GEOPHYSICAL
RESEARCH
LETTERS,
VOL.19,NO.9,PAGES
853-856,
MAY4, 1992
GLOBALCOORDINATES
wrDt CENTIMETER
ACCURACY
IN THE
INTERNATIONAL
TERRESTRIAL
REFERENCE
FRAME
USINGGPS
Geoffrey
Blewitt,
Michael
B.Herin,Frank
H. Webb,
Ulf J.Lindqwister,
andRajendra
P.Malla
JetPropulsion
Laboratory,
California
Institute
ofTechnology
Abstract.Using21 daysof GlobalPositioning
System
Of the 21 sites in this GPS solution, we have obtained
(GPS)
datafrom21 globally
distributed
receivers
operating local surveysfrom the GPS antennato an ITRF monument
during
early1991,we solvefor a 7-parameter
transformation for !3 "collocated"
sites. (Thesesurveysare completely
betweena GPS free-network solution and coordinatesof 12
independentof the GPS solution used in this study).
stations
listedin the InternationalTerrestrialReferenceFrame
ITRF'90coordinates
andstandard
deviations
at epoch1988.0
(ITRF).Standard
errorsof GPScoordinates
arederivedby
aregivenbyIERS[1991]. !TRF'90represents
a combination
applying
anorthogonal
projection
operator
tothefree-network of solutionsby variousgroupsusingvery long baseline
covariance.The weighted RMS difference between 33
interferometry(VLBI) or SatelliteLaser Ranging(SLR).
transformedGPS and ITRF coordinates is 12 mm in the
Sincethecoordinates
for YELLOWKNIFE havelargeerror
northern
hemisphere.Bestresultsareobtainedby mapping bars in ITRF'90, this site is excluded from the
ITRFcoordinates
to theepochof thisexperiment
assuming
no
intercomparison.
Theremaining
36 coordinates
havestandard
verticalsitemotions.Fixing selectedsitesin theGPS solution
deviations
no largerthan2.1 cm. Local surveysolutionsare
toITRF'90doesnot improvethe agreement.
We concludethe
useof fiducialconstraints
is unnecessary
for globalnetworks.
listed in Table 1 for the GPS monuments in our solution.
SiteVelocityModels
Introduction
To accountfor tectonic motion between ITRF'90 reference
In orderto useGPSfor monitoring
large-scale
geophysical
signals
suchasglobalsea-levelchangeandpost-glacial
crustal
rebound,
GPS must be capableof providingglobalstation
coordinates
with centimeter-levelaccuracy. The "fiducial
concept,"
in which a subsetof coordinatesare constrainedto
values previously determined by other space-based
techniques,
has been a successfulGPS techniqueover
regional
scales. Use of thefiducialconceptimpliesthatsuch
solutions
are no longer independentof other techniques;
fiducia! coordinate errors can bias the GPS coordinates and
evenstrain-rates
[Larsonet al., 1991]. Moreover,on a global
scale
it becomes
difficultto interpretGPSsolutions
whichare
strongly
constrained
a priori. In thispaper,we showthat(1)
high-precision
GPS coordinates
can be derivedwithoutuseof
fiducial constraints,and (2) fiducial constraintsdo not
improve
theaccuracy
of ourglobalnetwork
solution.
Heftinet al. [ 1992]appliedthefree-network
approach
in
ananalysis
of 21 daysof datafrom 21 globallydistributed
receivers
in the GIG'91 campaignbeginning
January22,
1991.Expanding
on this,we assess
coordinate
accuracy
by
performing
a similaritytransformation
of thisnetwork
intothe
epoch!988.0 and epoch1991.1 we apply two models.
ModelA usesempiricalsitevelocitiesfromsolutionGLB718
for VLBI sites [C. Ma, D.S. Caprette, and J.W. Ryan,
private communication,1991], and solutionSSC-(IERS)90C01for SLR sites[Boucherand Altamimi,1991]. Model
B is the same,exceptthatthe 4 collocatedsiteswhichhave
verticalvelocities
> 1 mm/yrin GLB718 (e.g.,MADRID at
TABLE 1. Local SurveySolutions(GPS- !TRF'90)
SITE
DELTA COORDINATE(ram) ANTENNA*
NAME
X
ALGONQUIN
FAIRBANKS
GOLDSTONE
PASADENA
94763
-74141
2879994
InternationalTerrestrial Reference Frame, ITRF'90. The
7941
-12461
-134193
-14128
125588
36288
39384
-18605
utilityoffiducialconstraints
for thisglobalnetwork
is tested.
YELLOWKNIFE
-327848
Data Usedin This Study
For this study,we use a free-network
GPS solution,
conveniently
represented
by Cartesian
sitecoordinates,
very
loosely
constrained
to theITRF. We referthereadertoHeftin
SIDNEY
etal. [1992]for detailsof the datareduction.In thisstudy,
USUDA
Z ,, HEIGHT
61017
49280
5222210
1987
KAUAI
KOOTWIJK
MADRID
MATERA
PINYON
TROMSO
WETTZELL
YARRAGADEE
CANBERRA
JOHANNESBURG
HONEFOSS
NY ALLESUND
¾
(not
(not
(not
(not
BC
SCRIPPS
166
0
-22•052
163
-23617
-37503
159660
-13229
117655
-33115
71978
-12471
-4407
23024
164320
18027
279910
-9219
-58362
-5835
163
156
0
352
1914
2543
1668
143
314599
82882
yet available)
yet available)
in ITRF'90)
in ITRF' 90)
Until
(not
(not
(not
184
-18063
From
SANTIAGO
6666
-31231
8549901
181
0
9754
4017
1810
Jan
28:3360
Jan
29:1349
yet
available)
in ITRF'90)
in ITRF'90)
183
1915
0
theEarth's
poleposition
andsatellite
orbitswereestimated
*GPSantenna
heightis to thetopof theRoguechokering.
independently
every24hours
toreduce
systematic
errors.
An additional
phasecenterheightof-20.7 mmwasassumed
forthedual-frequency
combined
phaseandpseudorange
observables.
Theseantenna
heightsmaynotbe validfor
datesotherthanJanuary22- February13, 1991.
Copyright
1992by theAmerican
Geophysical
Union.
Paper
number92GL00775
.
0094-8534
/ 92/92GL-00775
$03.00
853
854
Blewitt et al.: GlobalCoordinates
in ITRF usingGPS
TABLE 2. ITRF'90 Coordinates
of GPS Monumentsat Epoch 1991.1
SITE
NAME
ALGO
FAIR
GOLD
PASA
COORDINATE
ERROR(ram)* *
VELOCITY (mm/yr)
X
Y
g
.
X
Y
-4346071223
-1453595751
-4641385442
4561977794
5756961960
3676976512
-18.7
-24.1
-17.4
-3.6
-0.1
11.1
1.1
-10.3
-9.9
7
8
8
8
9
10
B*-2353614082
-4641385457
A
-4655215589
3676976523
3565497327
-18.5
-35.2
5.3
23.9
-5.2
3.4
8
13
10
18
3565497331
2387809490
5015078290
4114913118
-35.9
-13.1
-15.3
17.9
4.4
29.7
10.5
44.2
-10.2
13
9
18
15
15
18
9
19
10
10
14
4114913058
22.6
69.8
16.0
18.9
18.8
1393056751
-4761207198
4133280313
3511396053
-16.2
-25.7
11
11
11
11
15
13
-4761207226
3511396075
5958192044
A
918129630
-2281621298
-2353614080
-2493304032
A
-4655215595
-5543838061
3899225380
4849202647
-2054587585
396731754
-360329222
B* 4849202592
MATE
PINY
(ram)
Z
B*-2493304035
KAUA
KOOT
MADR
AT EPOCH 1991.1
MODEL
A
-360329223
4641950906
-2369510349
B*-2369510362
TROM
WETT
2102940468
4075579405
-2389025280
YARR
721569359
931807155
5043316823
9
9
9
9
14
10
17
17
23.7
13.6
12.1
-3.4
-29.3
-19.0
16.1
2.1
11
15
13
10.4
5.4
14
12
21
4801570931
-14.4
-3078530992
-51.5
12.5
8.0
9
12
9
12
13
18.1
9.5
53.0
17
9
*Model B hasbotha differentvelocityandepochvalueat 1988.0.In thecaseof Goldstone,
thevelocitysolutionfor stationMOJAVE12(7222)wasappliedfromGLB718.
**Errors aspublishedin IERS [1991]aremostlyunderestimated
sincetheyreferto 1988.0
41 mm/yr) were mappedto 1991.1assumingno vertical
motion. Since ITRF'90 implicitly containsGLB718, we
whereW is an assumed
weightmatrixdesigned
toreflectthe
relativestrengthof thevariousinputcoordinate
data.
were able to account for the correlation between !988.0 site
coordinates
andvelocitycomponents
in mappingcoordinates
to 1991.1. Table2 listsITRF'90 coordinates
mappedto GPS
monuments
at epoch!991.1 for bothvelocitymodels.
SimilarityTransformation
Model
The similaritytransformation
usedby IERSis:
s
y
z
=
y
Z
+
ty
+
O:
%
(1)
s
tz
-0•
O•
s
wherex, y, and z are GPS-derivedcoordinates;
X, Y, andZ
areITRF'90coordinates;
tx, ty, andtzrepresent
anoffsetin
origin;$ represents
a differencein scale;Ox,Oy,andOz
representdifferencesin orientation.Someparameters
have
physicalinterpretations:
(1) the offsetin originrepresents
a
discrepancyin the estimatedlocationof the Earthcenterof
massandshouldbestatistically
consistent
withzero;(2) since
both systemsadoptthe samespeedof light, gravitational
coefficientGMo, and definition of coordinatetime, the scale
differenceshouldbe consistentwith zero; (3) orientation
differenceshaveno physicalsignificance,
sincethey are
arbitrarilyconstrained
by loosea priorivariances.
For purposes
of estimatingthe transformation
parameters,
equation(1) canberearrangedandrewrittenin matrixform:
6x =
+v
(2)
where/Sxis thevectorof coordinate
differences
(GPS-ITRF),
[I is thevectorof 7 nanformation
parameters,
A is thematrix
of partial derivatives, and a vector of errors v has been
included.
Theweightedleast-squares
estimate
of • is
Constructing
theWeightMatrix
Wedonotmaketheusual
choice
ofweights
W=E'l where
2Eis the initial GPS covafiance,because2;is only savedfrom
beingnon-singularby loosea priori constraints.Thisarises
from (1) perfect correlationbetweenthe satelliteascending
nodesandthe originof longitudes,and (2) perfectcorrelation
between station coordinatesand pole (X,Y) coordinates.
Nevertheless,
we still use2; to computestandard
errorsin the
translation
andscaleparameters
to assess
theirsignificance.
In derivingcoordinatecovariances,
geodeticinvestigators
haveappliedvariousad hocmethods,including,for example,
constraininga number of coordinatesand/or directionsto
establish
a reference
framefor thetechnique
in question.
We
choseto allow ourGPS networkto remainfree-floating
until
performing the similarity transformation, however the
covariance
matrix was modified
so that covariances are
relative to an internally-definedframe, as opposedto thea
prioriframe.
Correlated
errorsarisingfromreference-frame
uncertainty
areremovedfrom F with the applicationof an orthogonal
projection operator B, which projects GPS coordinate
variations
ontoa spaceorthogonal
to a reference
framethatis
implicitlydefinedthroughtheinitial GPS coordinates:
rñ= BFB
where
B=I- A(ATA}'IA
T
(4)
(5)
and where all station coordinatesin the initial GPS solution
areusedto compute
matrixelements
of A, irrespective
of
whether
theyarein theITRF.Thiscovariance
transformation
is equivalent
to imposing
thecondition
thatATxhaszero
variance,
whichintuitively
corresponds
tofixingthedirection,
originand scaleof the coordinateaxesin an averagesense
to
• = {ATWA)'IATW•x
(3)
the initial GPS coordinate
solutionx [Koch,1987]. We
B!ewitt
etal.:Global
Coordinates
inITRFusing
GPS
ernphasise
thatonlycovariances
(notthecoordinates)
are
855
Results
transformed;
wearesimplyconstructing
a weightmatrix.
Fñishasa 7-rankdeficiency
because
B is indempotent. Equation (7) showsthe estimatedtranslationand scale
between
GPSandrrRF (ModelB), computed
usingequation
deviations
for theseparameters
derived
Fñusing
the3x3 covariance
submatrices
between
the (3), with standard
GPSeovariance
forZ.
coordinates
of individual
stations.
In thisstudy,
onlythe fromtheinitialloosely-contrained
variances
areused(forbothGPSandITRF)because
only
diagonal
termsfor ITRF wereavailable.In ignoring
offtx=(-7.5+_2.6)cm
Nevertheless,
a weightmatrixcanbeformedfromblocks
of
diagonal
terms,the relativeprecisionof verticalversus
horizontal
components
is notaccounted
for in theweight
%=(13.0
:k2.5)cm
tz=(-14.8
+_13.8)
cm
=(-3.6_1.3)x10-9
=(6.0:i:530)
x 10-9
matrix.We still retainthe informationthat somestationsare
better
determined
thanothers.Therefore,
theweightmatrixis
W--[diag(lrl.fl)]'x
(7)
30)x 10-
(6)
--(-299.s_+
20) x 10where
fl istheITRF90covariance,
B isdefined
byequation
Theframeorientation
hasno physicalsignificance.
The
(5),andF is ourGPScovariance
derivedusingveryweak
originis generally
believed
tobeaccurate
at
constraints.
Applyingequation(3), thisweighting
method ITRFgeocentric
the
2-cm
level,
therefore
the
geocentric
offset
deviates
produces
estimates
andeovarianees
whichareinsensitive
to
fromzero.At thispoint,wecannot
explainthis
loosea prioriconstraints.We scaledthe covariance
F such significantly
result.
The
estimated
offset
closely
agrees
with
Vigue
et al.
thatthemeanformalerrorin baseline
lengthwas2.8 x 10-9L
[1992],who,usingthe samedata,insteadheld 3 stations
(where
L is length)corresponding
to the empiricalstandard fixedin theSV5frameandexplicitlyestimated
a geocenter
deviation
in lengthdifferences
observedbetweenthef•t- and
parameter. The scaledifferencedoes not appearto be
second-half
of theexperimentfor lengths> 4000km.
significant
sinceITRF errorsareexpected
to be at the10-9
Tables2 and 3 list the standard deviations for ITRF90 and
GPS,definedby the diagonalelementsof fl and F•.. The
level. Results
fromModelA differfromModelB by <1 cm
in originand<10-10in scale.
larger
GPSerrorsin thesouthern
venusnorthern
hemisphere
The estimated GPS station locations are then transformed
aredueto the lack of stationsin the southduringGIG'91.
intoITRF'90usingtheestimated
parameters
from equation
The1TRFstandard
deviations
usedin thisstudy(asquoted
in
(7). Table 3 liststransformedcoordinatesfor Model B. Table
4 lists the differences between the transformed GPS
IERS[1991])do not includethe errorin propagating
the
solutionto epoch 1991.1, and are thereforeslightly coordinates and ITRF90 with combined standard deviations.
underestimatedin most cases.
The weightedroot-meansquare (WRMS) difference
between GPS and iTRF'90 coordinates is defined as:
,,i)
TABLE 3. GPS SolutionTransformedinto ITRF'90
WRMS
= .
(ModelB) atEpoch1991.1
SITE
.
(8)
GPS COORDINATE,EPOCH1991.1(ram) ERROR(m)
NAME
•
y
Z
X
Y
Z
where8xi is the (GPS-ITRF'90)deviationfor coordinatei,
Sites used in transformation:
ALGO 918129611
FAIR -2281621290
GOLD-2353614081
-4346071204
-1453595749
-4641385464
PASA-2493304014
KAU -5543838080
KOOT 3899225386
MADR 4849202601
MATE 4641950888
PINY -2369510357
TROM 2102940474
WETT 4075579393
-4655215599
-2054587609
396731738
-360329229
1393056766
-4761207232
721569348
931807157
YARR -2389025226
5043316891
4561977793
8
11
12
5756961960
3676976522
3565497323
2387809514
5015078315
4114913074
4133280284
3511396075
5958192071
4801570924
-3078531021
7
9
9
18
12
17
17
9
9
13
26
6
10
11
14
8
11
11
11
7
9
28
9
9
9
13
10
13
13
9
15
11
20
Other Sites:
YELL -1224452355 -2689216066
5633638282 5 6 9
CANB-4460987984 2682362398 -3674626569 28 27 20
JOHA 5084625596 2670366566 -2768494046 59 40 30
HONE 3132539046
566401749 5508609852 10 7 10
NYAL 1202431415
252626752 6237770882 6 5 12
SIDN -2327188023 -3522528954 4764832336 8 9 9
SANT 1769724664 -5044512291 -3468396472 41 56 36
SCRI-2455521635 -4767213472 3441654893 10 12 10
USUD-3855262601 3427432482 3741020871 23 17 22
CøordinateS
referstotheGPSmonument
and•/2isthecorresponding
combined
variance.
TheWRMS
between GPS and Model B coordinates is 15 mm. The chi-
squareof the 7-parameterfit is 33.8 with 29 degreesof
TABLE 4. GPSCoordinate
DeviationsfromITRF (ModelB)
SITE
NAME
ALGO
FAIR
GOLD
PASA
KAUA
KOOT
MADR
MATE
PINY
TROM
WETT
YARR
GPS-ITRF'
.......Y
-19+11
19+13
90 Deviations
Z
East
-1_+15
-15
8+11
1ñ12
21ñ16
-19+20
6ñ21
9ñ23
-18ñ20
5i15
6ñ17
-12ñ16
54ñ28
0ñ13
-1ñ13
-8ñ17
24_+16
25_+20
16+21
-29ñ17
0ñ16
27_+26
-7_+14
-29ñ24
X
2+11
-7ñ14
-4ñ21
-24ñ17
-16ñ20
-6ñ15
15ñ16
-6ñ18
-11ñ14
2ñ13
68ñ31
3
4
21
16
-16
-5
20
8
-12
5
-78
(mm)
North,
Up
15
7
-4
-3
13
12
6
-14
-2
7
4
-6
-16
-4
4
-10
13
22
18
-28
3
26
-13
48
856
Blewitt
etal.:Global
Coordinates
inITRFusing
GPS
freedom(DOF). ModelA givesa WRMSof 17mmanda
chi-square
of44.9.ModelB givesmuchbetter
agreement
in
free-network
solution
"externally"
without
anyaffect
on
sensitivity
togeophysical
signals.
theverticalcomponent
for the4 siteswhichhavetheirvertical
velocitysetto zero.
Best
results
areobtained
bymapping
theITRFcoordinates
totheepoch
ofourexperiment
under
thehypothesis
that
the
Of the12stations
wehavecompared,
YARRAGADEE
has vertical
sitevelocities
arezero(accounting
forcorrelations
the largestdeviationfrom ITRF (-78 mm in longitude). withepoch1988.0coordinates
usingtheGLB718
site
Without YARRAGADEE included in the fit, station
coordinates
changeat thecentimeter-level,
andtheWRMS is
12 mm, with a chi-squareof 21.2 with 26 DOF. In the
northern hemisphere,the RMS differencein the North
covariance
matrix).
Finally,we conclude
thatboththeITRF'90andGPS
standard
deviations
presented
in thispaperappear
tobeas
goodasclaimed(atthecentimeter
level).
componentis 9 mm, in the East is 13 mm, andin the Vertical
is 20 mm. (WRMS could not be computedfor local
coordinates
sinceweuseddiagonal
weights).Theremovalof
YARRAGADEEfromthecomparison
hasthebiggest
effect
on the tranformation
parameters,
butevensothesolutionis
robust:Thelargest
change
in theoriginoffsetis 12mmin the
X direction;
thescalechanges
from-3.6x 10-9to-4.6x 10-9.
Acknowledgements.
Wearegrateful
fortheimportant
contributions
ofthemany
agencies
participating
inGIG'91,
withspecial
thanksto coordinators
Bill Melbourne,
Tom
Yunck,
andRuthNeilan.Thisresearch
uses
sitesurvey
information
given
tousbyseveral
agencies.
Oneofus(GB)
thanksMark Fingerof MarshallSpaceFlightCenter
for
suggesting
the useof linearcombinations
of coordinates
to
intema!lydefinea GPSreference
frame.Theworkdescribed
Applicationof FiducialConstraints
in this paperwas carriedout by the Jet Propulsion
We askthequestion,
"Dofiducialconstraints
improve
the
agreement between the GPS and ITRF solutions?" The
Laboratory,
CaliforniaInstituteof Technology,
underconpact
withtheNationalAeronautics
andSpaceAdministration.
answerwould be yes if the GPS solutionhad someinherent
References
internalweaknessdue to the simultaneous
adjustment
of
satellite orbits and station locations. The answerwould be no
if the fiducial stationsare constrainedto coordinateswhich are
in errorat the levelof the abilityof GPSto determine
those
positions
independently.
We repeatedtheaboveanalysiswith the exceptionthatthe
coordinates of 3 stations, KAUAI,
WETTZELL
GOLDSTONE,
and
were constrained to ITRF Model B coordinates.
We usedthe sameweightmatrixasthefree-networksolution,
solvingfor thesimilaritytransfomation
in thesameway. The
resultsare very similarto thoseshownin Table 4 to within ~3
millimeters,andthechi-squarestatistic
is slightlyworsethan
the free-networkcase,with a valueof 36.9 as comparedto
33.8 with 29 DOF.
Other selections of fiducial
stations
producesimilarresults:Fiducialconstraints
do notappearto
improvetheGPSsolution.
Boucher,C. andZ. Altamimi, ITRF 89 andotherrealizations
of the IERS terrestrialreferencesystemfor !989,
TechnicalNote 6, Observatoirede Paris, 1991.
Herin, M.B., W.I. Bertiger,G. Blewitt, A.P. Freedman,
K.J. Hurst,S.M. Lichten,U.J. Lindqwister,
Y. Vigue,
F.H. Webb, T.P. Yunck, and J.F. Zumberge,Global
geodesyusingGPSwithoutfiducialsites,Geophys.
Res.
Lett., 19, 131-134, 1992.
Koch,K.R., Parameter
estimation
andhypothesis
testing
in
linearmodels,
p. 222,Springer-Verlag,
NewYork,1987.
IERS, Annual Report For 1990, iERS Central Bureau,
Observatoirede Paris, 1991.
Larson,K.M., F.H. Webb,andD. Agnew, Application
of
the Global PositioningSystemto crustaldeformation
measurement:
2. The
influence
of errors in orbit
Conclusions
determination networks, Journ. Geophys.Res.,96,
The RMS difference between transformed GPS and ITRF'90
Ray, J.R., C. MA, J.W. Ryan, T.A. Clark, R.J. Eanes,
M.M. Watkins, B.E. Schutz, and B.D. Tapley,
Comparison
of VLB! andSLR geocentric
sitecoordinates,
Geophys.Res.Lett., 18, (2), 231-234, 1991.
Vigue, Y., S.M. Lichten, and G' B lewitt, Precise
determination
of geocentricstationcoordinates
withthe
GlobalPositioningSystem,in AGU 1991 Fall Meeting
ProgramandAbstracts,
p. 114,AGU, 1991.
(B 10), 16567-16584, 1991.
coordinates
is 12 mm in thenorthernhemisphere,
whichis as
goodas any comparison
betweenVLBI andSLR [Rayet al.,
1991]. GPS thereforeappearsto be a viable,independent
technique
for monitoring
geodetic
signalsfromlocalto global
scales.Independent
confirmation
of geodeticsignalswould
greatly strengthen hypothesis testing in geodynamics
investigations,
so it behoovesus to assessthe long-term
stabilityof theGPSreferenceframe.
For the global GPS data set analyzedhere, fiducial
constraintsdo not improve the GPS solution, and are
therefore unnecessary.This gives GPS techniquesmore
flexibility and eliminatespotential sourcesof error. For
example,site tie errorscan lead to networkdistortionwhen
usingfiducialconstraints,
whereassucherrorsonlyaffectthe
G. Blewitt, M.B. Herin, U.J. Lindqwister,andR.?.
Malla, andF.H. Webb,Mail Stop238/625,JetPropulsion
Laboratory,4800 Oak GroveDr., Pasadena,CA 91109.
(ReceivedFebruary24, 1992;
acceptedMarch 16, 1992.)