,
short-term
continuity,
andlong-term
coverage.
In addition,
an
in-depth
study
ofdatafromtheweek
ofOctober
21-27,
1990
ispresented.
Thepurpose
hereistodemonstrate
thelevelof
short-term
precision
which
maybeobtained
withGPSduring
idealobserving
conditions.
Forthisparticular
weekthreedimensional
short-term
repeatabilities
atthefew-millimeter
levelwereobtained
forbaselines
upto 170kmin length.
Copyright1991 by the AmericanGeophysicalUnion.
Paper number 91GL01289
0094-8534/ 91 / 91GL-0128953 . 00
Fig. 1. The CaliforniaPGGAsitesareshownin relationto
nearbyfaults(dashed
lines).
1135
1136
Lindqwister
et al.:Few-Millimeter
GPSBaseline
Precision
over a day and was estimatedstochasticallyby using a
randomwalk stochasticmodel [Lichten, 1990]. The random
walk model is characterizedby the time derivativeof its
variance,sethereto (6 cmday-I/2)2.
The analysisof datafrom the PGGA andCiGNET sites
was performedwith the GIPSY orbit determinationand
baselineestimationsoftware [Lichten, 1990]. The two data
streams are processed separately, because of different
analysispaths.The Rogue data are edited with Turboedit
[Blewitt, 1990], which relies on dual band P-code
pseudorange
andcarrierphaseobservablesto performrobust
carriercyclerepairafterlossesof lock. BecausetheMiniMac
has no P-code capability, a codelessalgorithm has been
developedwhich single differencesdata betweenCIGNET
receivers
to eliminate
satellite clock offsets and effects of
SelectiveAvailability. A stationlocationand satellitemodel
is subtractedfrom the differenceddata, thusproducinga
smooth residual, which is input to a Kalman filter. If
discontinuitiesare found by the filter, then the receiverwith
the cycle slip may easily be determinedby comparingthe
residuals in each of three differenced
files. The correction is
Eachplottedpointcorresponds
to a single-day
solution
witha
para_meter-estimation
strategyessentiallyidenticalto that
usedin producing
theresultspresented
in detailbelowforthe
weekof October21. The verticalbar accompanying
each
pointreflectsthe formalerror.The areabetweenthe dotted
linesin Figure2 is theplus-or-minus
onestandard
deviation
regiondefinedby ratesfrom the VLBI solutionGLB659
[Caprette
et al., 1990],with an offsetadjusted
to minimize
theweighted
mean-square
difference
fromthePGGAresults.
The offsetswere21 mm east,-22 mm north,and24 mm
vertical.Thesebiasesmay be due to unknownVLBI-GPS
phasecenteroffsetsandgroundtie errors.
It is clearfromthefigurethattheshort-term
scatter
of the
PGGAresults
is oftenconsiderably
lessthanthelong-term
scatter.A simplemodelfor theerrorin themeasurement
ofa
baselinecomponent
[Daviset al., 1989]is
e=
+ s62,
(2)
where <• scalesthe low-frequency@weeks)errorsands
scales
theshort-term
errors.In (2), •5•and•52areindependent
finally appliedto the undifferenceddata. Alternatively,an
extra phaseambiguityparameteris introducedif the cycle-
randomvariables
withzeromeanandunitvariance.
61isa
slowly-varying
quantity,while•52is essentially
whitenoise.
slip cannotbe resolved.
Simultaneousprocessingof the RogueandMiniMac data
is theoretically better since it increasessatellite mutual
visibility. However, this would requirecarefultreatmentof
Thelong-termrepeatability
will be(G2+s2)
I/2.
The datain Figure2 can be dividednaturallyintofive
groups,with eachgroupcontaining
two or moreconsecutive
antennaphasevariationsbetweenthedifferentantennatypes
repeatability
sicanbeestimated
empirically
fortheithgroup
asthestandard
deviation
of themeasurements
in thegroup.
used in C!GNET
and the PGGA.
Parameter estimation is
or nearly-consecutive
daily measurements.
The short-term
performedseparatelyfor the PGGA andCIGNET networks. To estimatethe low-frequencycomponent(y, we usethe
To combine information (estimatesand covariances)from
eachnetwork,a weightedmethodis used.If x l and x2 are
independentestimates with covariancesP1 and P2, the
combined estimate is
x: P (P1-1Xl+ P2'1 x2),
(1)
where P = (P1-1+ P2-1)-•is thecombinedcovariance.
The orbital information(satelliteposition,velocity, and
solar radiation pressure coefficients) from the C!GNET
likelihood function
5
[ •xi2
]
L((•)o•
I-I
1
exp(3)
i:1%/271;(G2+$i
2)
2((j2+Si
2)
to indicate the level at which the data are consistent with a
givenvalueof (•. In (3), Si is the uncertaintyin the average
of
theithgroup(= sidividedbythesquare-root
of thenumber
of
daysin thegroup)and•Sxiis deviationof thei maverage
from
solutionis extractedand combinedwith the corresponding the weightedaverageof all points. (The resultsare largely
of rate.)The
solutionfrom the PGGA sitesusing an algorithmwhich insensitiveto reasonablea priori assumptions
effectively implements (1) [e.g. Bierman, 1977]. The value of <•which maximizesL((•) is the maximumlikelihood
resultingRoguesolutionsareconsiderably
improvedwith the estimate(MLE) of the true value of (• [e.g., Mathewsand
additionof thewider aperturecontinental
orbitaldata.For the Walker, 1970]. The resultsare summarizedin Table 1 for all
CaliforniaPGGA over 90% of all the carrierphasebiases baselinesin the array. The quotedlower (upper)number,
when addedto the MLE, is the value of <• below (above)
were resolved, with a cumulative confidence limit of 99%.
For the seven single-day solutionsobtainedwith data from whichtheintegralof L(<•) is 15.87%of its valuefrom0 to
wheretheMLE is zero,we quoteonlythe
October 21-27, 1990, ambiguity resolution resulted on In thoseinstances
upperlimit.'
averagein a factor of 3.6, 1.5, and 1.1 improvementin the 84.13%-confidence
daily repeatabilities in the east, north, and vertical
The PGGA datain Figure2 arenot inconsistent
withthe
componentsfor all three baselines,which is consistentwith
previousapplicationsof ambiguityresolutionin California
[B!ewitt, !989]. Note that the PGGA carrierphasebiases
could not be resolved without orbital enhancement from the
CIGNET
rates from the VLBI
GLB659 solution. However, because
Table 1 Low-frequencyErrors(• (millimeters)
data.
Baseline
East
Pinyonto JPL (172 km)
,:+5.8
.U_l.
0 < 1.9
Scrippsto JPL (171 km)
.z._0.8 z..t_0.5< 5.4
Results
Long-termRepeatability
Shownin Figure2 are the resultsof the analysisfor the
Pinyon-to-JPLbaselinefor a period spanning~8 months.
Scrippsto Pinyon(111 km)
,,+5.1
< 2.9
North
Vertical
8:5
+9'2
ß -2.5
.• •+3.3
• ,•+3.7
-'.'*-0.7
o ,•+9.8
o.:,-2.0
Lindqwister
etal.:Few-Millimeter
GPSBaseline
Precision
i
i
i
i
i
-'
for thisweek.EclipsingoccurswhentheGPS satellitemoves
in andoutof the shadowof the earth,eachtime experiencing
an impulseforce due to the changein the solarradiation
pressure,
whichis currentlynot modeledcorrectly.The daily
repeatabilities
for all baseline
components
fromthisweekare
summarizedin Table 2. The day-by-dayresultsfor the JPLPinyon baselineare plotted in Figure 3. An earlier high
precisionGPS study showed similar few-mm horizontal
repeatabilities
for 200-700-km baselines[Lindqwisteret al.,
•990]. In the current study, however, the vertical
repeatabilities
haveimprovedby a factorof-3 andare now
at,thehalf-centimeter
level. The improvementis mostlikely
dueto bettersatellitecoverageandtheuseof 8 channel,low
multipathreceivers(Rogue),as comparedto the 4 channel
TI-4100receivers
employedin thepreviousstudy.
The formalerrorbarsdrawnin Figure3 accountfor data
noise,the geometricstrengthof the satelliteconfiguration,
4O
2O
o
z
1137
0
-20
6O
40
20
and the numberof measurements.To assessthe consistency
of the observeddaily repeatabilities
with the formalerrors,
we showin Figure4 a plot of the formervs. the latter.Four
6O
components(east,north, vertical, and length) from each of
the threebaselinesare shown.The asymmetryof the vertical
40
errorbarsarisefrom the asymmetric
Z2 distribution,
which
2O
0
-20
-40
.
-60
-80
I
!00
150
I
200
i
250
i
300
,
,, i
350
......
400
1990 day
Fig. 2. Long-term repeatability for the Pinyon-to-JPL
baseline.The regionenclosedby the dottedlinesindicatesthe
VLBI solutionGLB659 extrapolated
to 1990.
their low-frequencyerrors are significant,estimatesof
baseline
ratesin this arraywouldbe premature
at thistime.
The results are neverthelessvaluable in that they imply
specific
observation
timesrequiredto measure
a givenrateat
somelevel of significance.If one assumes
that multi-day
averaging
caneffectivelyeliminatetheshort-term
errors,
and
thatthelow-frequency
errorsbecomeuncorrelated
oversome
time'•, then to measurea rate R to within 4"tJ
R wouldrequire
a period
describes
the degreeto which an rms can be estimatedfrom
only 7 measurements.
The magnitudeof the data noise, to
which formal errorsscale,has been set to 6.3 ram, giving a
slope of---! in Figure 4. The observed fluctuations in the
verticalare slightlylower than would be expectedfrom the
resultsof the othercomponents.
The 6.3-mmdatanoisealso
agrees with the post-fit residuals for the carrier phase
observable.For example, the 12-hour estimated standard
deviationof thecarderphasenoisebasedon post-fitresiduals
for the CIGNET sites averaged 6 mm on October 25.
Similarly for the PGGA sites on October 21 the residuals
averaged4-5 mm.
Sensitivity
of BaselineEstimatesto EstimationStrategy
The nominal estimationstrategycombinesC!GNET and
PGGA solutions,cutsoff low-elevation-angle
data,andtreats
the troposphere
as a randomwalk (RW). Many variationsof
this strategyare of coursepossible.For example, by using
only the first four or last three days of CIGNET data to
provideorbital data for the entire week, it was foundthat
baseline solutions varied at the level of 1-5 mm for all three
baselinesin all components.In addition, the troposphere
stochastic
modelwas changedfrom a randomwalk to a firstorderGauss-Markov(GM) process.With steady-state
sigmas
of 6-12 mm and time constants of 0.5-3.4
(4)
hrs the baseline
estimateswere virtually unchanged.However, comparing
GM and RW stochastic models revealed differences in the
(the expressionassumesuniform distributionof the
measurements
overT). With '• = 1 month,for example,the
(millimeters)
east
component
of thePinyon-to-JPL
ratecouldbemeasured Table2 Daily Repeatabilities
toanaccuracy
of2 mrn/yr
(orsignal
tonoise
ofabout
4 to1,
giventhe~8-mm/yrratefromGLB659)in T = 2 years.
Baseline
Pinyonto JPL(172km)
Scripps
to JPL(! 71 km)
Theweekof October
21 (7 days)provided
idealobserving Scrippsto Pinyon(111km)
DailyRepeatabilities
andFormalErrors
conditions
forGPS.Satellite
eclipsing
wasaltogether
absent
East
North
3.4
2.5
t.6
1.6
2.0
1.6
Vertical
4.1
5.3
7.7
Lindqwister
etal.:Few-Millimeter
GPSBaseline
Precision
1138
i
i
i
i
i
'
't"
i
from the formal-errorprediction,thus indicatingsomelowangiemismodeling.
Fins = 1.6 mm
20
Conclusions
lO
o
Basedon 23 daily measurements
spanning8 months,the
long-termrepeatabilitiesof baselinemeasurements
in the
PGGA are lessthan or approximatelyequal to 6 mm in the
+
o
horizontal and 10 mm in the vertical. The hi•h-precision G?S
analysisindicatesthat in the short-term(oneweek) precision
-lO
at the few-millimeter
level is achievable in all baseline
components,
includingthe vertical,for baselines
up to 170
km in length,with post-fit residualsthat are consistent
with
rms = 3.4 mm
20
the formal errors. These results are based on data from the
PGGA, using8-channelRoguereceiversequippedwith low
!0
multipathantennas.
Orbit enhancement
for theregionaldata
wasaccomplishe
d by combining
thePGGAsolutions
with
information from solutions based on CIGNET data.
0
Acknowledgments.
The authorsappreciatediscussions
with
-10
S. Lichten, K. Hurst, R. Treuhaft, and T. Yunck and would
also like to thank two anonymousreviewers for helpful
rms = 4.1 mm
comments. The work described in this letter was carried out
by the Jet PropulsionLaboratory, California Instituteof
Technology,undercontractwith the NationalAeronautics
andSpaceAdministration.
10
References
-!0
Bierman, G. J.,
20
21
2
2
24
25
26
27
28
1990 October
Fig.3.Dailyresults
forthePinyon-to-JPL
(172-kin)
baseline,
Factorization Methods for Discrete
SequentialEstimation,V128, Mathematics
in Science
and
Engineering
Series,AcademicPress,1977.
Blewitt,G., Carrierphaseambiguityresolutionfor theglobal
positioningsystemappliedto geodeticbaselinesup to
2000 km, JGR, 94, 10187-10203, 1989.
witheachcomponent
adjusted
to zeromean.Thermsscatter Blewitt, G., An AutomaticEditingAlgorithmfor GPSdata
GRL.,17, No. 3, pp. 199-202,March1990.
is 2-4 mm for all components.
vertical estimate of ~15 ram. Variations in the e!evation
cutoff angle were also studied.As the cutoff angiewas
lowered
and more
data
were
added, all
baseline
repeatabilities
improved,consistent
with whatwould be
expected
fromformalerrors.Below15ø(our
optimalvalue),
however,
thedailyrepeatabilities
started
increasing,
deviating
Caprette,
D. S., C. Ma, andJ. W. Ryan,"CrustalDynamics
ProjectData Analysis--1990",VLBI GeodeticResults
1979-1989,NASA TM 100765,December,1990.
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"Assessment
of Global PositioningSystemMeasurements
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Lanyi,G., Tropospheric
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Lindqwister,
U.J.,S. M., Lichten,andG. Blewitt,Precise
regional
baseline
estimation
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GRL.,
17,No.3,pp.219-222,
March
19.90.
Mathews,
J.,andR.L. Walker,
Ma.
thematical
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Physics,
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Menlo
Park,
1970.
G. Blewitt,U. Lindqwister,
F. Webb,andJ. Zumberge,
0
2
4
6
8
10
12
formal error (ram)
Fig.4. A comparison
of dailyrepeatability
withformal
errors.
Mail-Stop'
238-625,
JetPropulsion
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P•.gadena,
CA 9! 109.
(ReceivedJanuary25, 1991;
revisedApril 8, 1991;
accepted
April 25, 1991.)