Jul 10, 2001 - nal frequencies [Freedman et al., 1994; Herring and Dong, ... Herring and Dong, 1994; Gipson, 1996]. ...... Part D (compared to C and B) ben-.
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. B7, PAGES 13,711-13,738,JULY 10, 2001
High-frequencyvariationsin Earth rotation from Global PositioningSystem data M. Rothacher, 1 G. Beutler, 2 R. Weber, 3 andJ. Hefty4 Abstract. Using the dataof the global,denseGlobal PositioningSystem(GPS) network established by the InternationalGPS Servicea continuous,uninterruptedseriesof subdaily Earth rotationparameters(ERPs) with a time resolutionof 2 hourshasbeen generated at the Center for Orbit Determinationin Europe. The seriesstartsin January 1995 and has a lengthof more than 3 years. Startingfrom the 2-hour ERP valuesof this, to our knowledge,uniquetime series,the high-frequencyvariationsin UniversalTime (UT1) and polarmotion(PM) dueto oceantidesare studiedanda setof sineandcosinecoefficientsis estimatedfor all the major tidal terms at nearly diurnal and semidiurnalfrequencies.The GPS seriesis not very homogeneous (variousprocessingchangesduring the 3 years) and still shortcomparedto the lengthof very long baselineinterferometry(VLBI) and satellite laserranging(SLR) data sets.However,the resultsderivedfrom this seriesare alreadyof the samequality as the resultsfrom VLBI and SLR. A comparisonof the tidal coefficients
stemming fromall threespace-geodetic techniquek showsanagreement onthe 1/zslevel for UT1 and 10 microarcseconds(/zas)for PM, respectively.The RMS differencebetween the oceantide amplitudesestimatedfrom GPS data and from TOPEX/Poseidonaltimeter dataamountsto 0.7-0.9/zs in UT1 and 9-13/zas in PM. The residualspectrumthat remains after the removalof all tidal termshasa noiselevel of • 5-10/zas in PM and 0.5-1/zs in UT1 and containsnontidalsignals(up to 55/zas in PM and 3/zs in UT1) that might be due to the impactof the satelliteorbit modeling(12-hour revolutionperiod of the satellites)or, alternatively,due to atmosphericor oceanicnormalmodes.
1. Introduction
First calculations
The interactionof the oceansand the atmospherewith the solid Earth causeschangesin Earth rotation,that is, in the positionof the rotationaxis (polarmotion(PM)) andthe rotation rate of the Earth (length of day), and, consequently, in UT1. The variationsin Earth rotation seen at high frequencies(periodsof 1 day and shorter)are mainly the result of changesin the oceanheightsand currentsdue to the
of the effect of ocean tides on PM and
UT1 were performedby Yoderet al. [1981], mainly considering periods longer than 5 days. They also mention first rough estimatesfor subdailyvariations. Broscheet al. [1989] studied ocean effects on UT1 and similar effects in PM were consideredby Seiler [1991], Wanschand Seiler [ 1992], and Gross [ 1993].
Using long time spansof very long baselineinterferometry (VLBI) data variousgroupsestimatedthe coefficients tidal potentialgeneratedby the Sunand Moon. Apart from of the tidal amplitudesin PM and UT1 [Soverset al., 1993; thesetidal effectsof the oceanssomeatmosphericexcitation Herring and Dong, 1994; Gipson, 1996]. Similar computaof PM and UT1 is to be expectedat diurnal and semidiurtions were performedby, for example, Watkinsand Eanes nal frequencies[Freedmanet al., 1994; Herring and Dong, [1994] using satellitelaser ranging (SLR) data. Compar1994; Eubanks, 1993].
isons with more recent models derived from TOPEX/Posei-
don altimeter data [Chao et al., 1996] have shown that the
•Forschungseinrichtung Satellitengeodaesie, TechnicalUniversity of Munich,Munich,Germany. 2AstronomicalInstitute,Universityof Berne, Berne, Switzer-
agreementbetweenthe varioustechniquesnowadaysis at a level of ,-,, 10-30 microarcseconds(/•as) in PM and 1-3/•s in UT1.
The Center for Orbit Determination (CODE), a coopaDepartment of TheoreticalGeodesy, Universityof Technology eration of the AstronomicalInstitute, University of Berne Vienna, Vienna, Austria. (Switzerland),the SwissFederalOffice of Topography,Wa4Department of TheoreticalGeodesy, SlovakTechnicalUniver- bern (Switzerland), the Bundesamtftir Kartographieund sity,Bratislava,SlovakRepublic. Geodiisie,Frankfurt(Germany),andthe InstitutGdographiland.
Copyright 2001by theAmerican Geophysical Union. Papernumber2000JB900393. 0148-0227/01/2000JB 900393 $09.00
que National,Paris(France),and one of the sevenanalysis centersof the InternationalGPS Servicefor Geodynamics (IGS), startedto estimatesubdaily(2-hourly) Earth rotation parameters(ERPs) on a routinebasisin March 1996, using 13,711
13,712
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS
theGPSdataof the globalIGS network.With a reprocessing Here we first characterizethe global GPS data, the proeffort the serieswas extendedbackward in time to January cessingstrategyand modelsthat were usedto obtain2-hour 1995, which meansthat by now an uninterruptedERP se- ERP estimatesfor eachday sincethebeginningof 1995(secries with a time resolution of 2 hours is available that covers tion 2). In section3 we analyzethe resultinghigh-resolution ,-• 3 years. Sucha seriescouldbe generatedneitherfrom ERP seriesof 3 years, generateamplitudespectrafor the VLBI nor SLR data because of the limited amount of obserinterestingdiurnaland semidiurnalfrequencybandsanddevationsavailablefrom thesetechniques.In thecaseof VLBI, scribethe estimationof all the major diurnal and semidiurcontinuous24-hour observationsessionsover manydaysare nal oceantide amplitudesfrom the ERP series.In section4 onlyavailablefroma few specialexperiments like,forexam- we discussthe impact of various modeling and processing
ple, CONT94, CONT95, andCONT96. This situationwill aspectson the estimatedtidal coefficientsand comparethe changewith the startof CORE-like experiments (Continu- resultsof our four majorGPS solutionswith oceantide models derived from VLBI, SLR, and TOPEX/Poseidon altime-
ous Observationsof the Rotation of the Earth [Clark et al., 1997]).
ter data. Last,but not least,we studythe high-frequencypart So far, papersdiscussing subdailyEarthrotationparam- of the PM and UT1 spectrumthatremainsafter havingestietersderivedfrom GPS data were concentratingon results mated the coefficients of 57 tides in PM and 41 tides in UT1 fromatmospheric excitation stemmingfrom intensivecampaignsof 1 or 2 weeks(e.g., to detectpossiblecontributions or ocean normal modes. from GIG'91 [Lichten et al., 1992], Epoch'92 [Freedman et al., 1994], CONT'94 [Weber, 1996], or from other short time intervals[Grejner-Brzezinska and Goad, 1996]). These 2. GPS Data Analysis
investigations showedthat the high-frequency variationsof Earth rotationdue to the oceantides may be measuredby GPS. The time spanswere, however,muchtoo shortto allow the estimationof oceantide amplitudesor a model for subdailyERP variationsfrom GPS data. With the startof the routineestimationof subdailyERP valuesat CODE the situationhas changed. A first analysisusing• 1 year of subdailyERP estimates from CODE wasdescribed by Hefty et al. [2000]. PM and UT1 amplitudeswere estimatedfor • 20 periodsthere. With the ERP seriesnow covering3 years, an analysis goinginto muchmoredetailscanbe doneandthe impactof variousaspectsof theGPSprocessing andmodeling(e.g.,of the orbit model) on the subdailyPM andUT1 estimatesmay be studied.
2.1.
Global
Data
Set
The GPS data usedin this analysisstemfrom the global IGS network and include 40 to 90 sites. A map of the IGS networkas usedby CODE is given by Rothacheret al. [1997]. The sitesare well distributedand form a very stable reference frame for the determination
of Earth Rotation Pa-
rameters(ERPs). The time interval consideredstartsin Jan-
uary 1995 (day 002) and endsin February1998 (day 045). Thus it coversa periodof 1140 days. Using the latestversion of the Bernese GPS software [Rothacher and Mervart,
1996] and double-difference phaseobservations, global 3day solutionswere computedfor eachday in this time interval. By computingoverlapping3-day solutionswe could setup 3-day satellitearcs(only one setof initial conditions
9O 8O
._
o
70 6O
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06/96
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1
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06 f98
in Years
Figure1. Number of stations andphase double-difference observations included in the3-daysolutions when estimatingsubdailyERPs from GPS.
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS
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Table 1. Characterization of DifferentPartsof theSubdailyERP SeriesEstimatedFrom GPS Data for 3-day Solutions Series
Time Interval
Numberof Days
Orbit Model
A Priori Subdaily Cutoff, ERP Model deg
A A'
002/1995-126/1995 002/1995-126/1995
125 125
extended classical
RAY96 a RAY96
20 20
B C D
127/1995-181/1996 182/1996-272/1996 273/1996-277/1997
420 91 371
classical classical extended
no RAY96 RAY96
20 20 20
E
278/1997-045/1998
133
extended
RAY96
l0 b
SeeRay et al. [1994] or McCarthy [1996].
bWithelevation-dependent weightingof theobservations.
per satellitefor 3 days),therebyconsiderablyimprovingthe orbitquality.The resultsfrom the middledaysof each3-day solution are extracted
to form a continuous
series of satellite
positions,ERPs, etc. Figure 1 showsthe number of sites and phasedoubledifferenceobservations includedin the 3-day solutionsover the time interval of the 3 years consideredhere. We see a considerable
increase
in the number
of sites and observa-
accelerationactingon the GPS satellites.Both seriesare the resultof a reprocessingeffort aroundthe end of 1996 [Botton et al., 1997]. SeriesA is the moreup-to-dateseriesand hasthe samecharacteristics as therelativelyrecentseriesD. The seriesB and C were computedwhen still the old, classical orbit parameterization(describedin section2.3) was in use. At the end of September1996 the so-calledextended orbit model was introduced into the official CODE
tionssincethe startof the series,andwe may expectan improvementin the quality of the estimatedERPs over the 3 years (even without taking into accountthe modelingimprovementsimplementedduringthis period). Loweringthe satelliteelevationcutoff angle usedin the processingfrom
processingand from this date onwardall solutionsare based
20 ø to 10 ø in October 1997 resulted in an increase of the
the numberof observationsand stationsaroundday 020 in 1998 is dueto a problemat oneof the major operationaldata
the zenith angle of the GPS satelliteas seen from the receiver.The inclusionof dataat low elevationstogetherwith theaboveweightingschemewasshownto improvethequality of the results,in particularthe stationheightestimates
centers of the IGS.
[Rothacher et al., 1998].
number of observations
of ,-., 25%.
The drastic reduction
in
on thisimprovedmodel. In October1997, finally,the processingstrategywas changedto includelow-elevationdata downto 10ø usingan elevation-dependent weightingfunc-
tionw -- cos2(z)for thephaseobservations, wherez is
A varietyof differentparametertypeshasto be estimated 2.2. Earth Orientation Parameters in global3-day solutions.In the followingsectionswe will describetheseparameters andtheprocessing strategyto the For the entire seriesof 1140 3-day solutionsEarth Orienextentnecessaryto understandthe resultingsubdailyERP tationParameters(EOPs) were savedin the normalequation series. files with a high temporalresolution:for each2-hour interDuringtheroutineprocessing at CODE [see,e.g.,Rotha- val offsetsand rateswere setup for all five components of cher et al., 1997] the normalequationsystemsof the 3- Earth orientation,namely,the a;and !/pole coordinates,the day solutionscontainingsite coordinates, ERPs, and geo- difference UT1-UTC, and the nutation corrections A• and center coordinates as unknowns were saved. The series to A• in obliquityandlongitude.Startingfrom thesenormal be discussed herecouldthereforebe generatedstartingfrom equations, differentsolutiontypesmaybe produceddependthesenormalequationfileswithouthavingto gobackto the ing on the a priori constraints put on the Earth orientation raw GPS data. Becausethe routineprocessingstrategywas components[Brockmann,1997]. To generatethe subdaily changedseveraltimesduringthe 3 years,the entireseries ERP resultsof interesthere, continuitywas enforcedat the seemsratherinhomogeneous anddifficultto interpret.Un- boundaries of the2-hourintervals,thatis, thecomponents of fortunately,we are not yet in a positionto easilyreprocess Earthrotationwererepresented aspiecewiselinearfunctions the entireseriesstartingfrom the raw datadue to the huge over the 3 days. All the nutationparameterswere heavily amount of data involved. Table 1 summarizes the different constrainedto the valuesof the a priori nutationmodel,the partsof theserieswe shoulddistinguish, thetimeperiodthey IAU 1980 Theory of Nutation [see,e.g., McCarthy, 1996; cover,andthemajorprocessing characteristics usedfor each Seidelmann,1982; Wahr, 1981], that is, they were not estimated. The known deficiencies of the IAU 1980 nutation of the parts. We see that for the first time interval two series (A and model [see, e.g., Herring et al., 1991] have an important A') are availablethat differ only by the orbit model used, impacton subdailypolarmotionbut only on the retrograde that is, the parameterization of the solarradiationpressure diurnalfrequencyband, which will not be consideredhere.
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ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS
namely,of constantcoefFor a detailedanalysisof the nutationestimatesgenerated sistsof a totalof nineparameters, from the CODE series of normal equationfiles, we refer ficientsin threedirections (thedirectradiationpressure coto Rothacher et al. [1999]. Let us mention that although efficient,the y bias, and the so-called•; bias), and of onceoffsetparameters weresetup for all five components, only per-revolutionterms (each consistingof a cosineand sine
polarmotion(PM), thatis, the •; and/,tpolecoordinates,amplitude)for thesethreedirections.In the so-called"clasmodelonly two radiationpressure maybedetermined in anabsolute senseusingGPSdata.Be- sical"radiationpressure causeUT1-UTC is fully correlatedwith theorbitalelements parameters, thedirectradiationpressure coefficient andthe of the satellites(mathematicalrelationsbetweenUT1-UTC, the nutationcorrections,and the orbital elementsare given
bias,were estimated(as constantsover one arc). This model was usedwhen producingthe seriesA', B, and C (see Ta-
the by Rothacheretal. [1999])thefirstUT1-UTC offsetof each ble 1). After September1996, threemoreparameters, termsin the •; direction, 3-daysolutionhadto beconstrained to thea priorivalueob- z biasand the once-per-revolution tainedfrom VLBI. This means,that only the rate of change were addedto the setof radiationpressure parameters estiof UT1-UTC
is accessible to GPS. Therefore all our results
mated, now called the "extended" orbit model. This model
andcomputations concerning UT1-UTC variationswill be wasusedin the generationof all solutionssinceSeptember 1996 (seriesD and E) and in the reprocessing (seriesA).
basedon UT1-UTC rates(i.e., lengthof day (LOD) values). There is one more singularityto be consideredwhen setting up subdailyERPs in GPS data processing:A common tilting of all orbital planesin the inertialframe may easilybe compensated by an exactretrogradediurnalPM term. The simultaneousestimationof satelliteorbit parameters(especiallythe rightascensions of theascending nodes andthe orbit inclinations)andsubdailyPM will lead to huge retrogradediurnaltermsdue to this correlation.To avoid thissingularity,theretrogradediurnalcomponent in PM has been constrainedto zero using speciala priori constraints. The mathematicalderivationof theseconstraints is givenby Hefty et al. [2000]. Any exactlydiurnalretrogradesignal wasthussuppressed in the 3-day solutions. All the serieslisted in Table 1, with the exceptionof seriesB, wereproducedusingasa priori modelfor diurnaland semidiurnalERP variationsthe ocean tide model computed by Ray et al. [1994], whichis basedon altimeterdata.This model,which we call it RAY96 in the following,is identical with model C describedby Chao et al. [1996]). In section 4.7 we show that small artifactsin the high-frequency
The introduction
of the extended model resulted in a con-
siderableimprovementof the orbit quality and the station coordinates[Springeret al., 1996, 1999]. We will seethat it alsohasa significanteffecton the qualityof subdailyERP estimates.For the generationof the seriesstudiedhere, we neverestimatedthefull setof nineradiationpressureparameters because of the correlations
of some of these radiation
pressureparameterswith theLOD estimates[Springeret al., 1999].
To completethe descriptionof the orbit modeling,let us mentionthat small changesin the force field (e.g., the use of an improvedEarth tide model,of the JetPropulsionLaboratory(JPL) PlanetaryEphemerisDE200, etc.) tookplace on days 182, 1996, and 273, 1996 [Rothacheret al., 1997]. We expectthatthesemodificationsmainly resultin a change of the (more or less) constantbias in the UT1-UTC ratesand
thusonly affectthelong-termvariationsin UT 1-UTC, which are not of interest here.
2.4.
Station
Coordinates
and Reference
Frame
partof thesubdailyERP spectrum resultfromfittingthefull Two completeseriesof subdailyERPs,coveringtheentire subdailysignal(variationsof upto ,-.,1 milliarcsecond(mas) interval of 1140 days and differing only by how the terresin PM and0.6 ms/din UT1) by 2-hourpiecewiselinearfunc- trial referenceframe was realized, were generated,both with tions.The useof a gooda priori modelremovessucheffects. the characteristicslisted in Table 1. For the first series, series 2.3.
Orbit
Model
and Parameters
I, the global referenceframe was definedby fixing 37 sites with a well-establishedcoordinatehistory to their ITRF96 coordinatesand velocities[see,e.g., Boucherand Altamimi, 1997, http://lareg/ensg.ign.fr/ITRF/ITRF96-rep.html].The 37 sitesare a subsetof the 47 sitesselectedby the IGS to be usedby the analysiscentersbeginningMarch 1, 1998,
The orbit of each satellite over 3 days was represented by one setof initial conditions(positionand velocityvector at the startof the interval),by one set of radiationpressure parameters(moredetailswill be givenbelow),andby pseudostochastic pulses(small changesin the satellite'sveloc- to realize the reference frame for the routine orbit and ERP ity) [Beutleret al., 1996]. Suchpulseswere introducedin determination.The coordinatesof all othersiteswere freely the along-trackandradialdirectionsevery 12 hours,thatis, estimated. The second series, series II, was based on a ref5 timesin a 3-day interval. They were constrained(to zero) erenceframe given by network constraints(no-net-rotation conditions)setup for the 37 sites.In withana priorivariance of (10-s m/s)2 and(10-6 m/s)2 for andno-net-translation the along-trackandradialcomponents, respectively. thiscase,no particularsiteswere fixed; only the globalnetFor satellitesthat were difficult to modelon specificdays work as a whole was constrained. The two differentseries(I andII) were producedto assess (evenwith theestimationof pseudostochastic pulses),the3day arc was split up at the day boundariesinto two or even the impactof the stabilityof the referenceframeusedon the three arcsby estimatingadditionalsetsof initial conditions subdailyERP estimates. Let us mention in this context that no model for the coorand radiationpressureparameters. The radiationpressuremodelin useat CODE in its gen- dinatevariationsdue to oceanloading,the displacementof eral form hasbeendefinedby Beutler et al. [1994]. It con- sites because of the elastic deformation of the Earth crust in
ROTHACHERET AL.: HIGH-FREQUENCYEARTH ROTATIONFROM GPS
13,715
response to the surfaceloadexertedby the oceantides,was 3. Analysis of Subdaily ERP Series usedthroughoutthe entire series. Only startingMarch 1, 1998, we includedoceanloadinginto theroutineprocessing. 3.1. Subdaily ERP SeriesFrom GPS Althoughoceanloadingmainlyaffectsthe stationheights,it The entire seriesof 1140 daysof subdailyERP estimates, may still haveadverseeffectson the determinationof diur- morepreciselyseriesI (seesection2.4), is shownin Figure2 nal and semidiurnalamplitudesof PM and UT1 becauseit for the a: and !/ coordinateof the pole and for UT1 rates. is governedby exactlythe samebasicfrequencies aspresent To obtain this series,the following stepswere performed. in the subdailyERPs. Soverset al. [1993] discussthe influ- For thosepartsof the serieswherethe subdailyERPswere enceof the oceanloadingmodelusedfor sitedisplacements computedrelativeto the subdailyERP modelby Ray et al. on the estimatedsubdailyERP amplitudes.We will have a [1994] (RAY96, see Table 1), the Ray model valueswere closerlook at this problemin section4.2. Solid Earth tides addedto the GPS estimates.The low-frequencyvariations weremodeledaccordingto the IERS 1996 conventions[Mc- in the ERPs were removedby subtractinga smootha priori Carthy, 1996] startingin October1997. Beforethisdate,the poleseriessuchas the BulletinA series(rapidpoleseries model(step1) givenin theIERS Standards1992 [McCarthy, computed by the IERS Sub-Bureau) or theC04 seriespub1992] was used. We shouldkeep in mind that already small lishedby the IERS [Feisseland Castrique,1997], not conerrorsin the solid Earth tide model may have an impact on the estimationof oceantide amplitudes.
tainingsubdailyvariations.Finally,an offsetanddrift over theentireserieswasremovedfor eachcomponent.
The a: and !/pole variationsare of the orderof 1 mas. No major changesin the behaviorof the seriescan be detected In additionto the parametersdiscussedabove(ERPs, or- overthe3 yearsandfor anyof thepartsA throughE (dotted bit parameters,and site coordinates)a few other parameter verticallines). The UT1 ratesare typicallyon the 1 ms/d typeshaveto be estimatedsimultaneously in globalGPS so- level. Somelargerfluctuationsmay be seenin summer1996 lutions:troposphere zenithdelays,initial phaseambiguities, and in autumn 1997. These events cannot be associated with geocentercoordinates,and satelliteantennaoffsets. These anyof the processing changes.The largerscatterin autumn parameterswere treatedidenticallyin all the seriesdiscussed 1997 coincides, however, with the dramatic decrease in the below. numberof sites(seeFigure 1). The a posterioriuncertaintiesof the 2-hour estimatesare Site-specifictroposphere zenithdelayswereestimatedevgiven in Figure3. Theseerrorsof the 2-hourERP estimates, ery 6 hours,that is, four troposphere zenithdelayswere detheGPS data, terminedper siteandday. The mappingof the zenithdelays a samplingof 180 s wasusedwhenprocessing to the actualzenith anglesof the satelliteswas doneusing vary between• 0.02 and 0.12 mas in PM andbetween0.04 theSaastamoinen [1971] mappingfunction.The handlingof and0.12 ms/din UT1 drift. It is interestingto notethat the tropospheric delaysis of importanceherebecause,first, the uncertaintiesof the a: componentare smallerthan thoseof 6-hour samplingof the troposphere may not be sufficientto the!/component,mostprobablya consequence of the distrimodelrapidly varyingtroposphericconditionsand partsof butionof theIGS sitesoverthe globe.The strangesignature the troposphericdelay signalsmay propagateinto the ERP in the PM uncertaintiesis causedby the circumstancethat estimates.The relatively coarsesamplingfor tropospheric theywere only savedwith a 10-/•as resolution. A decreaseof theerrorsis visiblefor all threecomponents delaysmightbe one of the accuracylimiting factorsin the estimationof subdailyERPs andmight makeup a significant and is partly a resultof the increasingnumberof siteswith contribution to the RMS level of the ERP estimates. Second, time (seeFigure 1). The transitionfrom seriesA to seriesB the circumstancethat the binning interval of 6 hoursused (changefrom the extendedto the classicalorbit model) and for the troposphericdelay estimationis almostcommensu- theimprovements of orbitmodelingin autumn1996(change rate with the diurnal and semidiurnaltidal periodsmay be from the classicalto the extendedmodel) are clearly visianother reason for concern. If there are correlations between ble. The :r componentalsoseemsto haveslightlyimproved troposphere parametersandERPs,thismightgiveriseto un- throughthe inclusionof low-elevationdatainto the processdesiredeffects.Assuming,however,that the biasesthat may ing in October 1997. Some features,for example,the inbe introducedinto the ERP seriesdue to the relativelylow creasein the uncertaintiesin September1997, are common samplingof the troposphereare almost random over long to all three componentsand are probablycausedby the retime periods,we do not expectany major effectson tidal duction in the number of sites (and observations)available for the globalsolution. amplitudesestimatedfrom this series. As we will see in section 4.4, the actual noise level of Using the quasiionosphere-free (QIF) ambiguityresolution strategydescribedby Mervart [1995] about80-90% of the entireseries(computedfrom the differencesbetweenthe the initial phaseambiguitieswere resolvedto integernum- model RAY96 and our series) is of the order of 0.33 mas bersfor all baselinesshorterthan 2000 km. Taking into ac- and 0.27 ms/d in PM and UT1, respectively.This implies countthat no ambiguityresolutionwas performedon base- that the formal uncertaintiesare too optimisticby abouta lines longerthan 2000 km, • 50% of the total numberof factorof 4-5 on the average.This discrepancy betweenacambiguitiescouldbe resolved,strengthening the globalso- tualandformalerrorsis confirmedby thescalingfactorsthat lutions. haveto be appliedin theIGS solution-independent exchange 2.5. Other
Parameters
Estimated
13,716
ROTHACHERET AL.: HIGH-FREQUENCYEARTH ROTATIONFROM GPS (a)
B
A
C
2
D
',
-3
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3 (b)a
B
in Years
C
D
E
2
•
0
0
--
>.-
,, -2
--3
i 01/95
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(c)
01/98
06
in Years
C
A
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D
E
,
,
,
__
-2
I -3
01/95
•
I
I
t
I
06/95
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06 98
Time in Years
Figure 2. Two-hour ERP estimatescomputedfrom the global IGS network over a time interval of ,-- 3 years. A smootha priori ERP serieswas subtractedand only the high-frequencyvariationsare shown.The dottedverticallinesmarkthebeginningandendof the differentpartsA to E of the series(see Table 1): (a) x pole coordinate,(b) !/pole coordinate,and(c) UT1 rates.
(SINEX) format combinations(in this casea factor of • 7).
about90 sitesnowadays.We may alsoconcludethat a considerableimprovementmust still be achievablein our GPS around0.24 mas (PM) and 0.19 ms/d (UT1 drift). Accord- analysis.Modelingchangesthatmightreducethenoiselevel ing to C. Ma (privatecommunication,1998) the noiselevel are a highersamplingof the tropospherezenithdelaysand of high-frequency ERP determination fromVLBI CORE ex- the useof an adequateoceanloadingmodel. periments(ContinuousObservations of the Rotationof the To obtaina first impressionof the amountof signalconEarth [Clark et al., 1997]) is ,-- 0.27 masin PM and 0.016 ms tainedin the subdailyERP seriesobtainedfrom GPS, Figin UT1 (not UT1 drift), values that are similar to those of ure 4 showsan enlargementof 18 days (January1 to 18, the most recent GPS estimates with 0.24 mas and 0.016 ms 1998) of the mostrecentpart of the seriestogetherwith the (0.19 ms/d dividedby 12 to obtaina roughestimateof the values of the ocean tide model RAY96. The RMS differerrorfor one2-hourinterval).That VLBI is performingso encebetweenthe two seriesshownin Figures4 amountsto
For the most recentpart E of the seriesthe noiselevel is
well is astonishing whenconsidering thattheVLBI network 0.20 masin the :• and!/componentand0.20 ms/din UT1. for a typicalCORE experimentonly consistsof aboutfour In general,the signalpresentin our seriesseemsto be larger sites,whereasthe GPS networkusedat CODE comprisesthanexpectedaccording to theRay model.Quitea similar
ROTHACHERET AL.: HIGH-FREQUENCY EARTH ROTATIONFROM GPS
13,717
0.13
(a) A
'•'
0.12
E
0.11
•
._
B
C
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E
O.lO
0.09 0.08
0.07 0.06 0.05
0.04
0.03 0.02
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• o.12 '--'
O.ll
•
0.10
o o
0.09
._
O
_.e
0.08
nø 0.07 i ,• •
._
•
:•
0.06 0.05
.04
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E
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0.07 0.06 o.o5
0.04 '95
06/95
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01/98
06 •98
in Years
Figure 3. Uncertainties(lcr) of the 2-hour ERP estimatesover a time intervalof ,-, 3 years. The dotted vertical lines mark the beginningand end of the differentpartsA to E of the series(see Table 1)' (a) z pole coordinate,(b) !/pole coordinate,and (c) UT1 rates.
picturecanbe seenin thearticleby Chaoet al. [1996] when [Gipson,1996],thenoiselevelis ,-•2 •s. The low noiselevel comparingthe VLBI resultsfrom the CONT94 campaign (especiallyin UT1) notonly enablesusto clearlyseeall the with model RAY96 (called model C). The VLBI estimates majortidal terms,it shouldalsoallow the detectionof postendto havelargeramplitudesthanthe RAY96 model,too. sible nontidalsignalsif suchsignalsexist with amplitudes The spectrageneratedfrom the entire seriesare given in largerthanthenoiselevelpresentin Figure5. We will have Figure 5 for the diurnal and semidiurnal,progradeand ret- a closerlook at this issuein section4.7. The spectraof this rogradefrequencybands(seesection3.2 for definitions).In GPS seriesalsodemonstrate the potentialof this technique the case of UT1 the UT1 rate spectrawere convertedinto for thefuture,in particularin view of thefactthatthe series UT1 spectraby dividingthe spectralrate amplitudesby the analyzedhereis still veryinhomogeneous andfar from opticorresponding circularfrequency[Rothacheret al., 1999]. mal. The seriesis still far too short,however,to give insight The polar motion spectraexhibit a noise level of ,-, 5- into theproblemof the sidebands of the majortides. 10 •as, thoseof UT1 a level of 0.5-1.0 •s. This gives a The retrogradediurnalpart of the polarmotionspectrum first indicationof the type of precisionto be expectedwhen hasonly beenaddedfor completeness. The linespresentin estimatingtidal amplitudes.For a similarspectrumof thedi- thisspectrum are mainlydueto the deficiencies of the IAU urnal UT1 frequencyband obtainedfrom VLBI ERP series 1980 NutationTheoryusedin our GPS analysis.For more
13,718
ROTHACHER ETAL.'HIGH-FREQUENCY EARTH ROTATION FROM GPS
1'0 1(a) 0.4
A
0.8
0.2
',
.•
.
/•
.,.
..
.::
...
...":
-0.2
-0.4 -0.6
'
'
13
14
15
16
17
18
14
15
16
17
18
-0.8 -1.0
I
2
3
4
5
6
7
8
9
10
11
12
Day of Year 1998
I
GPS Estimates ........Ray Model
(b) 0.8
'•'
0.6
,---,
0.4
'•
0.2
.=_
• 8
o
o.o -0.2
nø
-0.4
I
>-
-0.6 -0.8
-1.0
....
i ....
2
i ....
3
i
i
4
5
i ....
6
i ....
7
i
i
8
9
11
10
12
13
Day of Year 1998 GPS Estimates
........
Ray Model
1.0 0.8
• 0.6 ,._.,
•,
0.4
0.2 00
• -0.2 -• -04 -0.6
-0.8
....
• ....
2
• ....
3
• ....
4
• ....
5
• ....
6
• ....
! ....
7
8
i ....
9
• ....
10
i ....
11
• ....
12
i ....
• ....
13
14
• ....
15
• ....
16
• ....
17
18
Day of Year 1998
GPS Estimates ........Ray Model Figure4. Eighteen daysof 2-hourERPestimates fromGPS.Forcomparison, thevaluesof theRaymodel (RAY96) areincluded,too' (a) x polecoordinate, (b) y polecoordinate, and(c) UT1 rates.
n
information on nutation estimatesfrom GPS, see Rothacher
et al. [1999].We alsoseethatmostof thesignalat a period of 24 hourshasbeenblockedby puttingconstraints on the 2-hourpolarmotionparametersas mentionedin section2.2 (for details,seeHefty et al. [2000]).
zxx(t)
- E [-P•cosej(t) + pjsin½j(t)], (la) j=l n
- E [pjsinej(t) +pjcosej(t)],(lb) j=l
3.2. Estimation of Ocean Tide Amplitudes
Thedriving forcebehind ocean tidesandthusbehind most
ofthehigh-frequency variations inEarth orientation isthe
XUT(t) -
cobOl(t) +
J=•
'
gravitational attractionexertedby SunandMoon. The tidal whereAX, AY, and AUT1 are the tidal variationsin the potentialgenerated by SunandMoon deformstheEarthand x and y componentof polar motionand in UT1, respecchangesoceanheightsand oceancurrents,whichin turn tively. The quantityn is the numberof tidesconsidered; cause variations inpolarmotion andUT1.In thefrequency p;c and s are the cosineand sine amplitudesof the tidal
domain thetidalpotential maybeexpressed asa series of variations in polarmotion, c and s are the corresponddiscrete tides[Cartwright andEdden,1973].Wemaythere- ingamplitudes inUT1.Thesigns forthecoefficients pj and
forerepresent thetidally driven variations intheERPas
pj in (la) and(lb) area consequence of thewell-known
ROTHACHER ET AL.' HIGH-FREQUENCY EARTH ROTATION FROM GPS 140
13,719
701 (b)
130 K1
120 110'
100' O1
70
30
20 ] 10 0' 22
23
24
25 Period
17 f
26
27
28
29
114
11 5
116
117
118
119 120
-126
-124
12 1 12 2 12 3 12 4 12 5 12 6 12 7 12 8 12 9 13 0
;n Hours
--
•6 (c) 15 14
•
13 11
•E 10 9 •
•
8
7 6
4 N2
0 I - 29
- 28
- 27
- 26
- 25 P•lod
- 24
- 23
- 22
- 21
-128
in Hours
-122 Pedod
-120
in Hours
18!( 0
16 ! (e)
161
12• 10
E-- 10i
41
2
o•;-• - 29
,._ - 28
---
N2
;- 27
- 26
-25 P•lod
-24
-23
-22
-21
-130
-128
in Hours
-126
-124
-122
-120
Period In Hours
Figure5. Amplitude spectra of thediurnalandsemidiurnal tidalfrequency bandsgenerated fromthe entiresubdailyERP series(seriesI). The majortidal termsare labeled(seealsoTables7a-7c). The UT1 spectra werecomputed fromtheUT1 rateestimates andsubsequently converted to spectra in UT1. Notethatthespectra aredrawnwithdifferentscales.(a) Prograde diurnalpolarmotion,(b) semidiurnal prograde polarmotion,(c) diurnalretrograde polarmotion(nutation), (d) semidiurnal retrograde polar
motion, (e) diurnal UT1, and (f) semidiurnalUT1.
definition of thepolex andy coordinate system (forming [1953]. For more detailsand a simple algorithmto convert a left-hand system).Theangleargument O;(t) denotes a between the Woolard and Doodson [ 1921] conventions,see linear combination of the five fundamental astronomical ar-
Gipson[1996]. Tidal termswith the multiplierN6 equalto
guments Fi (i = 1, 2,..., 5) andof F6 = 0 + zr(0 stands 1, 2, - 1, -2 are calledprogradediurnal,progradesemidifor Greenwich meansiderealtime)thatmaybewrittenas urnal, retrogradediurnal,and retrogradesemidiurnaltides, respectively.
6
(t)-
.,% (t). i=1
(2)
We decidedto estimatethe samesetof tidal amplitudes from the 2-hour ERP series as the one selected and deter-
minedby Gipson[1996] usingVLBI datafrom 17 years.
Ni; aretheinteger multipliers uniquely characterizing the Thissetwaschosen inorder toallowa careful comparison tidej. Thefundamental arguments Fi (alsocalled Delauneyof thedifferences between theresults fromVLBI andfrom variables) areusually labeled 1,1',F, D, andf• andrepresentGPS.It includesall termswith a tidalheightlargerthan5 themeananomaly of theMoonandtheSun,theargumentmm (for definitionsandconversions, seeMcCarthy[1996, of latitudeof the Moon, the elongation of theMoonfrom p. 49]). Three diurnal and three semidiurnalterms with estheSun,andthelongitude of theascending lunarnode[see, sentiallyzeroamplitudein the tidalpotentialwere addedto e.g.,McCarthy,1996]. The circularfrequency of tidej is the set,too, for comparison purposes andto checkthe size thengivenbywj := dO;/dt. of errorsto be expectedfor the amplitudeestimates.The Theuseof thenumbers Ni; asdefined abovetocharacter-completelist of termsis givenin section4.5. ize thetidesfollowstheconvention introduced by Woolard Apartfromthisseta muchsmallersetconsisting of only
13,720
ROTHACHER ET AL.' HIGH-FREQUENCY EARTH ROTATIONFROM GPS
the eight major tides,namely,K1, P1, O1, and Q1 in the
tion andUT1 rate values,respectively, of the 2-hourERP series,T5 is theperiodof theestimated tidalsignal,andn is was estimatedto assessthe impact of variouserror sources. thenumberof observations contributing to theestimation of We will call it the "small set" in the following. The terms thecoefficients. The factorof 2n in thedenominator of (5) belongingto this small setare markedin the resulttablesin accounts for thefactthatboththex andy polevaluescondiurnal and/(2,
S2, M2, and N2 in the semidiurnal band,
section 4.5. c
s
c and
Theestimation of thecoefficients pj, pj, uj,
s in
uj
c and PJ' s Looktribute totheestimation ofthecoefficients P5
ing at theentireseriesof 3 years,we haven = 27361,rYpM
(1a), (lb), and (1c) for the selectedtideswas performedus- = 0.33 masandat/t = 0.27 ms/d,andwe obtainformalering a least squaresalgorithmintroducingthe 2-hour polar rorsof 2/tas for polarmotiontermsand0.37/rs for diurnal motionandUT 1 rate estimatesof the globalERP series(cor- and0.19/rs for semidiurnal UT1 terms. If systematic efrected for the low-frequencypart as shownin Figure 2) as fectsare presentin the series,theseformal errorswill be too pseudo-observations. In thisprocedurethe pseudo-observa- optimistic,however. tions may or may not be weighted accordingto the formal Let us now seewhateffectsthe 2-hourresolution may errorsshownin Figure 3. The correspondingresultswill be haveon the estimatesof tidal amplitudesin the casewhere called "weighted"and "nonweighted".In order to estimate no a priori subdailyERP model was usedin the GPS data the UT 1 amplitudesdirectlyfrom UT 1 ratesthe time deriva- analysis.This happened for thetime intervalof seriesB (see tive of (lc) was usedin the leastsquaresprocedure: Table1). Because seriesB is quitelong,we wouldcertainly
like to includeit intotheestimation of tidalamplitudes. We
n
AU;I'l(t) -- Z wj[-ujsinckj(t) + ujcosq55(t)] . (3) j=l
By directly usingthe UT1 rateswe took into accountthat the rate of UT1 is accessible to GPS measurements,but UT1
itself is not. We thusavoidthe problemof havingto remove a random walk behavior from a GPS-derived
UT1 series. For
comparisonpurposeswe also usedthe PM ratesinsteadof the PM offsets in one of the main solutions to be discussed
have to make sure, however, that no biases are introduced
into the amplitudeestimates.The semidiurnal frequency bandwith its relativelyshortperiodscomparedto the estimationintervalof 2 hoursis particularlycritical. To study thisproblem,we generateda seriesof subdailyERP values with a 6-min samplinginterval for the time intervalof series B usingthe Ray model(RAY96). The 6-min valuesof the Ray modelwerethenusedin a leastsquares algorithm to fit piecewiselinear,continuous functionswith exactlythe samecharacteristics as in the real 3-day GPS solutions(2hour binning,overlapping3-day solutions).The resulting seriesof 2-hour subdailyERP valueswas thenusedto determinethemajortidalamplitudes present in theRaymodel andto comparethemwith theoriginalamplitudevaluesby
below. The relevantequationsin this caseareeasilyobtained by taking the time derivativeof (1a) and (1b). Simultaneouswith the tidal amplitudeswe alwaysdeterminedan offsetand drift for all threeERP components over the entire seriesanalyzed to remove systematicoffsetsor drifts presentin the subdailyERP series. Ray. Becausethe subdailyERP seriespresentlyavailablefrom The RMS of the postfitresidualswasaround15 •uasin GPS are much too shortto estimatethe major tidestogether PM and 1.0 •usin UT1. This is not muchcompared to the
with their sidebands (e.g., K1, K•, and K•), we imple- 330/tas RMS level of the real series,but it givesan idea mentedthe optionto constrainthe ratio of the sidebandam- of the "noise"increasewe may expectfor seriesB. Much plitudeandtheamplitude of themajortideasdescribed by moredangerousare possiblesystematiceffects. The RMS Gipson [ 1996]' of the differences betweenthe recovered andoriginalamaj, Hj, plitudes over all terms included in the Ray modelamounts = (4) aj Hj ' to 1.3/tas and0.1 /rs in PM andUT1, respectively. Obviously, the largest deviations of 5.1 /tas in PM and 0.31 /rs c s c and where aj represents oneofthecoefficients pj, pj, us, in UT1 and of 2.4/tas and 0.16/rs, respectively, were found uj forthemajortidej andaj, thecorresponding coefficient termswiththelargestamplitude, namely, of itssideband j•. Hj andH5, arethecorresponding tidal forthesemidiurnal the sine term of M2 and the sine term of 8'2, a systematic height.Seesection 4.5forthevalues Hi, Hi, usedtosetup increase of •0 2% of theoriginalamplitudes. SinceseriesB theconstraints, whichweretakenfrom Gipson[1996]. makes up about one third of the entire series we mayexpect Theformal errors o'(pj),o'(pj),o'(uj),ando'(uj)ofthe that the largest bias will affect the M2 tide in PM andUT1 tidal coefficients to be expected from a leastsquares algowith about +2/•as and 0.1/•s, respectively. rithmmaybeestimated by usingthesimpleformulas devela priorimodel, opedby Rothacheret al. [1999] (assuming equalweights It is obviousthatwhenusingan adequate such biases may essentially be removed: a clear argument for all observations)' for reprocessing all our GPS data. This raisesthe question whetherthe tidal amplitudesshouldbe estimateddia(pj) - a(pj)-o'pM•n (5) rectlyfrom the GPS measurements, avoidingthe problem of aninsufficient timeresolution. Theobvious disadvantage
•/2 -•T5
(6)
would be, however, that ERP variations outside the tidal fre-
quencies(e.g., atmosphere-driven) couldnot be detected.A
whereapMandat/t are the RMS scatterof the polarmo- combinedestimationof tidal amplitudesand 2-hourERPs,
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS
althoughpossible,wouldhavethenegativeeffectin practice of morethandoublingthe numberof unknownsassociated
13,721
c2+Aaj2)
with Earth rotation.
RMScrp - J=• •2
In section4 we will make variouscomparisonsbetween differentsubdailyERP models,eachconsistingof a number or, comparing(7) and (10), of tidal termswith their sine and cosineamplitudes.In the literaturethe mostcommonway to measurethe difference betweentwo models (each assumedto consistof the same rt termsfor simplicity)is to computethe RMS of thediffer-
'
(10)
1
RMScoeff - • RMSerp ß
ences Aa• andAaj ofthesineandcosine amplitudes:
j=l
5-•
difference
between the ERP values of the two
modelsmay thereforebe much larger than the actualvalue
of RMScoeff and,in addition, theratiobetween RMSerp and
(xaj+xaj
RMScoeff -
The RMS
(11)
RMScoeffdependson the number of terms included in the
.
(7)
models.
In section4 we will stickto thismethod,althoughwe should 4. Results be aware of the fact that RMScoeffdoes not reflect the RMS
of the actualdifferencesin the a:and•/componentsandUT1 betweenthe two modelsover a very long time intervalT.
4.1. Influenceof a Priori ERP Series,ReferenceFrame, Sampling,and Weighting
ThisRMSerpmaybe writtenas
In thissectionwe studytheimpactof variousprocessing optionson the estimationof tidal amplitudes, namely,(1)
RMSe2rp
the choice of the reference ERP series used to remove all the
low-frequencyvariationsin the 2-hour series,(2) the real-
izationof the terrestrialreferenceframe,(3) the impactof using a samplingin the analysisof the 2-hour series,and (4) the influenceof weightingthe 2-hour ERP values. The impactof the oceanloadingmodel and the orbit model on
_1/ y•[Aaj cos(coj t)+ Aaj sin(coj t)]dt. (8)
Becausethe integralover a long time intervalof products theresultswill be discussed in greaterdetailin subsequent of theformcos(co/t) cos(co dt), sin(co/t)sin(co• t) fori • j sections.For studies1-4 we only estimatethe smallsetof andof theformsin(co/t)cos(coj t) for arbitrary i andj will majortides,butwe usetheentiredataseriesof 3 years. vanish, we have To check the impact of the referenceERP series(which we haveto subtractfromthe2-hourseriesto obtainthehighRMS erp 2 frequencysubdailyvariations),we usedthe official Bulletin T A seriesand the IERS seriesC04 [Feisseland Castrique, 1997]. The changesin the estimatedtidal amplitudesdue
=7 0 •j=l Aa•cos 2
a• sin2(•t) dr.(9) to a different reference series are summarizedin Table 2,
which lists the RMS differenceof the amplitudes(RMScoeff
Usingelementarycalculus,we finally get
as defined in (7)) between the two solutions, the two tidal
Table 2. Impactof DifferentProcessing Parameterson the Tidal AmplitudesEstimateda Comparison
Reference ERP series: Bulletin
A minus C04
LargestDifferences
RMScoeff PM
UT1
Tide
0.5
0.0
K• + P• + K• + K• + ,5'2K2 M2K2 -3P• + K• -3-
Reference ERP series: Bulletin A minus CODE
0.9
0.1
Reference frame: 37 fixed minus network constraints
2.3
0.1
Samplingof series:nonsampled minussampled
1.8
0.7
Observation weighting:weightedminusnonweighted 6.1
0.6
cos/sin
PM
Tide
cos/sin
UT1
cos
-1.6
Q•-
sin
0.1
cos cos sin cos sin sin sin sin sin
- 1.3 -3.3 - 1.2 5.3 -4.2 4.7 -4.2 13.8 - 11.7
O1 K1P• M2 ,5'2P•K• O•K• -
cos cos cos sin cos cos sin cos cos
0.0 -0.2 0.2 -0.2 -0.2 -1.6 1.4 1.3 - 1.3
aOnlythesmallsetof majortideswasestimated for thisstudy.Thesignafterthetidedesignation refersto prograde (positive) andretrograde (negative) terms.Unitsare•as for PM and•s for UT1.
13,722
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS
Table 3. Comparisonof Three Setsof AmplitudesDerived From the GPS 2-hour Series With the Models
RAY96
and GIPSON96
a
RMS Differences
SolutionType
Weighting Yes Yes No
RAY96
GIPSON96
ReferenceFrame
PM
UT1
PM
UT1
network constraints 37 sites fixed 37 sites fixed
12.1 11.8 13.3
0.7 0.7 1.1
14.2 14.0 14.9
1.5 1.5 1.8
aOnlythe smallsetof majortidesis compared.The RMS differences (RMScoeff) aregivenin unitsof/•as for PM and/•s for UT1.
coefficientsfor PM andUT1 with the largestdifferencesand the amplitudedifferencesthemselves.Clearly,the type of referenceERP seriesusedonly has a marginaleffecton the amplitudes,causingdifferenceson the 0.5/•as and 0.03/•s level in PM and UT1. An additionaltestusinga reference seriesby CODE, whereeachERP componentwas modeled by only one offset and drift parameterover the 3-day solutions,did not changethe resultsfor any term by morethan twice its formalerror(seeTable2). In thefollowing,we will alwaysusethe Bulletin A seriesas a reference. As stated in section 2.4, two series (series I and II) were generatedwith a different realization of the referenceframe
valueandusingdifferentstartingvalues.The comparison of thesereducedserieswith the completeseriesis alsoincluded in Table 2. The changesin UT1 (differenceof the orderof 0.7/•s) arepredominant.The termslistedin Table2 indicate
that thereare systematiceffectspresentmainly at periods near24 hours.This is not really astonishingwhenwe keep in mind thatthe satelliterevolutionperiodis aboutonesidereal day, that is, near 24 hours, too. Correlationsbetween the subdailyERP variationsand the satelliteorbit parameters, especiallythe radiationpressureparameters,may be responsiblefor a propagationof orbit modelingbiasesinto the subdailyERP estimates,a difficulty that has to be dealt
(37 sites fixed or network constraints). Tidal amplitudes were estimatedin the sameway in both series.The differencesbetweenthe two solutionsare also given in Table 2. The changesin PM are muchlargerthanthosein UT1. This may be expectedfrom the fact that day-to-daychangesin the realizationof the referenceframe directlymap into PM, whereasthe ratesin UT1 are quite "immune"to the referenceframe. The deviationsin PM areof theorderof 2.3/•as, and the largestdifferencein PM reaches5.3/•as and is still
2-hourERP valueswereweightedusingtheformalerrorsof the 2-hour estimatesresultingfrom the originalGPS 3-day solutions.Theseformal errorsare shownin Figure 3 for all three components.Becausethe formal errorsare decreasing with time, the mostrecentpartsof the series,whereimprovedmodelsandstrategies were applied,will geta higher weight. A secondsolutionassignedequal weightsto all
within
ERP
2.5 times the formal
error.
Therefore
we conclude
that thereare no systematiceffectscausedby the reference frame stabilityin our results.It is interestingto note,however,that the impactis largestin the retrogradesemidiurnal band.This is not clearlyunderstood yet. The referenceERP seriesin comparison, asyou may expect,mostlyaffectsthe lowerdiurnalfrequencies.Comparingthetwo solutionswith the Ray model (RAY96, derivedfrom altimeterdata [Ray et al., 1994]) and the modelby Gipson[1996] (GIPSON96,
with in all satellitegeodetictechniques. In study4, finally, one solutionwas computed,wherethe
values.
The
numbers
listed
in Table
2 demonstrate
the importanceof a correctweightingof the ERP pseudoobservations when estimatingthe tidal amplitudes.Differencesof up to 14/•as in PM and 1.3/•s in UT1 are far outside the confidenceinterval of 99% (3a error band). To decide whetherthe weightingreally improvesthe results,we comparedthe two solutionswith the modelsRAY96 and GIPSON96. The amplitudeRMS differenceswith respect to thesemodelsare givenin Table 3.
derivedfrom VLBI observations;unlessotherwisestatedwe
The weightedsolutionagreesmuchbetter,in particularin UT1, with both, the Ray and Gipsonmodel, than the nonweightedsolution.All the analysesto follow will therefore be basedon a correctweightingof theERP observations. As we will showin section4.4 when lookingat the individual into account the variance-covariance information of the site partsof the series,the betteragreementof the weightedsocoordinates. Stations with bad coordinate estimates on one lutionis mainlythe resultof the deweighting of theERP esday (e.g., due to a deficiencyof data)may thenaffectthe timatesstemmingfrom the classicalorbitmodel,whichare
referto the "constrained" modelby Gipson[1996]), we see that seriesI (fixed sites)givesslightlybetterresultsin PM (seeTable3). This may be due to the way the systemconstraintsare set up in the solutions,namely,withouttaking
referenceframe. In the following,seriesI will alwaysbe muchworsein qualityandseemto includelargesystematic used.
effects.
In study3 (of the four mentionedabove)the original2From Table3 we alsoget a firstideaof the differences hourserieswasreducedby only takingeverythirdor fifth between thethreemodelsstemming fromthreeindependent
ROTHACHERET AL.' HIGH-FREQUENCYEARTHROTATIONFROMGPS
13,723
1.0 0.8
0.6 0.4
0.2 0.0
-0.2 -0.4
-0.6 -0.8
Day of Year 1998
............ GPS withoutOcean Loading
........
GPS with Ocean Loading
............ GPS without Ocean Loading
GPS with Ocean Loading
Ray Model
1.0
0.8 0.6
0.4 0.2 0.0 -0.2 -0.4
-0,8
Day of Year 1998
........
Ray Model
0.6 0.5 0.4
0.3 0.2 0.1 0.0 -0.1
-0.2 -0.3
-0'4 I• -0.5 .-..,..'.
-0.6 -0.7
61
62
63
64
65
Day of Year 1998
............
GPS without Ocean Loading
........
GPS with Ocean Loading
Ray Model
Figure6. Effectofocean loading onthe2-hour ERPestimates fromGPS.Forcomparison thevalues of theRaymodel(RAY96)areincluded: (a)z polecoordinate, (b)1/polecoordinate, and(c)UT1rates.
techniques.The GPS-derivedtidal amplitudesare in bet-
erneck[1991]usingtheoceantidemapsfromLe Provost etal. [1994].Additional information isgivenbyMcCarthy GPSagreeswith theRay modelat a levelof 12 yas in PM [1996].Figure6 shows the2-hourvalues forthez and!/ and0.7 ys in UT1. Comparison with modelsfrom different component of PM andfor theUT1 rates.Comparing the space-geodetic techniqueswill be discussed in more details z and1/components totheRAY96modelweonlyseesmall ter agreementwith the model RAY96 than with GIPSON96.
in section 4.6.
differences whenoceanloadingis activated, butin UT1 rates someof thepronounced peaksin theGPSseriesarereduced. 4.2. OceanLoading Model To give a quantitativemeasureof the differences,Table 4 of the two GPS series(with No oceanloadingmodelfor sitedisplacement wasused containsthe RMS differences to themodwhencomputing thesubdailyERP parameters. Becausere- andwithoutoceanloadingmodel)withrespect processing the entireseriesis presentlyout of reach,we els RAY96 and GIPSON96. computedfive global3-day solutions(days060 to 064 in Althoughthe differencesseemsmall comparedto the 1998)with andwithoutan oceanloadingmodelto get an noise of our series, we have to take into considerationthat impression of its impacton thesubdailyERP values.The thesedifferences aresystematic andnotrandom.We may oceanmodelcoefficients werecomputed according to Sch- thereforeexpectchanges in the tidal amplitudes dueto an
13,724
ROTHACHER ET AL.: HIGH-FREQUENCY
EARTH ROTATION FROM GPS
Table4. Comparison of a GPSSeriesof 5 daysWith andWithoutOceanLoading
Model Used and the ERP Values of the Models RAY96 and GIPSON96 a
SolutionType
RAY96
Pole Without oceanloading With oceanloading
GIPSON96
•/Pole
UT1 Rate
a:Pole
•/Pole
UT1 Rate
172 170
125 121
178 175
169 166
133 126
167 163
aThe RMS differencesaregivenin unitsof ktasfor PM andkts/dfor UT1 rates.
erroneousoceanloading model for stationmotion that are
to the middle day for the classicalorbit model. In the case of the extendedmodelthe agreementwith RAY96 is still slightlyworsefor theboundary daysbutonlyby ,-.,15-20%, andin UT1, almostno difference existsbetweenthequality of themiddleandthefirstandlastday. Figure7, showing
of the order of the differences between the two solutions, or
about2-4/.tas in PM and0.3 to 0.6/.rs in 2 hoursin UT1. 4.3. Impact of Orbit Modeling A considerablepart of the subdailyERP series(abouthalf of the entire series,namely, parts B and C, see Table 1) was basedon the classicalorbit model (see section2.3). It is thereforeimportantto know whetherthere are any significantdifferencesin the quality of the ERP valuesstemmingfrom solutionsusingthe classicalor theextendedorbit model. In the beginningof 1995 125 dayswere processed twiceusingthe two differentorbit models.The resultingseries are calledA andA' in Table 1. They allow us to study the impact of the orbit modelingin more detail. For both seriesA andA' the differenceswere computedbetweenthe respectiveseriesandRAY96. In addition,two alternativeseriesof 2-hourERP valueswere composed by extractingthe ERP valuesof the firstandthe third day,respectively, of the 3-day solutionsinsteadof the middle day commonlyused. The seriesgeneratedfrom the ERP estimatesof the first or last daysare interestingto look at, becauseit is well known that the orbit quality of a 3-day arc deterioratestowardthe
the differences between the GPS series and RAY96 for the x
component of PM, illustratesthequalitydifferencebetween
theorbitmodels.Results forotherparameter types(like,for example,the stationcoordinates)confirmthisconclusion. We may also see from Table 5 that the differencesin the
ERPsstemming fromthetwoorbitmodels arenotverylarge for themiddleday,the valueswhichwe includein theusual analyses.We might thereforeconcludethat we shouldbe ableto includepartsB andC of the seriesin our final analyseswithoutdegrading theoverallqualityof theresults.In section4.4, whenstudyingthedifferentpartsof the series
in moredetail,we will see,nevertheless, thatsomesystematiceffectsin thesubdaily ERPestimates arecaused by the classical orbitmodel.A detailedstudyof thecorrelations of orbitparameters withsubdaily ERPsshouldbeperformed to get a betterpictureof the interactionbetweenthesetwo sets of parameters.This is left to the future.
Let us not forgetto pointout thatthe qualityof results obtainedin thebeginningof 1995(witheitherorbitmodel)is bestprecisionis achieved.Table5 summarizesthe compari- notrepresentative for theentireseries.Comparing thevalues sonof the six series(first, middle,andlastday valuesfor the inTable5 withthose inTable4, werecognize thatthequality classicaland the extendedorbit model) with model RAY96. of theseriesimprovedovertimeby a factorof 2 in PM and We see that the differenceswith respectto RAY96 in- 2.5 in UT1 rates.Oneof thereasons for thisdevelopment is creaseby 40-50% and 70-80% for the x and •/component certainlytheincrease in thenumberof globalsitesandthus of thepole,respectively, whencomparingtheboundarydays the numberof GPS measurements (seeFigure1); another boundaries of the arc, whereas in the middle of the arc the
Table5. Comparison of GPSSeriesof 125daysEachwith RAY96a Day of 3-Day
Solution
ClassicalOrbit Model
x Pole •/Pole UT1 Rate
Extended
Orbit Model
x Pole •/Pole UT1 Rate
First
546
641
478
441
407
313
Middleb
376
377
315
368
353
301
Third
518
679
452
421
411
311
aGPSvaluesweregenerated by extractingtheERP valuesof thefirst,middle,and
lastdaysof the3-daysolutions. TheRMSdifferences, in unitsof gasforPM andkts/d for UT1 rates,arelistedfortheclassical andextended radiation pressure model. bThemiddle-day series areidentical withseries A (extended) andA' (classical).
ROTHACHER ET AL.' HIGH-FREQUENCY EARTH ROTATION FROM GPS
"-'
0.5
13,725
ß ,•. ,,
•
•?;:, , •..',•: ;.,• , ,-, ; :,,,,' ', • !•,;', .•.
:•..'• '•'..' "'. :
•
no - 0.5 x
:
It
'
: .'•:
r.:
..:
•,'
--.
-1.0
115
116
117
118
119
120
121
122
123
124
125
Day of Year 1995
............
First Day of 3-Day
Arc
MiddleDay of 3-Day Arc
........
LastDay of 3-Day Arc
1.5
(b) 1.0
....:,: ,
....
œ.•...
,,
I
o.o
-o.5 -1.5
11•
,,.
\?1... ,.
. i:,,,...:... ¾ :... ' '
,•
-1.0 1 115
:, :'7 '
,
•
•
•
•
•
,
117
118
119
120
121
122
123
• .... 124
125
Day of Year 1995
I ............ First Day of3-Day Arc
Middle Day of 3-Day
Arc
........
Last Day of 3-Day
Arc
Figure 7. Effect of the radiationpressuremodelon the a:componentof polar motionfor the first, middle, and last day of the 3-day arc. Differenceswith respectto the oceantide model RAY96 are shown. (a) Classicalradiationpressuremodeland (b) extendedradiationpressuremodel.
reasonistheimprovements in theprocessing strategy andthe GPS observation modeling.Our experiments with different orbitmodelssuggest thatthe qualityof the subdailyERPs couldalsobe usedasanindicatorfor theorbitquality.
is encouragingand indicatesthat the entire seriescouldbe
significantly improved by usingthemostrecentstrategy for theentireseries.The qualityincrease of seriesE (compared to seriesD) is dueto theloweringof theelevationcutoffan-
gleto 10ø (seeTable1). PartD (compared to C andB) benefitsfromtheimprovedorbitmodel.We shouldnotforget,
4.4. Quality of Individual Parts of the Series
Eachof thesubintervals (A, B..... E) of thesubdaily ERP seriesdefinedin Table1 wasanalyzedseparately to check the performance of the strategyandmodelsusedin the respectivetime interval. As a firststep,the differences between each sub-interval and the model RAY96 were formed.
The RMS of thesedifferences maybe foundin Table6. SeriesE is in bestagreementwith theRAY96 model.This
however, that there was a constant increase in the number of
sitesduringthe 3 years(seeFigure 1). SeriesA andD, for example,wereproducedusingidenticalprocessing options. Thedifference in performance reflectsthedenserglobalcoverageduringperiodD (almosttwiceasmanysitesasduring period A).
Table 6. Comparisonof the GPS SeriesA, B, C, D, and E
We alsoestimated the smallsetof majortidalamplitudes from eachof the individualseriesand comparedit to the amplitudes of the Ray modelusing(7). In this analysis
Listed in Table 1 With the Values of RAY96 a
we also included a few combinations of intervals like, for
Series
RMS With Respectto RAY96 Pole
A B C D E
368 364 345 294 233
•/Pole
UT 1 Rate
353 361 351 331 253
301 277 347 265 185
•The RMS differencesare givenin unitsof/zas for PM and/zs/d for UT 1 rates.
example,the serieswe name ADE, consistingof the three partsA, D, andE, and the entire 3-year series,oncewithout (All) and oncewith weighting(weighted)the ERP pseudoobservations(see section4.1). All the other serieswere not
weightedin thisexperiment.To accountfor thefact thateach of theserieshasa differentlengthin time andthereforea different numberof ERP pseudo-observations, we normalized the RMS differences RMScocffwith
RMSnorm - RMScoeff /All' V/1
(12)
13,726
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS 17•
1.8 -•
I (a)
1.71 (b)
ßa
1.6
1.5 1.4
1.3 1.2,
1.1
• 13t •
12
ß Weighted
ßA
0.7 •
100
200
ß DE
ß AOE
o.5•
ß AOE' ACOE
91 ,E
ß Weighted
ß CDE
0.6 ß O
ACDE
0.4
87 ..... 0
ß c ß A
0.9
0.8
ß COE ß DE
z• 10
1.0
0.3
300
400
500
600
700
800
900
1000
1100
1200
1300
Length of Series in Days
0
100
200
300
400
500
600
700
800
900
1000 1100 1200 1300
Length of Series •n Days
Figure8. RMS differences (reducedto a commonserieslengthusing(12)) betweenvarioussetsof GPSderivedtidal amplitudesand the amplitudesof RAY96. See text for details. (a) Polarmotionand (b) UT1.
where1 and/All are the lengthsof the seriesconsidered and schemecouldbe derived,whichmighthelpto improvethe the entire series,respectively,in days. With this normaliza- resultsof the entire series. This was not done so far, howtion the RMS of the entireseries(All) is not changed.If the ever. differences betweenthe modelswerecausedpurelyby ranFigure 8 also demonstrates that the qualityof the estidom errors, the value of RMS,o•n should be almost identical matedamplitudes criticallydepends on thelengthof thesefor eachseries(errorsdecreasingwith the squarerootof the ries.This is notamazingwhenlookingat thenoiselevelof, numberof observations or daysused). Figure 8 showsthat for example,0.2-0.3 masin PM of thesubdailyERP series. this is not the case. Quitea longtime seriesis necessary to endup with formal We immediatelysee that seriesB and to a certainextent uncertainties of the order of a few microarc seconds even seriesC are much poorer in quality than the other series. from a purelystatisticalpointof view (see(5)). Not only the noiseof theseseriesis larger(aswe haveseen in Table 6) but also the systematiceffects,which bias the 4.5. Four Main G PS Solutions amplitudeestimates,are obviouslymuchworse.We would On the basis of the studies described in sections 4.1-4.4 like to pointout thatthebadperformanceof seriesB cannot be explainedby the artifactsdiscussed in section3.2 (the fit- we decidedto producefour GPS solutions: tingof thefull subdailysignalwith a 2-hourpiecewiselinear 1. Estimationof the completeset of 57 tidesin PM and function).The comparison evengetsworsewhencorrecting 41 tides in UT1 listed in Tables 7a-7c. We call this set for this effect using the simulationresultsof that section, "complete".The amplitudesof the sidebands(markedwith thusleavingtheorbitmodelastheonlyconvincing explana- a primein Tables7a-7c) wereconstrained according to (4). tion. The recentseriesD andE areperformingbestin both 2. Estimationof a reducedset of tides(47 and 33 tides component,PM and UT1. The value of 0.4 ps in UT1 for in PM andUT1, respectively) called"reduced", namely,all the seriesE meansthat providedwe would haveat our dis- thetermslistedin Tables7a-7c withtheexception of all the posala homogeneous 3-yearseriesof typeE, we couldreach sidebands. No constraints hadto be appliedin thissolution. the 0.4 •s levelin UT1. Not the sameimprovement is to be 3. Estimationof thesmallsetof majortides(seefootnote expectedin PM. There we couldreach,-09 pas according b of Tables7a-7c) or a totalof 12 termsin PM and8 in UT1. to Figure 8 with an E seriescoveringthe entiretime span This set is called "small". of 3 years. The actual (not normalized)RMS valuesof the 4. Sametypeof solutionasfirstsolution(thecomplete set comparison of seriesD andE with themodelRAY96 are,by tidesin Tables7a-7c,addingconstraints) butbasedonpolar the way, 16 and 25 pas in PM and 0.9 and 1.2/zs in UT1, motionratesas pseudo-observations andnot polarmotion offsets. It is therefore called "rates". This solution differs respectively. Weighting the 2-hour ERP values (the solutionlabeled from the firstsolutiononly in PM amplitudes becausethe weighted)helpedto improvethe agreementwith RAY96 in UT1 estimates are identical. PM andUT1. Becausethe ERP valuesstemmingfrom soluAll foursolutions arebasedontheentireseriesof 3 years tionsusingthe classicalorbit modelhavelargerformalun- (actuallyseriesI, seesection2.4), andthe ERP observations certainties,theweightingoption"deweights" theERP values wereweighted. of seriesB andC, resultingin a bettercomparison. Looking Theresultsof three(twoin UT1) of thesesolutions may at Figure 8, we might be temptedto excludeseriesB from be found in Tables 7a-7c. The results from the small soluthe main solutions.We shouldbe awareof the fact, however, tionwerenotincludedbecause theyareverysimilarto those
that the seriesweighted(includingseriesB) still performs betterthanseriesACDE or ADE (notnormalized according to (12) asin Figure8) because of theconsiderable lengthof seriesB. On thebasisof Table6 a morerefinedweighting
of thereducedset. The formaluncertainties of the amplitudeestimatesare 1.9 pas for all polar motionterms(with theexception of thetideswithsidebands in thecomplete solution)whenusingERP offsetsaspseudo-observations. For
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATIONFROM GPS
13,727
Table 7a. Tidal Amplitudesof UT1 From GPSa Tide
Multiple of
I
l'
F
D
Period,
fl
0 + •r
H,
Reduced
hours mm
Diurnal
c
Complete c
uj
uj
uj
uj
UT1 Terms
2Q•
2
0
2
0
2
-1
28.01
-7
-0.6
-2.3
-0.6
-2.3
rr
0
0
2
2
2
-1
27.85
-8
-0.6
-1.1
-0.6
-1.1
Q•
1
0
2
0
1
-1
26.87
-9
Q•b
1
0
2
0
2
-1
26.87
-50
-2.0
-10
-0.6
p•
O• O• b
-1
0
2
2
2
-1
26.72
0 0
0 0
2 2
0 0
1 2
-1 -1
25.82 -49 25.82 -262
0
-1.0
-4.6
-2.3
-5.6
-0.5
-0.6
-0.5
- 11.8 - 14.5
-1
0
0 1
0 -2
2 0 2
24.85
1 0
2 0 2
-1
M• N•
-1 -1
24.83 24.13
p• b
0
0
2
-2
2
-1
24.07
$•
0
1
0
0
0
-1
24.00
K•' K• b K•
0 0 0
0 0 0
0 0 0
0 0 0
-1 0 1
-1 -1 -1
23.94 23.93 23.93
-7 369 50
7.8
•I,• •I,• J• OO•
0 0 -1 0
-1 0 0 0
0 -2 0 -2
0 2 0 0
0 -2 0 -2
-1 -1 -1 -1
23.87 23.80 23.10 22.31
3 5 21 11
-1.1 -0.3 0.8 0.8
OO•
0
0
-2
0
-1
-1
22.30
7
r/•
-1
0
-2
0
-2
-1
21.58
2
r/•
-1
0
-2
0
-1
-1
21.58
1
Semidiurnal
-0.4
-2.7
-3.3
-14.2
-17.8
7 21 -7
0.3 -0.2 -2.2
1.6
0.3
1.6 1.6
-0.2 -2.2
1.6
- 122
-3.0
-6.0
-3.0
-6.1
-2.4
1.6
-2.4
1.7
-0.2
-0.4
8.9 1.2
18.3
3
-0.6
16.4 2.1 -0.1 0.4 0.3
0.3
1.6 1.6
2.5
-1.2
2.1
-0.2
-0.2
0.8
0.4
1.4
0.3
0.9 -0.9 -0.4
0.2
0.6 0.3
UT1 Ter•ns
2N2
2
0
2
0
2
-2
12.91
16
-0.4
0.9
/•2
0
0
2
2
2
-2
12.87
19
-0.3
0.6
-0.4 -0.3
0.9 0.6
1
121
-1.8
4.0
-1.8
4.0
23
-0.2
0.7
-0.2
0.7
0.3
-0.6
-6.8 0.4 -0.2
15.9 -0.2
N2b
0
2
0
2
-2
12.66
-1
0
2
2
2
-2
12.63
0 0
0 0
2 2
0 0
1 2
-2 -2
12.42 12.42
-24 632
-7.1
16.4
-1 0
0 1
2 2
0 -2
2 2
-2 -2
12.19 12.02
-18 17
0.4 -0.2
-0.2 0.6
,.•'2 b
0
0
2
-2
2
-2
12.00
294
-0.5
R2
0
1
0
0
0
-2
11.98
1
-0.8
K2b K•
0 0
0 0
0 0
0 0
0 1
-2 -2
11.97 11.97
80 24
-1.3
v'2
M• M2b L2 T2
7.7 -0.5
1.8
0.6
-0.5
7.7
-0.8
-0.5
-1.7
2.6
-0.5
0.8
Small UT1 Terms
Ternary
0
0
3
0
3
-3
Small Small Small Small Small Small
0 1 0 3 1 0
0 0 0 -1 1 0
0 4 0 2 2 0
4 -2 1 0 0 -2
1 2 0 2 1 2
-1 -1 -1 -2 -2 -2
8.28 27.67 27.04 24.77 13.14 12.68 11.58
8
0.0
0.0
0.0
0.0
0 0 0 0 0 0
-0.5 0.3 0.6 0.2 -0.2 0.0
-0.6 0.8 0.2 -0.1 0.1 0.2
-0.5
-0.6
0.3
0.7
0.6 0.2 -0.2
-0.1
0.0
0.2
0.2 0.1
aAmplitudesin
bIncluded in thesmallsetof theeightmajortides.
the fourthsolutionusingpolarmotionratesthe formaluncertainties areincreasing withtheperiodconsidered (see(6)) andamountto 4/zas for diurnaland2/zas for semidiurnal PM terms.The ternary(M a) amplitudes havea formalerror of 1.4/•as.Theuncertainties of theUT 1 termsalsodepend ontheperiodaccording to (6) andlie between0.4 and0.5 tzs
for diurnalbetween0.20 and 0.23 tzsfor semidiurnalterms and at 0.15/zs for the ternaryterm. Let us first havea look at the consistency of the four GPS solutionsbefore comparingthem to other modelsin section 4.6. The RMS differencesbetweenthe solutionsmay be found in Tables8a and 8b for UT1 and PM, respectively.
13,728
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS
Table 7b. Tidal Amplitudesof ProgradePolar Motion From GPSa Tide
Multiple of
1
l'
F
D
Period,
•2 0 + •r
H,
Reduced
hours mm
C
pj
S
pj
Complete C
S
Rates C
S
Pd
P•
P•
P•
Diurnal ProgradePolar Motion Terms
2C2•
-2
0
-2
0
-2
1
28.01
-7
-4
-1
-4
-1
-5
-1
0
0
-2
-2
-2
1
27.85
-8
-6
1
-6
1
-6
0
Q•
-1
0
-2
0
-1
1
26.87
-9
-6
2
-6
2
Q•b
-1
0
-2
0
-2
1
26.87
-50
-27
9
-33
11
-31
9
1 0 0
0 0
-2 -2
-2 0
-2 -1
0
-2
-3 -25 -134
0 8
0
-4 -25 -132
0 7
-2
26.72 -10 25.82 -49 25.82 -262
-4
0
1 1 1
1
0
-2
0
-2
1
24.85
7
3
-10
3
-10
0
-19
-1 0
0 -1
0 -2
0 2
0 -2
1 1
24.83 24.13
21 -7
-8 - 14
-8 -5
-8 - 14
-8 -5
-13 - 18
-9 3
-72
30
-72
29
-66
30
36
39
37
39
40
26
a
p• O[
O• b M• N•
p• b
0
0
-2
2
-2
1
24.07 -122
S•
0
-1
0
0
0
1
24.00
K•' K• b K•
0 0 0
0 0 0
0 0 0
0 0 0
1 0 -1
1 1 1
23.94 23.93 23.93
-7 369 50
ß• ß• J• OO•
0 0 1 0
1 0 0 0
0 2 0 2
0 -2 0 0
0 2 0 2
1 1 1 1
23.87 23.80 23.10 22.31
3 5 21 11
OO•
0
0
2
0
1
1
22.30
7
r/• rh
1 1
0 0
2 2
0 0
2 1
1 1
21.58 21.58
2 1
!
3
-109
32
37
127 -79
-3 2 143 -89 19 -12
-12 7 -1 5
-45 -0 -7 -3
-13 6 -1 10
-44 -1 -7 -10
6 -1
-3
-5 -2
45
-3 2 139 -88 19 -12 -8 9 -1 12
-35 2 -6 -8
-6
7
-5
-5 -2
-4 -2
-3 -2
2
SemidiurnalProgradePolar Motion Terms 2N2
-2
0
-2
0
-2
2
12.91
16
2
3
2
3
4
tt2
0
0
-2
-2
-2
2
12.87
19
-3
0
-3
0
-2
0
-1
0
-2
0
-2
2
12.66
121
14
-7
14
-7
12
-8
t22
1
0
-2
-2
-2
2
12.63
23
2
-4
2
-4
1
-4
M• 3f2b
0 0
0 0
-2 -2
0 0
-1 -2
2 2
12.42 12.42
-24 632
46
-50
-2 44
2 -48
-2 43
2 -52
L2 T2
1 0
0 -1
-2 -2
0 2
-2 -2
2 2
12.19 12.02
-18 17
-3 -13
4 -13
-3 -13
4 -13
-2 -18
4 -13
294
-1
-31
-2
-31
-7
-31
1
8
-14
8
-14
2
-14
80 24
13
-12
16 5
-17 -5
15 4
-14 -4
N2b
S2b
0
0
-2
2
-2
2
12.00
R2
0
-1
0
0
0
2
11.98
K2b K•
0 0
'0 0
0 0
0 0
0 -1
2 2
11.97 11.97
Small ProgradePolar Motion Terms
Ternary
0
0
-3
0
-3
3
8
3
-1
3
-1
1
-2
Small
0
0
0
-4
-1
1
27.67
8.28
0
3
-2
3
-2
0
-2
Small Small Small Small Small
-1 0 -3 -1 0
0 0 1 -1 0
-4 0 -2 -2 0
2 -1 0 0 2
-2 0 -2 -1 -2
1 1 2 2 2
27.04 24.77 13.14 12.68 11.58
0 0 0 0 0
- 11 -0 0 0 4
-0 -7 3 -1 -3
- 11 0 0 0 4
-0 -7 3 -1 -3
-8 -5 0 -3 4
3 -12
2 -0 -3
aAmplitudesin ttas.
bIncludedin thesmallsetof theeightmajortides.
The small and reducedsolutionsare almostidenticalexcept nored;in the latter solutionthey were inferredby putting for the fact that many more termswere estimatedin the re- constraintson the amplituderatio betweenmain tide and ducedsolution.The maximumamplitudedifferencesoccur sidebands. (An unconstrained estimation is notpossiblebecauseof the shorttime spanof the GPS series.)If we comin P• for PM (6 ttas)andin K• for UT1 (0.4 tts). The main difference
between
the reduced and the com-
paretermswithoutsidebands only, the two solutionsgive pletesolutionis the treatmentof the tideswith relevantside- almostidenticalresults(RMS differenceof only 0.23 ttas bands.In the former solutionthe sidebandswere simplyig- in PM and0.03 its in UT1 compared to 3.8 ttasand0.6 tts
ROTHACHERET AL.' HIGH-FREQUENCYEARTHROTATIONFROMGPS
13,729
Table 7c. Tidal Amplitudesof RetrogradePolarMotion From GPSa Tide
Multiple of
I
l'
F
D
Period,
•
0 + •'
H,
Reduced C
hours mm p.•
$
p.•
Complete C
p.•
$
p.•
Rates C
$
p.•
p.•
SemidiurnalRetrogradePolar Motion Terms 2N2
2
0
2
0
2
-2
12.91
16
3
1
3
1
3
3
•2 N2b
0 1
0 0
2 2
2 0
2 2
-2 -2
12.87 12.66
19 121
0 3
5 44
0 3
5 44
-1 1
5 45
-1
0
2
2
2
-2
12.63
23
3
1
3
1
1
3
0 0
0 0
2 2
0 0
1 2
-2 -2
12.42 12.42
-24 632
-1
265
0 -1
-10 256
1 -19
-10 255
-1 0
0 1
2 2
0 -2
2 2
-2 -2
12.19 12.02
-18 17
2 12
-3 3
2 12
-3 3
5 7
-2 3
S2b
0
0
2
-2
2
-2
12.00
294
-67
112
-67
112
-69
110
R2
0
1
0
0
0
-2
11.98
1
-5
-13
-5
-12
-2
-10
K2b K•
0 0
0 0
0 0
0 0
0 1
-2 -2
!!.97 11.97
80 24
2
30
! 0
43 13
•2
M} •f2 b L2 T2
-3 -1
43 13
Small RetrogradePolar Motion Terms
Ternary
0
0
3
0
3
-3
Small Small Small
3 1 0
-1 1 0
2 2 0
0 0 -2
2 1 2
-2 -2 -2
8.28 13.14 12.68 11.58
8
0
-2
0
-2
1
-2
0 0 0
2 3 0
4 4 0
2 3 0
4 4 0
-0 0 -1
3 3 1
aAmplitudesin ttas. bIncludedin the smallsetof theeightmajortides.
when comparingthe full set of commonterms). However, whensidebandswere setup in the completesolution,the solution differed significantlyfrom the reducedsolution.The amplitudesof the main tideschangeby aboutthe sizeof the sidebandamplitudesdeterminedin the completesolution, which are particularlylarge for the tidesO• and K•. The cosinetermsof thesetwo tides differ by 23 and 16 ttas in PM between
the two solutions.
In UT1
in very good agreementwith the formal errors mentioned aboveexceptfor diurnal polar motion, wherethe formal er-
rorsstemmingfrom the completesolutionare too smallby a factorof • 2. The formalerrorsof thesolutionusingPM ratesseemto be more realistic. It is worthnotingthat also in VLBI [Gipson, 1996] the semidiurnalcoefficientsof the small terms show a lower scatter than the diurnal ones.
we have corrections
No signalcan be detectedat the frequencyof Ma, the of 2.4 and 3.3 tts in the sineand cosineamplitudesof O•. ternary.None of the ternarycoefficientsis significantlydifThe divisionof the signal at thesefrequenciesinto a main ferentfrom zero. Also, the 2-hourbinningwould lead to tide and sidebandsis solely determinedby the tidal height a dampingof the ternaryamplitudes[seeRothacheret al., values, since the GPS series does not contain additional in1999, Figure9]. formation.Withouta muchlongertime series(18.6 yearsare neededfor a full decorrelationof main tide and sidebands)it 4.6. ComparisonWith Models From Other Techniques will notbe possibleto obtainany insightinto this"sideband problem"from GPS results. The solutionusingpolar motion rateswas generatedbecausethe effect of remaininglow-frequencysignalsin the ERP seriesshouldbe reduced. It agreesreasonablywell with the completesolution(see Table 8b). There is a relatively large differenceof 19 ttas in the cosineterm of the largesttide M2, whichis far outsidethe 3o-limit of the com-
By now, modelsfor high-frequencyERP variationsare availablefromfourdifferentobservation techniques, namely, modelsderivedfrom the dataof threespace-geodetic techniques(VLBI, SLR, and GPS), and oceantide modelsbased on, for example,TOPEX/Poseidonaltimeterdata. Eachof thesetechniqueshasits own advantages and dis-
advantages.GPS, in particular,with its big spacevehicles (wherethemodelingof radiationpressureis a realchallenge) bined formal errors of the two solutions. This indicates that but alsoSLR suffersfrom correlations betweenorbitalpatherearestill quiteimportantsystematic biasespresentin the rametersand ERPs. GPS has the advantages,however,that GPS series(in the PM offsets,rates, or both). it hasa very denseglobalnetworkandthatsimultaneous obThe "small"
terms estimated
in three of the four solu-
tions(alsoincludedin Tables7a-7c) give an estimateof the "noise"in the coefficientsandmay be usedto checkthe formal errors.In UT1 we find amplitudesof typically0.15 and 0.5 tts for the semidiurnaland diurnal band, respectively. The numbersfor PM are 2.6 and 5.5 ttas. Thesevaluesare
servations to several satellites from each site are available.
Modeling the observablemay not be as refinedyet for GPS as in the case of VLBI
and SLR and GPS-based
time se-
ries are much shorterat presentthan thoseestablishedby VLBI andSLR. In VLBI andGPS themodelingof thetroposphereis a majorissuewhenstudyinghigh-frequency ERPs,
13,730
ROTHACHERET AL.: HIGH-FREQUENCY EARTHROTATIONFROMGPS
Table8a. RMS Differencein UT1 Amplitudes BetweenTidalModelsFromDifferentTechniques a GPS Solutions
Model
Small
VLBI Solutions
Reduced Complete 0.2
1.2 0.6
SLR
GipR
GipU
GipC
H&D
1.4 1.5
2.0 1.4
1.5 1.2
1.5
1.3 1.4
1.0 1.0 0.8
Small Reduced
0.2
Complete GipR GipU GipC
1.2 1.4 2.0 1.5
0.6 1.5 1.4 1.2
1.5 1.3 1.0
1.4 1.0
0.8
H&D Eanes A B C S
2.0 1.5 2.1 2.4 0.7 1.1
1.9 1.2 2.1 2.3 0.7 1.1
1.7 1.1 1.9 1.8 0.9 1.1
0.9 1.6 2.9 2.6 1.4 1.6
1.6 1.4 3.4 2.7 1.8 2.1
1.3 1.2 2.2 2.1 1.1 1.2
Ocean Models
Eanes
A
B
C
S
2.0 1.9
1.5 1.2
2.1 2.1
2.4 2.3
0.7 0.7
1.1 1.1
1.7 0.9 1.6 1.3
1.1 1.6 1.4 1.2
1.9 2.9 3.4 2.2
1.8 2.6 2.7 2.1
0.9 1.4 1.8 1.1
1.1 1.6 2.1 1.2
1.8
3.3 2.8
2.8 2.4 2.8
1.9 1.2 1.6 2.0
2.1 1.8 1.1 1.9 0.8
1.8 3.3 2.8 1.9 2.1
2.8 2.4 1.2 1.8
2.8 1.6 1.1
2.0 1.9
0.8
aRMS Differencesin/•s.
andall threetechniques haveto dealwith residualdeficienciesin the modelingof solidEarth tidesandoceanloading thatmaypropagate intothesubdailyERPresults.The ocean tidemodels,finally,arebasedoncomplicated hydrodynamic modelswith difficulties,for example,to modeltidesin shal-
Threeof the four VLBI modelsstemfrom the analysis by Gipson[1996].The namesGipR,GipU, andGipC stand for the "restricted,""unconstrained,"and "constrained"so-
lutionsgivenby Gipson[1996].ThemodelGipRwithonly a limitedsetof tidaltermsbeingestimated is verysimilarto
low waters.
the smallGPS solutionandGipC is identicalwith the completeGPS solutionfrom the pointof view of thetermsestitween these different models. Tables 8a and 8b contain the matedandtheconstraints onthesideband terms,makingthe RMS differencebetweenany pair of modelscomputedac- interpretationof the differencesbetweenVLBI andGPS eascordingto (7). In thecomparison we includedourfourprin- ier. The fourthVLBI modelby Herring and Dong [1994], ciple GPS solutionslabeledsmall,"reduced,""complete," denotedH&D, is basedon a muchshortertimespanof data and "rates" as described in section 4.5, four VLBI models, andusesa differentanalysis technique. Detailsaregivenby one SLR model, and four ocean models. Older VLBI and Herring and Dong [1994]. SLR models,for example,by Soverset al. [1993] (VLBI) The SLR solutionwascomputed by R. Eanesin 1995(priand Watkinsand Eanes [1994] (SLR) were not considered vatecommunication, 1998). Insteadof estimatingindividual here (they showrelativelylargedeviationswith respectto tidaltermstheresponse functionof theEarthto thetidalpothe models included in Tables 8a and 8b). tentialwasdetermined with a limitednumberof parameters In Tables 8a and 8b we summarizethe comparisonsbe-
Table 8b. RMS Differencein Polar Motion AmplitudesBetweenTidal Models From Different Techniques a GPS Solutions
Model
Small
Small Reduced
2.2
Complete
VLBI
Reduced Complete Rate 2.2
7.6 3.8
SLR
GipR
GipU
GipC
H&D
9.1 5.3
14.3 14.5
17.0 13.1
14.0 12.0
24.2 23.1
3.4
14.6
13.3
10.4
13.3
12.5
9.4
10.2
5.6 8.9
7.6
3.8
Rate
9.1
5.3
3.4
GipR GipU GipC
14.3 17.0 14.0
14.5 13.1 12.0
14.6 13.3 10.4
13.3 12.5 9.4
10.2 5.6
H&D Eanes A B C S
24.2 13.1 13.6 11.5 11.8 17.0
23.1 11.8 13.8 10.7 11.5 16.7
21.5 11.7 11.3 13.1 10.2 14.8
19.8 12.0 12.0 11.0 9.1 11.8
17.0 11.7 11.8 11.8 7.5 12.6
aRMS Differencesin/•as.
Solutions
8.9 16.9 12.2 14.9 14.5 12.6 16.7
17.0 9.6 9.0 11.8 6.5 9.8
A
B
C
S
13.1 11.8
13.6 13.8
11.5 10.7
11.8 11.5
17.0 16.7
21.5
11.7
11.3
13.1
10.2
14.8
19.8
12.0
12.0
11.0
9.1
11.8
17.0 16.9 17.0
11.7 12.2 9.6
11.8 14.9 9.0
11.8 14.5 11.8
7.5 12.6 6.5
12.6 16.7 9.8
20.9
24.2 11.4
23.1 13.7 14.8
18.7 9.0 8.9 9.7
19.8 14.6 14.2 11.5 9.6
20.9 24.2 23.1 18.7 19.8
Eanes
Ocean Models
11.4 13.7 9.0 14.6
14.8 8.9 14.2
9.7 11.5
9.6
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS
[Munkand Cartwright, 1966]. This methodavoidsthe problem of the high correlationsbetweenmain tides and sidebandsthat has to be copedwith when individualterms are
13,731
calculated byCartwright etal. [1992].According toGipson [1996]theoceanmodelC isexpected tobethemostaccurate of the four models considered.
estimated. On the other hand, deviations of individual tides
Whenlookingat thevaluesin Table8aand8b,we should
from the responsefunctionrepresentation are not seenwith thisresponsefunctionapproach.
keepin mindthattheactualmodeldifferences (in termsof ERPvalues, notamplitudes) aremuchlargerthantheRMS differences of thetidalamplitudes (seesection 3.2,equation(11)).Thisisinparticular truewhencomparing models
The oceanmodelsA, B, and C are documentedby Chao et al. [1996] (which they also namedA, B, C). They were
generated usingthe sameTOPEX/Poseidon satellitealtime- whichincludemanyterms. try measurements but with differentmodelingapproaches. Thebestmodelsof eachof thefourtechniques (VLBI, In model A, corrections to the model of Schwiderski [1980]
St,R,GPS,andaltimetry) agreeto within,-,,10/•asin PM
were computedusing a harmonicfit to the altimetry data. and,-,,1 /•s in UT1. In thecaseof VLBI themodelGipC Model B wasdeterminedusingtheresponsefunctionmethod shows thebestagreement withtheothertechniques; in the mentioned
above.
The third model
C is based on tidal
caseof the oceanmodelsit is modelC, followed(at least
heightsandoceancurrentsfoundby minimizingthemisfitto both,the sealevel observations by TOPEX/Poseidonandthe classicalhydrodynamicalequationof Laplace.A modelfor shallowwater hydrodynamicswas includedin this analysis. Model S, finally, is a tide model developedby Schwiderski [1980] prior to the era of TOPEX/Poseidon,with currents
in UT1)bytheSchwiderski modelS. Whencomparing the GPSresults withtheothertechniques, themodelusingPM ratesshowsthesmallestdiscrepancies in almostall thecom-
parisons.Thisis in particular truewithrespect to thebest VLBI model (GipC) and the bestoceantide modelC. This
seems toindicate thatsystematic effects atlowfrequencies
200
(a)
(b)
ß *•
150
•N2
-I-
ß
+
100
8
-25
S2
*
Ol
50o
.(2_
E
ß-•
- 50
-5O K1 +
M2. ß
-150
- 75
-150
-100
-50
0
50
100
150
200
-25
0
Cosine [microarcsec]
[][]cGPS (complete) o(•ßGPS (rates) ---
+ + + GipsonC
25
5O
Cosine [microarcsec]
* * * Eanes
ß * * C (RAY96) 300
IJ•cGPS (complete) •aßGPS (rates) + + + GlpsonC
* * * Eanes
ß ' ' C (RAY96)
20-
(c)
(d)
M2
e•
25O
M2
&•
+ •*
.
* + S2
8 200-
*
N2
• 1501
.o_
S2
P1
._c lOO -10
N2
5o
Q
Ol ß
-150
-100
-50
0
50
100
150
Cosine [microarcsec] n c]c]GPS (complete) ß ß a)GPS (rates) + + + GipsonC * * * Eanes ß ' ' C (RAY96)
-2O
-20
-10
0
10
20
Cosine [microsec]
[][][]GPS (complete) +++Gipson C
* * * Eanes
' ' ' C (RAY96)
Figure 9.Phasor diagrams comparing themajor tidalamplitudes asestimated fromGPS("complete" and "rate" solution), VLBI,andSLRdata withtheocean model C.Mindthedifferent scale ofeach panel. (a) Prograde diurnal polar motion, (b)semidiurnal prograde polar motion, (c)semidiurnal retrograde polar motion,and (d) diurnaland semidiurnalUT1.
13,732
ROTHACHERET AL.: HIGH-FREQUENCY EARTHROTATIONFROMGPS
mightstill be presentin thePM offsetsaftersubtracting the diagrams of themajorocean tides,namely, K1, P1,O1,and a prioriERP seriesandmightbe reducedby usingtherates. Q• inthediurnalandK2, S2,M2, andN2 inthesemidiurnal It would be worth a specialstudyto see how each of the band(seefootnoteb of Tables7a-7c). two types of subdaily estimates,PM offsetsand rates, is As we alreadyknow from Tables8a and 8b the three correlatedwith theorbitradiationpressure parameters. The space-geodetic techniques aregenerally in goodagreement. modelderivedfrom SLR (R. Eanes,privatecommunication, In diurnalprogradePM the differencesare of the orderof 1998) is performingas well as the VLBI and GPS models. This is quiteremarkableconsidering the fact that the SLR dataset,althoughcoveringa verylongtimeinterval,is relativelysparsecompared to thatof GPS.Theresponse function method,whereonlya verylimitednumberof parameters has to beestimated,mightin partbe responsible for thissuccess.
10-20yas; in thesemidiurnal prograde bandtheGPSresultsdeviate in M2 andK2 byasmuchas20 yasfromthe otherresults;andfor SLR it is the$2 tidethatseemsmore
problematic. If thedifferences forthesemidiurnal retrograde amplitudes givenin Figure9cseemtobesmallthatisjust because of thelargescaleusedfor thisplot. All thetechIt is interesting to notethatthe GPS modelsagreeabout niques showdeviations of upto • 30 yasin theM2, K2, or
equallywell in bothcomponents, UT1 andPM (9-12 yas and0.7-0.9 ys, respectively)with the oceanmodelC (convertingUT1 valuesgivenin ys to yas). For the othertwo space-geodetic techniques, VLBI and SLR, the qualityof polarmotion(in yas) andUT1 (in ys) indicatesthatthe coefficients in PM seem to be better established than those in UT1.
N2 tidal amplitude.It is clearthatwith RMS differences of
theorder of 10yas(seeTable 8b),differences of20-30yas may be expectedfor a few terms. The differencesbetween
thecomplete andtheratesGPSsolution (10-20yas)maybe usedasa measure forthesizeof systematic effects present in theGPSsubdailyERP series. In UT1 we seean excellentagreement betweentheGPS
In orderto comparethe tidal termsobtainedby the dif- results andaltimetrymodelC. Thediscrepancy between the ferenttechniques in moredetailFigure9 showsthephasor M2 tidederivedfromVLBI andoceanmodelC should,ac-
3! (a)
31(b) 1N1
I
1i
1t
0• 1
-1] -21
e2Q1
1
-31 t -4
-3
-2
-1
0
1
2
3
-4
21' -3
-2
Cosine [microsec]
-1
0
1
2
3
Cosine [microsec]
50 •,
50*
(c)
(0)
25
25
21't2Q1 L1 21 -25
-25
- 5O - 50
-5O
- 25
0
25
50
-50
Cosine[rnicroarcsec]
- 25
0
25
50
Cosine[rnicroarcsec]
Figure10.Phasor diagrams (UT1andprograde diurnal PM)showing thesmaller tidalamplitudes andthe sidebands of themajortidaltermsfromGPS(complete solution) andfromVLBI (GipCsolution). The circles indicate the3o'uncertainties of theamplitude estimates (diurnal andsemidiurnal forUT1). (a) GPS,diurnal andsemidiurnal UT1,(b)VLBI,diurnal andsemidiurnal UT1,(c) GPS,prograde diurnal
polarmotion,and(d) VLBI, prograde diurnalpolarmotion.
ROTHACHERET AL.' HIGH-FREQUENCYEARTH ROTATIONFROM GPS (a)
13,733
(b) 10
5
5•
0
0
-5
-5
-10
-lO
-15
-15
-10
-5
0
5
10
15
-10
-15
Cosine [microarcsec]
-5
0
5
10
15
Cosine [microarcsec]
(c)
(d)
•2 2
N2
1
-lO
-lO
-15
-15•
-15
-10
-5
0
5
10
15
Cosine[microarcsec]
-15
-10
-5
0
5
10
Cosine[microarcsec]
Figure 11. Phasordiagrams(semidiurnalPM) showingthe smallertidal amplitudesand the sidebands of the major tidal termsfrom GPS (completesolution)and from VLBI (GipC solution). The circles indicatethe3oruncertainties of the amplitudeestimates.(a) GPS, semidiurnalprogradepolarmotion,(b) VLBI, semidiurnalprogradepolarmotion,(c) GPS, semidiurnalretrogradepolar motion,and (d) VLBI, semidiurnalretrogradepolar motion.
cording to Chaoet al. [1991],largelybeexplained by the tudes. More details about the formal errors of the GPS amM2 spinlibration.BothSLRandGPS,however, donotsup- plitudeswere given in section4.5. The circleswere drawn to portthisexplanation. TheGPSamplitude of M2 andK• seewhich tidal terms(apartfrom the major tidesdealt with areactually veryneartotheoceanmodelvalues. Weshould above)are significantlydifferentfrom zero. keepin mind,however, thateachof thethreespacetech- Figures10 and 11 revealseveralinterestingaspects.Most niques maystillsufferfromsignificant systematic errors. In of the tidal terms lie inside the 3a circles and are therethis contextit is interestingto seethat for the O• tide in UT1 theamplitudes of all threespacetechniques arein good agreement but differ from the oceantide modelby ,-• 3/•s, which suggests that the oceanmodelC is probablynot perfect either.
forenotreallysignificantlydifferentfrom zeroat thepresent level of precision.Judgingfrom the size and distributionof the small tides,the quality of the estimatesfrom both techniquesis comparable. Outside the 3a circles we find the sidebands of some of
Figures10 and11 showtheamplitudes andphases of the the majortides. Obviously,thesesidebandsgive significant smallertidesandthesidebands of majortidesobtainedfrom contributionsto the high-frequencyERP variations. The amplitudes fromVLBI andGPSarein verygood thecompleteGPS solutionon the left andtheresultsof the sideband VLBI GipC solution[Gipson,1996]on theright-hand side. agreement,whichis notreally amazingbecausein bothanalThe circlesin Figures10 and11 specifythe3a uncertainties yses exactly the same constraints(basedon the same tidal of thetidalamplitudes.For bothtechniques, GPSandVLBI, height values)were usedto tie the main tides and the sidethe sameerror circles(sameradii) were used,namely,those bands. Apart from the sidebandsthere are a few other tides with fromtheGPS solution,to facilitatethecomparison between thecorresponding panelsfromGPSandVLBI. In thecaseof amplitudesfar outsidethe en'orcircles. In UT1 the largest UT1 thetwocirclescorrespond to the3o'uncertainties of the discrepanciesoccur for GPS in the $•, N•, and • terms. diurnal(bigcircle)andthesemidiurnal (smallcircle)ampli- In PM the large GPS-derivedamplitudesof the $• and •
13,734
ROTHACHER ET AL.: HIGH-FREQUENCY EARTH ROTATION FROM GPS
tides in the diurnal and the R2 and T2 tides in the semidi-
forthequalityof thetidaltermscloseto periods of 12and
urnal band are particularlystriking. Accordingto the tidal 24 hours derived from GPS. heights(seeTables7a-7c) all thesetidesshouldhaverather This latteraspectis an important argument for the insmallamplitudes(with somecontributions to $• from the clusionof data from GLONASS satellitesinto a combined atmosphere).We easilyverify that thesetideshaveperiods GPS-GLONASSdataprocessing. Because of theloweralvery closeto 12 and 24 hours. This suggests that orbital titudeof theGLONASS satellites theirrevolution periodis biasesor correlationsbetweenthe subdailyERPs andthe or- about11hours16min,andtheyaretherefore outsidethe2:1 andtheymayhelpto distinguish betweenorbital bital parameters mightbe responsible for the unexpectedly resonance largeamplitudes.Becausetheacceleration of theGPSsatel- effects propagating intotheERPestimates andgeophysical in high-frequency Earthrotation.Moreover,with lites due to solarradiationpressureis varyingwith a period variations of 12 "solar" hours (one revolution of the satellite with re-
Etalon-1 and -2, there are two laser cannonball satellitesin
spectto the Sun),a "contamination" of theERP spectrumat GLONASS-typeorbitsavailable. periodsof 12 and24 hoursis well possible.The possibil4.7. Residual Spectrum and Nontidal
ity thatthelargeamplitudes arecaused by theatmosphereVariations is not veryprobablebecausethe corresponding amplitudes estimated from VLBI are much smaller. To a certain extent,
alsothemajortidesK• andS2 fromGPSmightbeaffected in thisway.Furtherstudies will beneeded to clarifythisaspect.GPSorbitbiases andthe2:1commensurability of the orbitperiodandthesidereal daymightbethelimitingfactor
in Earth
Orientation
Because the GPS series consists of continuous 2-hour val-
uessincethebeginning of 1995,it is easilypossible to generatedetailedspectraof theseries(like thoseshownin Figure 5) and to look for nontidalsignals.Figures12 and 13 show the residual spectraof the seriesfor the main tidal
141
21
22
23
24
25 Penod
26
27
28
29
11 4
115
11.6
11 7
11 8
119
12 0
in Hours
12 1
12 2
Period
12 3
12 4
12 5
12.6
12 7
12 8
12 9
13 0
in Hours
t(d)
16 (c) 14
13
11
,_ 10]
5
- 29
- 28
- 27
- 26
- 25 Period
- 24
- 23
- 22
- 21
-13.0
-12.8
-12
6
-12.4
-12 Pe•
