JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. C4, PAGES 8113-8127, APRIL 15, 1998
Precise orbit determination and gravity field improvement for the ERS satellites Remko
Scharroo and Pieter Visser
Deffi Institutefor Earth-OrientedSpaceResearch,Delf• Universityof Technology,Delft, Netherlands
Abstract. Theradialorbiterrorhaslongbeenthemajorerrorsourcein ERS-1altimetry, crippledby havingonlysatellitelaserrangingforprecise trackingandrelyingoninsufficiently accurate general-purpose gravityfieldmodels.Altimetercrossovers areusedveryeffectivelyas additional trackingdatato laserranging.TheERSTandemMissionevenprovides theunique possibility to simultaneously determineorbitsof two similarsatellitesflyingthesameorbit. Altimeter crossovers between the two satellites then link the two orbits into a common reference
frame.Tailoringof theJointGravityModel3 (JGM3) is anotherstepto reduceorbiterrors.This technique is aimedat thereductionof thegeographically anticorrelated orbiterror(observed in the crossover heightdifferences) throughtheadjustment of selected gravityfieldparameters. The resultingDelft GravityModel (DGM)-E04 hasreducedthispartof theorbiterrorby a factorof 2, performs evenbetterwithrespect to theESA-provided orbits,andalsooutperforms therecent EarthGeopotential Model EGM96 in thisrespect.ERS-1 andERS-2 orbitsfor the entireTandem Missionarecomputed andstudiedin detail,andorbiterrorsdueto thegravityfieldand nonconservative forcesareidentified.Analyses systematically showthattheorbitscomputed with JGM 3 havea radialroot-mean-square orbitaccuracyof 7 cm, with DGM-E04 5 cm.
1.
Introduction
budgetrequiredby the oceanographic community[Tapley,1992] with its appliThe Tandem Mission of the two EuropeanRemote Sensing just couldnotbe met. Recently,SAR interferometry, cations of mapping land surface elevation and elevation changes,is Satellites ERS-1 and ERS-2 started soon after launch of ERS-2, now also demanding ever increasing accuracy of the orbit compuApril 21, 1995. Sincethen, both satellitesare flying in the same near-circularorbit with an averagealtitudeof 800 km and an in-
clinationof 98.5ø. The orbitsareSunsynchronous andarephased suchthat the satellitescrossthe equatorat 2230 local solar time (LST) in northern direction and 1030 LST in southerndirection.
Generallyspeaking,the satellitesobservethe Earth at nightfallon ascending tracksandat dawnon descending tracks.After making 501 revolutionsin exactly35 daysthe satellitesrepeattheir ground tracks.Until today,eventhoughERS-1 is no longerprovidingdata, ERS-2 is followingit at a 32-minlag in the sameorbitplane,which causesERS-1 to precedeERS-2 by 24 hoursalongthe sameground track.
Satellitetrackingis the only meansto tie the (at itself relative) radar altimeterrangemeasurement into a well-definedglobalreference frame. Precise orbit determination (POD), on the basis of
thesetrackingdata and variousdynamicalmodels,ensurescontinuity of this link also at locationswhere there is no immediate tracking. The accuracyat which the absolutesea level, land, or ice elevationis inferredby differencingthe orbital altitudeandthe altimeterrangemeasurementis alwayslimited by the accuracyof the orbit computation.The radial orbit error has been one of the largererrorsin recoveringthe seasurfaceheightfrom ERS altimeter measurements and,becauseof their long-wavelength character, disqualifiedERS altimeterdata from applicationin global ocean circulationstudies,oceantide modeling,and monitoringof seasonalandsecular(climate-related)change.The 10-cmoverallerror
tation in cross-track direction.
The bulky satellitesERS-1 and ERS-2 were never designed for high-accuracy orbit determination, and the lossof the Precise RangeandRange-RateEquipment(PRARE) trackingsystemleft ERS-1 even more poorly equippedfor orbit determination. Yet, subdecimetric orbit accuracyis not of academicinterestonly. The ERS altimetricsystemhasperformedwell aboveexpectationsand is uniquebecauseof its multidisciplinarycharacter,samplingnot only oceanbut alsoland andice surfaces,in combinationwith the suiteof instrumentson board, providing,e.g., simultaneousmeasurementsof wet troposphericcontentand surfacetemperature. Undoubtedly,ERS will alwayslag behindon the 2-cm root-meansquareorbitaccuracyof theTOPEX/POSEIDON altimetermission [Marshall et al., 1995], so only when the preciseorbit determination is stretchedto its very limits, ERS altimetrywill be regarded a reliable sourceof information. Only then will ERS be able to demonstrateits additivevaluein oceanresearchand its uniquecapabilitiesin, e.g., monitoringof the ice sheetmassbalance.
Copyright1998by the AmericanGeophysicalUnion. Papernumber97JC03179. 0148-0227/98/97 JC-03179509.00 8113
In section 2 we will discuss the numerous advances made in ERS
operationalandpreciseorbit determinationoverthe yearsup to the currentstate-of-the-artmodeling. An importantstepin this is the development of the ERS-tailoredDelft Gravity Model (DGM)-E04 (section3). In section4, DGM-E04 demonstratesthat it constitutes a remarkableimprovementon the ERS orbitsbut also has a more
generalapplicability.Section5 examinesthe effect of nonconservativeforcesactingon the satellite. Section6 combinesall results andattemptsto give a realisticerrorbudgetfor the radial accuracy of the variousorbit solutionsdiscussedin this paper and demonstratesthat,with the selectionof the right orbit solution,the overall errorbudgetfor the ERS altimetricsystemis well within the origi-
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SCHARROO AND VISSER: PRECISE ORBIT DETERMINATION
nal goalssetfor TOPEX/POSEIDON. Finally, we will discussthe achievements
and limitations
in section 7.
2. Operational and PreciseOrbit Determination
FOR ERS
JGM I and JGM 2 (NASA/University of Texasat Austin) [Nerem et aL, 1994], and in the GeoForschungsZentrum (GFZ) PGM035 gravitymodelusedto producetheofficialESA orbitsfrom the German ProcessingandArchivingFacility (D-PAF). The introduction of these models in the ERS-1 POD reduced the orbit error to a level
of 15-20 cm [Aksneset al., 1994; Massmann et al., 1994; Schar-
In December1991 the Delft Universityof Technology(DUT) producedthe first operationallydeterminedorbit for ERS-1 and has been doing so ever since at the regular pace of two orbital arcsperweek. With theseorbitsandadditionalgeophysicalmodels the UnitedStates'NationalOceanicandAtmosphericAdministration (NOAA) has been upgradingthe EuropeanSpaceAgency's (ESA) ERS fast-deliveryaltimeterproductto interim geophysical data records(IGDRs). Major advanceshave been made during ERS' lifetime in the accuraterestitutionof the orbit. Starting at radial orbit errorsof around 140 cm in 1991, the best available orbits are now believed
roo et al., 1993b, 1994], at which the surfaceforce modelingerrors(dragandsolarradiation)startedto dominate[Le Traonet al., 1995]. A new modelfor the satellitegeometrywas developedand successfullyimplemented,but incorporatingaltimeterrange measurementsas an additionaltrackingtype scoredlittle successin reducingtheremainingorbiterrors[Scharrooet al., 1993a,b; Visser, 1993; Shumet al., 1994]. Nondynamicorbit improvementusing TOPEX/POSEIDON
as a reference, with all its associated limi-
tations, seemedthe only way to provide subdecimetricorbits for ERS-I [Smith and lhsser, 1995; Le Traon et al., 1995; Le Traon and Ogor, this issue].
to be accurateup to about5 cm. This could only be obtainedby 2.2. Second-Generation Precise orbits adoptingimprovedmodelsfor the gravityfield,the satellitesurface Simultaneous with the launch of ERS-2 and the start of the forces,and addingaltimetercrossoverdatato the POD as long as the PRARE trackingdata were not generallyavailable. ERS TandemMission(May 1995),DUT introducedJGM 3 [Tapley et al., 1996] in the operationalorbit determination.This brought the radial orbit error down to • 10 cm, but satellitelaser ranging 2.1. History (SLR) to ERS-1 and ERS-2 simply remainedinsufficientlyabunAt the time of the launchof ERS-1 the NASA GoddardSpace dant and lacked a regularglobal distributionrequiredto provide FlightCenter(GSFC) GEM-T2 gravitymodel[Marshet al., 1989] subdecimetricaccuracyeverywhereon the globe. With mostof the was the most commonlyused model. Becausethis model lacked trackingstationslocatedin Europeand North America (Figure 1), sufficienthigh-inclination orbitinformationin its development, the orbit accuracyremainedpoor, especiallyat southernlatitudes. InresultingERS-1 orbitsstartedat about140-cmradialaccuracy.Afclusionof additionaltrackingdata was imperativeand foundin the terGSFCindluded trackingdataof SPOT-2,whichrunsin anorbit form of altimetercrossoverheightdifferences,nailing downthe orvery similar to ERS, orbit errorsreduceddrasticallyto •30 cm bits everywhereover the oceansand paving the way to introduce [Scharrooet al., 1993a]. ERS-1 trackingdatawere firstincludedin additionalparametersin the POD to absorbremainingunmodeled thefirstJointGravityModelsfor theTOPEX/POSEIDON mission, or inadequatelymodeledforces.
Figure 1. Satellite laser rangingcoveragefor ERS-1 and ERS-2 during the TandemMission Locationsof SLR stationsthat have trackedERS-1 or ERS-2 during the TandemMission are indicatedby open trianglesand site numbers.
$CHARROO AND VI$$ER: PRECISE ORBIT DETERMINATION
FOR ER$
8115
In spring1996, DUT produceda full set of second-generation Table1. Coordinates of SomeERS Reference Pointsin a Body preciseERS-1 andERS-2 orbitsfor theperiodApril 1992until Au- Fixed Frame gust 1995, on the basisof the JGM 3 gravitymodeland SLR and Xs Ys Zs altimetercrossover(XO) trackingdata. Of these,the ERS-1 orbits ReferencePoint for missionphasesC, D, E, and F (multidisciplinaryphase,secAltimeter -3786.4 570.0 -840.4 ond ice phase,andthe two geodeticphases)were madeavailableat Laser retroreflector -2850.4 -700.0 -995.0 http://deos.lr. tudelft.nl/ers/precorbs/. The radial orbit accuracyof ERS-I nominal CM -1827.0 11.8 11.9 -1853.0 -9.0 -3.0 about7 cm [Scharrooet al., 1996a,b] wasa significantimprove- ERS-2 nominal CM ment over the 10-cm accuracyof the GFZ/D-PAF orbits featured Coordinates are in millimeters. CM is Center of Mass. on the officialESA altimeterproducts.This wasmainly due to the superiorityof JGM 3 overtheGFZ PGM035 gravitymodelusedin 2.5. Tracking Data thosedaysto computetheseorbits. SLR 15-s normal pointsare collectedfrom the EurolasData
2.3.
Third-Generation
Precise Orbits
Center(EDC) and CrustalDynamicsData InformationSystem (CDDIS). All rangesare correctedfor a distanceof 4.3 cm be-
A naturalstepwas to developa gravitymodel,tailor-madefor ERS orbit determination,startingfrom JGM 3, and so to reduce tween the effectivesphereof reflectionand the laserretroreflector (rss)of a the gravity-inducedorbit error. The resultingDelft Gravity Model referencepoint. The dataweightis the root-sum-square system-dependent noise value (ranging from 1 to 20 cm) and an DGM-E04, discussedin section3 now forms the basisfor the thirdestimate of the overall solution error (5 cm). generationpreciseERS-1 andERS-2 orbits,currentlyall accessible
at http://deos.lr. tudelft.nl/ers/precorbs/. In thispaper(section4 and
All ERS altimeter data are retrieved from the ESA altimeter
further) we will limit ourselves to the POD for the ERS Tandem
oceanproducts(OPR). Unfortunately,for the actualorbit determi-
Mission,exploitingtheuniquesituationof havingthetwo altimeter missionsflying the sameorbit. During this period,ERS orbitsare computedsimultaneously, usingERS-1/2 dual-satellitecrossovers to link the orbitstightly in a commonreferenceframeandto have oneorbitbenefitfromtheotherwhenSLR trackingis sparse.At the sametime, altimeterheightdifferencesalongcollineartracks(section 5.2) and ERS/TOPEX dual-satellitecrossovers(section4.4) remainasindependentindicatorsfor the radialorbiterror.This situationis significantlydifferentfrom the ERS orbit determination
nationwe hadto harmonize severaldifferentversions of thisproduct. Version30PRs were availablepriorto the TandemMission [CentreERS d'Archivageet de Traitement(CERSAT),1994], followed by a numberof intermediateversions.The verificationof the orbits of the Tandem Mission, however,is all basedon the current Version6 data [CERSAT, 1996].
All 1-Hz altimetricseaheightsare screenedandcorrectedfor the followinggeophysical andinstrument corrections: (1) range corrections for biasjumps andoscillatordrift (ESA, publicdata,
bitsare merelyusedas a referenceandare not simultaneously adjusted.
1996); (2) GFZ/D-PAF preciseorbitsbasedon the GFZ PGM055 gravitymodel[Gruberet al., 1997]; (3) meteorological dry tropospheric(ECMWF) and ionosphericcorrections (Bent); (4) wet
2.4. Applied Models and Constants
tropospheric correctionfrom the ERS MicrowaveRadiometer;(5) solid Earth and pole tides; (6) oceantides (GrenobleFES95.2.1)
basedon ERS/TOPEX dual-satellite crossovers,where TOPEX or-
The proceduresand modelsusedfor the POD are basedon the most up-to-dateknowledgeof gravity and nonconservative force modeling. In this study,three differentgravity models,JGM 3, EGM96 (NASA/NationalImageryandMappingAgency,truncated to degreeand order 70) [Lernoineet al., 1997], and DGM-E04 (DUT), havebeen usedto describethe gravitationalfield of the Earth (includingtides),and resultsare intercompared.The satellite surfaceforcesare accuratelymodeledby meansof satellitespecificmacromodels consistingof 10 panels.Additionalunmodeled forcesare parameterizedthroughso-calledempiricalaccelerations.Becauseof the inclusionof SLR and single-and dual-
andtidalloading;(7) sea-state bias;OPR version3, 5.5% of SignificantWaveHeight(SWH); [GasparandOgor,1994];OPR version6, BM3 model [Gasparand Ogor, 1996]; (8) 100% inverse barometercorrection;and (9) mean sea surface(Ohio State Uni-
versity(OSU) MSS95). Forthwith,the 1-Hz relativeseaheights areconvenedto altimetercrossover heightdifferences(XDs).
First, the locationof the crossingpointof all ascending and descending passesis computed.Combinations of passes with extremeshallowangles,passeswith a time intervallargerthan 17.5 days,andXOs with too few surrounding 1-Hz measurements are rejected.(Theoretically,17.5 daysis the largestpossibletime intervalbetweenconsecutive ascending anddescending passes in a satellite crossovers, sufficient data are available to estimate these XO is half therepeatperiod.)Relativeseaheightsarefilteredand additionalparameters. interpolatedwith a quadraticpolynomialat thelocationof theXO. Other estimatedparametersare the stationlocation for some The two relativeseaheightsin the XOs are convenedback to (mainly mobile) SLR stationsand time tag and rangebiasesfor rangemeasurements by subtractingthem from the orbital altitude. stationsthat are notoriousfor producingSLR measurements with The measurement weightvarieswith thegeographical positionand significantoffsets.Sincethedatationof the altimeterdatamay not is basedon a combination of a posterioriorbiterrorandseasurface be the sameasthe SLR standard(UTC), it is importantto estimate variability determined fromanearlierorbitsolution (valuesranging a time tag biason the altimeterdata. If not, the altimeterranges from 4 to 40 cm). Becausewith XDs we are only concerned with appearcorruptedby a signalwith a frequencyof two cyclesper a heightdifference,theusageandchoiceof thereferencemeansea revolution(cpr) introducedby shiftingthe altimeterrangesforward surfaceis irrelevantbutfacilitatesthedatascreening. or backwardon a flattenedEarth; a signalthat may well aliasinto the orbit [Schutzet al., 1982]. Especiallyrelevantto the orbit determinationare the locationof the satellite center of mass, and the laser retroreflector and altimeter
Model
referencepoints,listed in Table 1. A concisedescriptionof the modelingis givenin Table 2.
The perceptionthat much of the radial orbit error in secondgenerationorbits(section2.2) was still causedby deficienciesof
3. Developmentof an ERS-TailoredGravity
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Table 2.
SCHARROO AND VISSER: PRECISE ORBIT DETERMINATION FOR ERS
Summaryof the DynamicalandMeasurementModels Used for the ERS Third-GenerationPreciseOrbit Determination.
Item
Description Measurement Models and Constants
SLR observations
globalquick-lookSLR data(1 per 15 s normalpoints)retrievedfromEDC andCDDIS; correctedfor 4.3-cmdistance to LRR referencepoint; 10ø elevationcutoffandeditingof spuriousmeasurements; weightis rsscombination of overallmodelaccuracy(5 cm) andsystemnoiselevel(1-20 cm)
Station coordinates
LSC(DUT)95L02 LAGEOSI/II solution (September 1983- December 1993,epochJanuary 1, 1988),advanced to
Tidal displacement SLR geometriccorrection
epochby three-dimensional motionsincorporated in the coordinate solution Love model,includingfrequencydependent andpermanent tides(h2 = 0.609, 12= 0.0852);poletide offsetof LRR opticalcenterwith respectto LRR referencepoint(4.3 cm) andto thespacecraft nominalcenterof mass (see Table 1)
Crossover observations
globalERS radaraltimeterdata,retrievedfrom the ESA ERS OPR products, plusadditionalgeophysical corrections (seesection2.5); convertedto single-and dual-satelliteXOs; weightis basedon localrms XDs comingfrom an
Speedof light
c = 299792.458 km s- 1
Gravity model
Delft GravityModelDGM-E04, completeto degreeandorder70, includingsecularC21 and$2• anddynamicpolar
Tidal gravity Third bodyattraction Atmosphericdrag
Wahr solidEarthtides;backgroundoceantides:JGM 3* Sun,Moon, Mercury,Venus,Mars, Jupiter,Saturn,Neptune,accordingto JPL DE200 ephemeris
Radiation Orbit maneuvers
solarradiationincludingumbra,penumbra,andoccultationby Moon;Earthalbedo a prioriinformationaccordingto ESOC predictions;adjustedduringPOD
Mass
ERS-1, 2377.13 kg; ERS-2, 2502.00 kg satellite-specific macromodels, eachconsistingof eightfixedandtwo rotatingpanels(seesection5.1)
earlier solution
Force Model
motion; GM = 398600.4415 kma s- 2, ae= 6378.1363 km,1/ f = 298.2564 FrenchDensityTemperature Model(DTM) [Barliefet al., 1977]withdailyF•o.? and3-hourlyKp values
Satellite Model Cross-sections
ReferenceFrame Polar motion
Coordinatesystem
Earthorientationandlengthof day from IERS EOP 90 C 04 solution J2000;precisionIAU 1976(Lieskemodel);nutationIAU 1980(Wahrmodel) Estimated Parameters Per Orbital Arc
State vector
Nonconservative forces
positionandvelocityat epochover5.5-dayorbitalarcat 3.5-dayintervals 6-hourly drag coefficients;22-hourly 1-cpr along-trackand cross-trackaccelerations;orbit maneuvers(threedimensionalaccelerations)
Measurement
offsets
coordinates of some(mobile)stations;rangeandtimingbiasfor someSLR systems; timingbiasfor bothaltimeters; relativerangebiasbetweenERS altimeters
* In fact,this modelwasnot adjustedsinceJGM 1 [Neremet al., 1994; Tapleyet al., 1996].
JGM 3, led to the attemptto developa modelspecificallytailored to ERS orbit determination.This is doneby first isolatingthe gravity-induced orbit errorobservedin XOs andthenadjustinga well-chosenset of gravity field coefficientssuchthat the error is
variableaboutthe component 6r 8, whichis just as time invariant asthe"mean"component 6r c, withtheonlydifferencethat6r * has an oppositesignon the two passes.Thuswe write for the gravityinducedradialorbit errorin a crossover point
minimized.
6r = 6r • 4- 6r • 3.1. Linear Perturbation Theory
(1)
wherethe plus sign is usedfor the ascendingpassand the minus
pass. Gravity-induced orbit errorsare, becauseof theirorigin,geo- signfor the descending graphicallycorrelated;that is, they are repetitivealongthe same groundtrack,repeatcycleafterrepeatcycle.This meansthatalong two suchcollineargroundtracksgravity-inducedorbit errorsare identical and cancel when differencingthe altimetfic sea surface profilesalong thesetracks. This is not the casefor two crossing tracks;the different"history"of gravitysensedalongthe ascending and descendingpassescausesthe orbit error to be essentially differentalongeachpass. Figure 2 showshow thesetwo gravity-inducedradial orbit er-
rors(Jrascand6rdes)canalternatively bedecomposed intoa geographicallyfully correlated andanticorrelated component (Jr c and 6r8). The firstcomponent is identicalon bothpasses; the secondis of equalmagnitudeon both passesbut of oppositesign. In literature [e.g.,Tapleyand Rosborough, 1985;Rosborough, 1986] these
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6r •c = 6r • + 6r •
6r• = •1((jwasc + ) •Sr•= • (•Sr• - •Sr )
components are oftengiventhe confusingindications"meanerror" Figure2. Geographically correlated orbiterrorsalongtwocrossand "variabilityerror." It shouldbe stressedthat there is nothing ing passes.
SCHARROO AND VISSER: PRECISE ORBIT DETERMINATION FOR ERS
Linear PerturbationTheory (LPT) [Kaula, 1966] describesthe three orthogonalcomponentsof the orbit error as a linear combination of terms due to (commissionor omission) errors in the gravitymodelcoefficients.Eachcoefficientproduceserrorsat variousfrequencies, dependingon their harmonicdegree1 andorder rn. It can be easily shown[e.g., Schrama,1989] that on the nearcircular,Sun-synchronous, and "frozen" ERS orbit, the dominant errorshavefrequencies
•lmp= (l - 2p)cpr- rncpd
(2)
wherep ranges from0 to I andcp.dis cycles perday.Somecom-
8117
or more, or are not observed at the maximum time interval of 17.5
daysin our setof XDs. The procedureof generatinglocal averageXDs can easily be performedby griddingthedataon a regularlatitude-longitude grid. However,thiswill removesomesignalat high latitudeswhere,becauseof the closureof XO locations,the spatialscalesare much shorterthan aroundthe equator. A more sophisticated approachtakesadvantageof the fact that XOs are alreadylocatedon a more or lessregular "grid" which densitiestoward the poles. XOs can be stackedtogetherat the "grid points." Followinga 3.5-crediting in each stack,the average of the XDs in eachgrid point (• and or:) representstwice the
binationsof I - 2p andrn cause•bt,,pto be closeto theresonance geographically anticorrelated orbiterror(26rS), while the variance frequencies0 and 1 cpr. For ERS' 35-day repeatthis happensfor resembles v• timestherssof thenongravitational orbiterror,sea coefficientsof order43 because43 cpdis very closeto 3 cpr. surfacevariability, and altimeter correctionerrors. An additional Rosborough[ 1986] focuseson the spatialdistributionof the radial orbit error and convenientlyformulates
•r = • 1--1
•r?•+ •r•
(3)
rn---0
with
= •t• (A•t,,•cosra• + A•t,,•sinrn•) = -
(4)
3.5-•r editingis performedover all •rx valuesto root out areasof extremevariability. We ignorethe fact that the XO locationsactuallyvary from repeat cycle to repeatcycle within a radiusof about 1 km because of drifting of the groundtrack aroundthe nominalpositionby the
sameamount.This is allowedsincethe signalwe areisolatinghas muchlongerwavelengths than1 km. Also, we arenot hamperedby cross-track geoidslopessincewe haveconstructed the XDs at their true locations and not nominal locations, and if we had, this would
havebeentackledby takingthe OSU MSS95 modelasreference. This techniquewas usedfor a total of about 1,000,000 ERS-1 where •t• and•t•,• areboth functions oflatitude onlyandfurther and ERS-2 single-mission XOs for the periodof April 1992 undependon the orbit meansemimajoraxis,eccentricity,andinclinatil August1995, createdfrom the OPR altimeterproductsas detion. scribedin section2.5. All XDs were first adjustedby replacing This yieldsthe following:(1) whencomputingXDs, •e,the fully the GFZ orbits by DUT second-generation JGM 3 orbits, simulcorrelatedpart cancels,but the anticorrelatedpart is observedat taneouslyaccountingfor an estimatedtime tag bias of-1.3 and doubleefficiencysince -1.1 ms for ERS-1 and ERS-2, respectively.The averageand the varianceof the XDs at the remaining45,000 locationsare sketched (6) in Plate 1. The varianceclearly resembleswhat we know of sea surfacevariability(high in WesternBoundaryCurrents)and areas (2) the zonal coefficientsdo not contributeto XDs sincesin rn3, = with largeoceantidemodelingerrors(e.g.,EastChinaSea,Indone0 and$tm = 0 for rn = 0; (3) nonzonalcoefficients of thegravity sian Archipelago). Clearly noticeableare a few latitudebandswith field do contributeto XDs, eachwith a distinctglobalpattern;and remarkablelow local variance;thesepertainto thoseXOs with a (4) viceversa,whenwe canisolateandobservetheeffectof gravity time lag of about12 hours,duringwhich seasurfacevariabilityis model deficiencies in XDs, the nonzonal coefficients can be tuned minimal. Note also the significanceof the averagescomparedto to reduce the observed effect. the variances: in areas with little or no variance (central Pacific)
The LPT has been usedbefore to cancelgravity-inducedrathe averagemay be a coupleof timeslargerthanthe variance.This dial orbiterrorsbutneverin sucha way thatit actuallyprovideda indicatesthat the isolationof the gravity-inducedcontributionto well-tunedgravitymodel. Engelis[1987, 1988] and Ksser [1992] XD functionsquite well andthat the gravity-induced orbit error is useLPT to simultaneously improvedynamictopography andgeoid sizeablecomparedto nongravitational errors. fromSeasatandGeosataltimeterdata.The link betweengeoidand orbiterrorsprovidethe meansto partiallyseparategeoidanddynamictopography.Ksser[1995] extendsthe techniqueby includ- 3.3. Tailoring the Gravity Model ing SLR andsingle-anddual-satelliteXDs. Becausethe gravityTailoringa gravitymodelinvolvesthe tuningof its coefficients induced orbit error was not isolated from other orbit errors, each suchthat residualsof observationsfrom a single satelliteare reof the additionalestimatedparameters in orbit determinationalso duced. Usually, only a subsetof coefficientsis adjustedbecause had to be readjusted.Novel in the approachproposedhereis the othersmay not affectthe residuals;like, in our case,we cannotobuseof crossover heightdifferences insteadof altimetricseaheights serve errors in the zonal coefficients in the XDs. We limit ourselves (thuscancelinggeoidanddynamictopography errors)andthe use to thosecombinations of degreeandorder(l, rn) that,according of the two-stepapproach:firstisolatingthe gravity-induced signal to LPT, producea global root-mean-square (rms) XD of 2 mm or andthenadjustingthe gravityfield coefficients. more,assuming a 1crerrorin eitherJGM 3 coefficientCtm or Stm. 3.2. Isolating the Gravity-Induced Orbit Errors The geographicallyanticorrelatedorbit error is but one of the many contributions to XDs. The assumptionwe make is that all contributions,exceptthe constantgravitational,are time variant and averageout to zero over a sufficientlylong period,like a year
This leadsto a setof 550 pairsof Ct• andSt• coefficientsto be adjustedwhile othergravitycoefficienterrorsaredeemedto be too poorlyobservedin the XDs for any adjustmentto be realistic. Solvingthe gravitycoefficientadjustments ACt,,., and /•Slrn from (5) and (6) leadsto a setof linear equationswith 45,000 observations and 1100 unknowns.
Because the number of observa-
tions far exceedsthe number of unknowns, we solve the unknowns
8118
SCHARROOAND VISSER:PRECISEORBIT DETERMINATIONFOR ERS
(b) Variances:Rms = 11.31 cm
(a) Averages:Rms= 5.97 cm
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Plate1. (a) Averages and(b) variances of stacks of ERS-!andERS-2single-mission altimeter crossover height differences fortheperiodof April1992till August1995.GravitymodelJGM3. Apparent altimeter timetagbiasis applied. in a Bayesianlinearleastsquaresfashion
the solutionis to be constrained. The optimalvaluefor f (= 10) was foundexperimentally,weighingbetweenan almosttotal reduction (7) (A:VWA + f N)c = A:VWx of the geographicallyanticorrelatedorbit error but unrealistically largegravityfield adjustments (smallf) andsmalladjustments but wherec is the vectorof unknowns(ACi•,ASi•); x is the insignificantreductionof the orbit error(largef). vectorof averageXDs (• • ha:); A is the matrixof partials Plate2 showsthe adjustments to the 1100 C1,.•andS1,.•coeffim
m
(2•1• sintaX,-2•1• costaX)'W isadiagonal matrix ofobser- cientsof JGM 3 dividedby theirformalstandard deviations (A/a). vation weights, givenbyn/(a:) 2, where n isthenumber andor: The rmsof all valuesA/a is 0.52, whichmeansthatthe newsois the varianceof XDs in eachstack;andfN is theJGM 3 normal matrixN (= inverseof theerrorvariance-covariance matrix,kindly providedby JohnRies,CSR) multipliedby a weightingfactorf. Becausewe only haveobservations overoceansandonly up to a certain latitude, we need to constrain the solution elsewhere. This
is done by adding the relevantpart of the JGM 3 normal matrix
to thenormalequations. The factorf furtherdetermines howtight
lution falls well within the error budgetof JGM 3. Largestadjustments are to some coefficients around orders 16, 33, and 41 but are
still within acceptablelimits. Thus we have generatedthe ERStailoredDelft Gravity Model DGM-E04. With (5) and(6) it is quiteeasyto determinethe globaldistribution of the geographically anticorrelated orbiterrorimpliedby the coefficientadjustments, asshownin Plate3a. Note thattheimplied
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70
1.5
C-coefficients
S-coefficients
a/o
A/or
0.5
...-
4O
:
o
I
ß I ,,
....
: m
,• ..... •o..... •'"
'"
I1'
-0.5
11. mmlmmm i ' '""'
m ' "•'""
•
I
-1.5
•
'
•
....
""m .,;ß.ram •o m
.....
2o
•o
m
m
"
•o
degree
•o
5o
•
degree
Min = -2.60, Max = 3.70, Mean = 0.06, Rms = 0.52
Plate2. Estimated gravitycoefficientadjustments to JGM 3 to obtainDGM-E04. All adjustments (A) arenormalizedthroughdivisionby theformalerror(a) of theoriginalJGM 3 coefficients.
•o
SCHARROO AND VISSER: PRECISE ORBIT DETERMINATION FOR ERS
(b) Residual: Rms -- 3.13 cm
(a) Predicted: Rms = 5.14 cm
•.
r•
I 1;0
1 ,
1 ,
:',
-,
,
,
.•
-15
•
-10
8119
1 O
'
,'
1
1 O
-5
5
10
15
cm
Plate 3. (a) Geographically anticorrelated orbiterrorimpliedby thedifferences betweentheJGM 3 andDGM-E04 gravitymodels.Valuesaremultipliedby 2 to resemblethe averageXDs in Plate la. (b) ResidualaverageXDs when subtractingPlate3a from the averageXDs in Plate I a. errorsoverthe continentsare not significantlydifferentfrom ocean areas,which again indicatesthat the constraintwas successfully applied.From this it is an easystepto projecthow the remaining averageXDs would look when replacingJGM 3 for DGM-E04 in the POD (Plate 3b). It reducesthe globalrms of the averageXDs from 5.98 to 3.13 cm, which impliesthat the geographicallyanti-
of a global oceandynamictopographyfrom TOPEX/POSEIDON seasurfaceheights.The resultingtopographydiffersfrom the numericaloceanmodelPOCM-4B by 12.55 cm rms. The sameanalysis with JGM 3 gives a similar rms residualof 12.62 cm. This suggests thatfor this application,DGM-E04 indeedactswithin the error marginsof JGM 3.
correlated radial orbit error reduces from about 3.0 to 1.6 cm.
4. 3.4.
Independent Assessmentof DGM-E04
Since the French Earth observation satellite SPOT-2 is in a simi-
Gravity-Induced Orbit Errors
The third-generationorbits (section2.3) cover the entire ERS TandemMission,startingon April 29, 1995, with the switchon of
lar orbitasERS-1 andERS-2, orbitdeterminationoughtto alsoimprovefor thissatellitewhenadoptingDGM-E04 insteadof JGM 3. Table 3. Statisticsof TOPEX Orbit Computation The only meansof verificationis, in this case,the data fits of the Rms Difference Doppler orbitographyand radiopositioning integratedby satellite Data Fits With GPS Arc, cm (DORIS) trackingdata. Becausetheserange-ratemeasurements aremostlysensitiveto along-trackorbiterrors,it givesus somefeel Gravity SLR, DORIS, Radial Cross Along of the orbit error in this direction as well, in contrast to the vertical
mm s
--1
Model
cm
JGM 2 JGM 3 DGM-E04
4.58 4.60* 4.63
0.580 0.577* 0.577*
JGM 2 JGM 3 DGM-E04
4.42 4.24* 4.29
0.551 0.546* 0.546*
JGM 2 JGM 3 DGM-E04
3.05 2.86* 2.89
0.539 0.537* 0.537*
JGM 2 JGM 3 DGM-E04
3.00 2.73* 2.74
0.565 0.563* 0.563*
sense of ERS altimeter crossovers. The observed reduction of rms
of theDORISdatafitsfrom0.70to0.66mms- • isencouraging. Frank Lemoine(GSFC, personalcommunication,1997) tested the suitabilityof DGM-E04 for TOPEX orbit determination.Having a completelydifferentorbit from ERS, its sensibilityto the gravityfield is likewisedissimilar.This meansthat adjustments to somecoefficientsthat had little effect on the ERS orbit might producesignificanterrorsin the TOPEX orbit. Table3 showsthe resultsof computinga randomset of four TOPEX orbitsbased on SLR and DORIS trackingdata and eachusingthreedifferent gravitymodels:JGM 2, JGM 3, andDGM-E04. The datafits appear leastsensitiveto the choiceof gravity model. Comparisons of the SLR/DORIS orbitswith Global Positioning(GPS) reduceddynamicorbitsproducedby the Jet PropulsionLaboratory(JPL), however,showa clear preferencefor the DGM-E04 model. Becausethe GPS reduced-dynamic orbitsare virtuallyunaffectedby gravity model errors,this test would indicatethat the one that comesclosesthas the leastgravity-inducedorbit error. In 11 out of the 12 cases it is the DGM-E04
orbit that fits the GPS orbit best
in radial,cross-track,andalong-trackdirection. RichardRapp (OSU, personalcommunication,1997) verified thelowerdegreeandordercoefficients (up to 14) of DGM-E04 by applyingit asa long-wavelength referencegeoidfor theextraction
Cycle 10 3.02 2.18 2.14'
6.85 6.45 5.99*
11.89 8.97 8.28*
3.71 3.00 2.98*
5.17 3.52 3.43*
10.51 7.66 7.45*
3.20 2.46* 2.48
5.40 5.03 4.73*
7.93 5.86 5.83*
2.48 1.74' 1.74'
6.76 4.53 3.63*
7.80 5.11 4.76*
Cycle 19
Cycle 21
Cycle46
TOPEX orbit computationswith the JGM 2, JGM 3, and DGM-E04 gravitymodelswasperformedat GSFC(FrankLemoine,personal communication,1997). Parametefizationof the orbitsis the sameas in the second-
generation TOPEX preciseorbits,asdocumented by Marshallet al. [ 1995]. Listedarethe SLR andDORIS trackingdataresidualsandorbitdifferences with theJPL reduced-dynamic orbitsbasedon GPS trackingdata. * Lowestvaluesfor eachcycle.
8120
SCHARROO AND VISSER: PRECISE ORBIT DETERMINATION FOR ERS
Table 4.
Results of the Simultaneous ERS Orbit Determination for 104 Orbital Arcs of the Tandem Mission Orbit Solution
DataType
Number
SLR residuals(rms), cm XD residuals(rms), cm
112,144 196,325
JGM 3
EGM96
DGM-E04
5.63 9.58
5.59 9.16
4.94 8.06
ERS-1
104
Altimetertime tag bias(mean-t-variance), ms
-1.24+0.21
-1.45+0.20
-1.52+0.19
ERS-2
110,428 187,212
SLR residuals(rms), cm XD residuals(rms), cm
Altimetertime tag bias(mean+variance),ms
104
Altimeterrangebias(mean+variance),cm
104
5.49 9.62
5.50 9.20
- 1.054-0.22
ERS-1 Minus
4.78 8.07
- 1.254-0.23
- 1.304-0.22
-2.144-0.65
-2.144-0.65
ERS-2
-2.14-t-0.65
The XD residuals pertainto eachsinglecomponent of a XO, soeachXO is countedtwiceandno differentiation is madebetweensingle-anddualsatelliteXOs. The altimetertime tag biasesandrelativerangebiasareestimatedindependently for eachorbitalarc. Only arcswithoutsignificantorbital maneuvers are considered.
the ERS-2 altimeterand endingon June2, 1996, with the switch off of theERS-1. We havechosenthisperiodto comparetheresults obtainedwith the DGM-E04, JGM 3, andEGM96 gravitymodels andto analyzetherespectiveorbitaccuracies.Becauseonly 30% of theTandemMissionoverlapswith thedataperiodof the DGM-E04 tailoring,the verificationcan be consideredto be nearly independent. Naturally,we expectthe gravity model tuning to have the sameeffecton the orbitsfor this particularperiodas on any other. In thissectionwe will firstshowtheimpactof the gravitymodel selectionon the trackingdataresiduals,followedby an assessment of the geographically correlatedorbiterrorsbasedon ERS singleand ERS/TOPEX dual-satelliteXOs, and finally look at the orbit differences.In section5, collinearaltimeterprofilesare analyzed to determine the orbit errors due to nonconservative
4.1.
Precise Orbit
Determination
forces.
Results
The tracking data residuals(measuredminus computedSLR rangesandcrossover heightdifferences)are a measurefor the orbit accuracy.Theseresidualsshouldbe interpretedwith care,sincethe datathat havebeenusedin the POD are likely to underestimatethe
actualorbit error. Moreover,the SLR rangeresidualsare a measureof the orbit accuracyin all threedirections,whereasthe XDs dependon the radialorbit erroronly. Table 4 presentsthe resultsfor 104 arcsof 5.5 days,in which ERS-1 and ERS-2 orbits are determinedsimultaneously, alternativelycomputedwith the JGM 3, EGM96, andDGM-E04 gravity models. Listed are the statisticsof the SLR rangeresiduals,the statisticsof the satellite-specific components of the XD residuals (so eachsingleandeachdual are countedtwice), andthe estimates of the apparentaltimetertime tag biasandrelativealtimeterrange bias.
Note that the XD residualsare clearly the smallestin the DGM-E04 solutions.This is to be expectedsinceXDs havebeen used to tailor the gravity model, startingfrom JGM 3. More relevantis the fact that DGM-E04 even performsbetterthan the new EGM96 model, not only in terms of the rms XD but also judgingfrom the SLR residuals,andthiswhile SLR datahavenot been used to tune the model.
The POD alsoprovidesestimatesfor deficiencies in thedatation of thealtimeterdata.Althoughtheestimatedparameters couldalso
Table 5. Statistics of ApparentSLR RangeBiasesof Near-VerticalOverflights JGM
Number
of Passes
Site
Location
7090 7105 7109
3
EGM96
Mean
DGM-E04
Mean
Mean
Asc
Des
Asc
Des
Var
Asc
Des
Vat
Asc
Yarragadee, West-Australia Greenbelt,Maryland Quincy,California
69 23 34
46 13 15
-1.0 -0.7 -0.2
0.4 -1.4 3.0
1.7 1.5 2.1
-1.0 -2.4 -0.4
0.8 -1.0 3.4
1.4 2.0 1.9
-0.2 -0.6 0.2
7110
Monument Peak, California
74
38
0.7
1.9
2.7
0.1
2.7
3.1
0.9
7403
Arequipa,Peru
59
30
-0.9
-0.7
1.5
-2.5
0.3
2.2
0.5
7835
Grasse,France
38
7
-4.0
-9.2
2.1
-2.7
-5.8
2.4
-3.1
-5.2
1.8
7836 7839 7840 7843
Potsdam,Germany Graz, Austria Herstmonceux, England OrroralValley,Victoria
89 61 60 10
5 18 42 9
-3.3 -3.9 -1.7 3.0
-4.8 -5.5 -2.7 5.1
2.5 1.8 1.8 2.7
-2.9 -3.0 -1.6 1.0
-2.1 -4.8 -1.7 3.5
2.1 2.1 1.9 1.8
-2.4 -2.9 -2.0 -0.4
-1.7 -2.7 -1.6 4.6
2.3 1.8 1.6 1.6
517
223
2.4
4.3 3.3 1.2
2.0
2.0
3.1 2.4 1.1
2.1
1.7
2.6 2.0 1.0
1.8
Total (rms) (Des+Asc)/2 (rms) (Des-Asc)/2 (rms)
Des
0.7 0.0 0.2 -0.5
2.6
Vat
1.3 1.4 1.6 2.7
1.9
The apparentrangebiasis estimatedsimultaneously with a timingbiasfor eachpasswith at leasteightmeasurements beforeandaftertheculmination pointanda highestelevationof at least65ø. For all threeorbitsolutions theaveragerangebiasesalongascending passes (Asc)andalongdescending passes (Des)andthevariance(Vat) of therangebiasesaroundtherespective meansareshown.Valuesarein centimeters.
SCHARROOAND VISSER: PRECISE ORBIT DETERMINATION FOR ERS
absorbpart of the 2-cpr errorsin the restitutedorbital altitudeas well as constantalong-trackerrors,the largerpart will indeedbe relatedto the altimeterdatation. Table 4 clearly showsthat the OPR altimetertime tagsare systematically early by 1.3 to 1.5 ms for ERS-2 andERS-1, respectively. 4.2.
High-Elevation SLR Passes
8121
under"Var") and isolatesthe time invariantgeographically correlatedpart (under"Mean"). The mean of a station'sascendingand descending averages would resemblethe geographicallyfully correlatedorbit error (-6r•), if it were not for constantsystembiasesand station coordinateerrorspersistingin this value. Computinghalf the differencebetweenthe descending and ascending averageseliminatesthe constanterrors and providesa better estimatefor the geographicallyanticorrelatedorbit error (-6r•). The rms of
Despite the aforementionedlimitations,the SLR residualsof high-elevationoverflightsof laser rangingstationscan provideat leastsomeindicationof the orbit's verticalaccuracy.Along nearvertical overflightsthe cross-trackorbit error does not affect the residuals,andsystemnoiseandalong-trackorbit errorcan be separatedfrom the radialby fittinga two-parametercurvethroughthe SLR residuals.The two parameters,rangebias and timing bias, more or lessdirectly relate to the radial and along-trackorbit error. Table 5 lists the statisticsof theseapparentrangebiasesfor near-verticaloverflightsof somehigh-performance SLR stations. When we distinguishbetweenascending anddescending passes, theaverageof the apparentrangebiasesperstationbecomesa measurefor the localgeographically correlatedradialorbiterroralong
To demonstratethat the reductionof the geographicallyanticorrelatedorbit error is also apparentin the actualcomputedorbits for the TandemMission, we have generatedlocal averages and variancesof XDs like in section3.2 and alternativelytook the
eachpass, i.e.,-6r ascand--•r des,where theminus signs come
GFZ PGM055
these linear combinations are listed at the bottom of Table 5 and
are indicativeof the improvementof DGM-E04 over JGM 3 and even over EGM96 but shouldnot be regardedas an accurate representation of the actualgeographicallycorrelatedorbit error. 4.3.
Geographically Anticorrelated Orbit Error
orbit from the OPR data or substituted our third-
from the fact that the SLR range residualsare "observedminus generationorbits. In each casethe ,-.,870,000XOs coveringthe computed"and the orbit errorsare "computedminustrue." The Tandem Mission are reduced to some 48,000 locations. station-by-station averagingremovesthe time variantpart of the The statisticsandgraphsin Plate4 indeedshowthatthe average orbit error associatedwith nonconservativeforces(listed in Table 5 XDs are by far the smallestwith DGM-E04. Both PGM055 and, (a) PGM055: Rms = 7.79 cm
(b) JGM 3: Rms = 6.18 ½m
ß--•ß-••, -. •. ß, ..... ,i•. &. , !. ß•. . •. . ! •o \
t
.•
•'
'
.) • ß
. I .
.i!.....
.
• lit} '.
'•o' '•o • 'eb ø' •o" ' go' ' • ' ' •o' ' •o' ' •' (c) EGM96: Rms = 5.42 cm
-15
,,
-10
..
"
'•o' '•o' '•
L
-. •l
.• , ,
' ß .'-'•
./ •,•.,
-,%, - , ' ':'•, , ,*-..', •::. ,. I'.:
(d) DGM-E04: Rms = 3.00 cm
-5
Plate 4. Averages of stacksof ERS-I andERS-2 single-mission XDs (TandemMission).The variousgraphspertain to the following differentERS orbit solutions:(a) GFZ PGM055 and (b) DUT JGM 3 and (c) EGM96 and (d) DGM-E04. In eachcasethebestfittingapparentaltimetertime tagbiasis applied.
8122
SCHARROO AND VISSER: PRECISE ORBIT DETERMINATION
DGM-E04:
FOR ERS
thattheresultfor DGM-E04 is virtuallythesameaswaspredicted
Rms = 9.31 cm
in section 3.3 and Plate 3b.
ß
The varianceof the XDs displayedin Plate 5 is for DGM-E04 but is quite indifferentbetween the variousorbit solutions. The
.•
largerpartof thevariancecanbe attributedto sealevelvariability. 4.4.
Until now we have mainly consideredthe geographically anticorrelatedorbit error, which is observedin single-satelliteXDs and is usedto tunethe gravity model. But, as a rule of thumb,the fully correlatedorbiterror(6r ½)whichis invisiblein XDs hasabout the samevariance[Rosborough,1986]. To verify whetherthe tuning alsoreducedthe geographicallyfully correlatedorbit error, or at leastdid not increaseit, we have computedERS/TOPEX dualaltimetercrossoversfor the sameperiod during the TandemMis-
''
.
5
Geographically Fully Correlated Orbit Error
10
15
sion as before. The TOPEX
20
data are extracted from the recent re-
leaseof mergedgeophysical datarecords(cycles98-136) [Archiving, Validationand Interpretationof SatelliteOceanographic data Plate 5. Local varianceof stacksof ERS-I and ERS-2 single- (AVISO), 1996], applyingthe appropriatecorrections. missionXDs basedon DGM-E04 orbits(TandemMission). A justifiableassumption is that TOPEX orbit errorsare small comparedto thoseof ERS [Le Traonet al., 1995; Marshall et al., to a lesserextent,JGM 3 displaylargegeographically correlated 1995], suchthat ERS/TOPEX dual satelliteXDs displayERS' ororbit errorsin distinctivenarrowpatterns,10ø to 20ø wide. These bit errorsratherthanTOPEX's. Again, local averagingof XDs can patternscan be associatedwith errorsin the near-resonantterms of be usedto root out time varianterrors. Becauseascendingand dethegravityfield. Thesetermsappearslightlybetterconditioned in scendingpassesof ERS do not meet in the samecrossoverwith theEGM96model,in whichbroader features, however, stillpersist. TOPEX passes,it is actuallymorepracticalto averageon an equiIn the DGM-E04 solution all of these features are absent. What rectangulargrid. The acquiredaverageERS/TOPEX XD is repreremainsis a chaoticpatternof small spatialscalesof which the sentativeof the geographicallyfully correlatedradialorbiterrorfor ERS. origin is not clear. The rms of the geographicallyanticorrelated radial orbit error Plate 6 showsaveragesof ERS-1/TOPEX and ERS-2/TOPEX equalshalf thermsof the averageXDs displayedin Plate4. Note XDs for four different ERS orbit solutions: PGM055, JGM 3, (a) PGM055: Rms= 5.41 cm
o " " ':'
'
(b) JGM 3: Rms = 4.90 cm
' " '"-
', . •"
(c)EGM96: Rms =4.•cm
•
'
-
.'2•
•
'
,
• -10
'
t • /,½ .•',' ' ß
"
. . ,: . •d)•GM-E•: Rms= 3.27 cm
•'• -5
,
' "
• 0
.
, 5
.'
-, ,
,
•,•
'•
'
•.."
10
cm
Plate 6. Locally averagedERS-TOPEX dual-satellitecrossoverheightdifferencesfor the ERS TandemMission. The graphspertainto the followingdifferentorbitsolutionsfor ERS: (a) GFZ PGM055and(b) DUT JGM 3 and(c) EGM96 and(d) DGM-E04. In eachcasethebestfittingapparentaltimetertimetagandrangebiasesfor ERS-1 and ERS-2 is applied.
,
SCHARROOAND VISSER: PRECISEORBIT DETERMINATION FOR ERS
8123
EGM96, and DGM-E04. Appropriatetime tag biasesand range biasesfor ERS-1 andERS-2 are appliedto give a goodmatchwith TOPEX. PGM055 is againthe one to showthe mostevidenttracklike patternassociatedwith near-resonant terms. Striking is the broadstructurethat seemsto persistthroughoutJGM 3, EGM96, andDGM-E04 orbit solutions.It evenappearsto be theonly significantstructureremainingin the DGM-E04 solution.The reduction
gravityfieldcoefficients (ACtre, AStm). Theobserved spectra are
absolute error. This makes the DGM-E04
DUT DGM-E04 orbit solutionsdiffer quite a bit more. Yet the radialorbit differenceof only 7 cm is a very encouraging figure, likely to indicatethat either orbit solutionis at least as accurateas that. Encouragingis also that, comingfrom JGM 3, the EGM96 andDGM-E04 solutionsappearto converge.
periodograms of the actualdifferencesbetweenthe orbit solutions. The JGM3/DGM-E04 difference has a major peak at 0.93014 cpr (= 1 cpr- 1 cpd), and corresponds to I - 2p - 1 and m - I in (2), i.e., gravity coefficientsof odd degreeand order 1. This 1-cprsignalwith a daily modulationis preciselythe near-resonant signalwe heldresponsible for thetrack-likepatterns of the variancefrom 5.41 (JGM 3) to 3.27 cm (DGM-E04) remains in the averageXDs with JGM 3 (section4.3). This peakis indeed a remarkableachievement,sincethe fully correlatedorbit errorwas markedlysmallerin theEGM96/DGM-E04 difference,explaining not involvedin the gravityfield tailoringprocess.It demonstrates the reducedtrackpatternin the averageXDs with EGM96. againthat the tailoredmodelindeedconstitutes an improvement, Because we have not excluded the near-resonant terms in the not merely a reductionof oneobservable. predictedspectra,theyhavelargepeakscloseto 0 and 1 cprcaused It is not certain that the remainingpattern for DGM-E04 is a by the harmonicsof order 43. Having periodsor a modulation truereflectionof the actualgeographically correlatedorbit errorof longerthan the lengthof the orbital arc, they are effectivelyabERS. The ERS/TOPEX XDs will be partly corruptedby TOPEX sorbedby the statevectoror the daily empiricalalong-trackaccelorbit errorsand any time invariantor geographicallycorrelateddif- erations,as a resultof which they do not showup in the observed ferencebetweenERS andTOPEX altimetry.Even thoughwe have spectra.The JGM 3/DGM-E04 spectrashow a one-to-onematch attemptedto harmonizethe altimeterbiases,seastatebias,andthe betweenpredictedand observed. Near-resonantterms are absent oceantide corrections,persistentdifferencesin modelingpropaga- herebecausethey are simplycopiedfrom JGM 3 into DGM-E04. tion correctionsmay causepart of the effect. More likely though, Orbit differencescausedby nonconservative forces,with a moreor what we observeis due to referenceframe offsetsor is causedby lesscontinuous distributionof poweraround1 cpr,areminute. errorsin a few (zonal) gravitycoefficientsthat have not been adTable6 showsthat the orbit solutionsbasedon the threegravjustedin the developmentof DGM-E04. ity modelsare quite close. Irrespectiveof which combinationof Finally, we would like to emphasisthat for many applications DUT orbit solutionsis compared,the rms orbit differenceis around of altimetry,suchas the monitoringof oceancurrentsand compu- 4 cm in radial and around15 cm in along-and cross-trackdirectationof marinegravity,the slopeerroris moreimportantthanthe tion. Becausethey are fully independent,the GFZ PGM055 and orbits even more favor-
able becauseslopesin the orbit error are almostan orderof magnitudesmallerthanin thePGM055 orbits(order0.2 versus1/trad, respectiveJy). 4.5.
Orbit
Differences
5.
Nonconservative
Forces
Figure 3 depictsthe predictedand observedspectraof the radial orbit differences between the three DUT
orbit solutions. The
predictedspectraare accordingto LPT and the differencesin the
The dominantnonconservative forcesactingon the satelliteare atmosphericdrag and solarradiation. Both forcesare the sumefObserved
Predicted
JGM 3 DGM-E04
i ß ßi.. 0
05
1
15
2
25
3
0
,., 05
. . . ! .... 1
i - ß , I:h 15
2
25
2
25
frequency
frequency(c¾c,/rev)
EGM96
DGM-E04
,
0
05
•
-5
frequency(c¾cl/rev)
2
25
3
o
05
1
15
frequency(cycl/rev)
Figure 3. Predictedandobserved spectraof theradialdifferences betweenERS orbitscomputedwith theJGM 3, EGM96, andDGM-E04 gravitymodels.
8124
SCHARROO AND VISSER: PRECISE ORBIT DETERMINATION
Table 6.
aft side of the solar array. At first each of the panelshad been givena size,orientation,andreflectiveproperties,whichwererepresentative of the their true geometryand the propertiesof various subelements definedin a moredetailedmicromodel,consisting of 48 panels.A majoradvancewasmadeby adjustingtheproperties of thepanelsof the macromodel,suchthateachof themmightnot be conformthetruegeometryor collectivereflectiveproperties of thatpanelbutthattheiroverallconsistency with themicromodelin describingthe actualforcesactingon the wholesatellitewasoptimal,judgingfrom Monte Carlo ray-tracingexperiments on both
Differences Between GFZ (PGM055) and DUT Orbit
Solutions Orbit Differences rms
OrbitSolutions
Radial
Cross
JGM 3 JGM 3 EGM96 PGM055 PGM055
-
EGM96 DGM-E04 DGM-E04 JGM 3 DGM-E04
ERS-1 Orbits 3.93 15.18 4.29 11.72 3.81 16.04 8.00 19.79 6.69 19.52
JGM 3 JGM 3 EGM96 PGM055 PGM055
-
EGM96 DGM-E04 DGM-E04 JGM 3 DGM-E04
4.04 4.40 4.03 8.30 6.99
Along
Total
14.48 17.53 14.55 28.50 23.69
21.34 21.51 21.99 21.34 31.42
15.01 17.29 14.47 29.19 23.70
20.75 20.91 21.46 21.34 30.03
the macromodel and micromodel.
All valuesare in centimeters.A 3.5-a editingis imposedon the total
5.2.
orbit difference to remove bad orbits.
Collinear
Tracks
As indicatedbefore, gravity-inducedorbit errorsare the same alongtwocollinearpassesandcancelwhendifferencingthetwo altimetricseasurfaceprofiles.The remainingnonconservative forces causeorbiterrorsaround1 and2 cprandareeasilyseparated from the short-scale seaheightdifferencesassociated with measurement and correctionerrorsand sealevel variability. Figure4 givesanexampleof a pairof collineartracksof ERS-1 and ERS-2 with a time interval of only 1 day. Figure 4a shows the locationsof the measurements. Figure4b givesthe relativesea surfaceheightprofile with respectto the OSU MSS95 meansea
fect of particlesor photonsexchangingmomentumon the (quite complex)satellitesurface,and this requiresaccuratemodelingof the satellitegeometryandsurfaceproperties. 5.1. Modeling of the NonconservativeForces The satellitegeometryof ERS-1 and ERS-2 is modeledby two macromodels consistingof 10 flat panels.Six panelsform thesatellite busandpayloadmodule,two panelsform the front andaft side of the SAR, and a furthertwo rotatingpanelsmodelthe front and Alon (m) : rms =203.4
Because of the attachment of the
GlobalOzoneMonitoringExperiment(GOME), the ERS-2macromodeldiffersslightlyfrom the oneof ERS-1. Further deficienciesin the modeling of the nonconservative forcesareabsorbed by estimating6-hourlydragcoefficients, which scalethe atmospheric dragforces,anda setof 22-hourlyempirical accelerations with a 1-cprmodulation.
ERS-2 Orbits
13.73 10.90 15.33 17.53 17.06
FOR ERS
correction
mox =320.1 0.2
a
0
--0.2
80
'•
70
.Z•
60
-b' ' ' i c '
[d, ißt]''
ß
_
....
:
? I-
...
50
I
40
r
,'
/
,'
I
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, '
30
"•
20
•::•"•
•
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..... •'"'••
,
10
0
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,o \
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20
'
I1%
'
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3o
- t) ).
•
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40
rio
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,
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o
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o
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; ;
ß.
•
•
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6o
80
.....
0.2 pO48OoOl .ps/cps
0
SSH-MSS95(m)
-0.2
, .... 4
SWH(m)
•
2
I
0
0.2
ß
0
I
-0.2
El-E2:SSH-MSS95
Figure4. A collinear pairof seasurface heightprofilesof ERS-1(grey)andERS-2(black)witha timeintervalof 1 day.Ascending pass480,May 13-14,1995.(a) Location, (b) seasurface heightwithrespect to MSS95,(c) SWH, and(d) heightdifference,includingfive-parameter fit.
/
SCHARROO AND VISSER:PRECISEORBITDETERMINATIONFORERS
surfacemodel.A runningaveragefilteris appliedto removethe altimeternoise.Figure4c givesthe SignificantWaveHeight(SWH) for both collinear tracks derived by the two altimeters. Here, a
runningaveragefilter is alsoapplied.Figure4d showsthe residual differencebetweenthe smoothedseasurfaceprofiles. A five-
parameter orbiterrormodel(displayedasthethin solidline) is fittedthroughtheresiduals andabsorbs constant, 1-cprand2-cprsignals. The deviationsfrom the fit are a resultof sealevel variability
anderrorsin the geophysical corrections to the altimeterdataand amount(in this case)to 4.7 cm rms;the orbit errormodelitself has
8125
tion of correctionerrors(mainly wet troposphericcorrection)and increasingsealevel variability. But, also, the orbit errorstend to be highly correlatedover a 1-daytime lag becauseERS-2 thenstill sensesaboutthe sameatmospheric conditions asERS-1 1 daybefore.Thiscouldbe usedto an advantage by couplingthenonconservative forceparameters betweenthe two satellitesduringthe orbit determination.For 35-day time intervalsthe rms valueof the five-parameterfit thusincreases to about5.0 cm (for DGM-E04, Table7). Assumingfull decorrelation, this indicatesthat the radial orbit error due to nonconservative
forcesis x/• timessmaller, 3.5cm.
an rms of 1.3 cm.
The top part of Table 7 givesthe statisticsof over 10,000 collinearpairs of ERS-1 and ERS-2 with a 1-day time interval. Obviously,because of theclearseparation betweenorbiterrorand 6. Error Budget short-wavelength errorsthe rms value of the residualsea height Table8 summarizes theresultsobtainedin the previoussections differences(5.62 cm) is independent of the gravitymodelusedin and lists the three components of the radialorbit error(geographithe orbit computation.The rms valueof the five-parameter fits is for each significantly higherfor theGFZ/PGM055orbitsthanfor the DUT cally anticorrelated,fully correlated,andnonconservative) of the orbit solutions discussed in this paper. Table 8 also gives the (JGM 3, EGM96, and DGM-E04) orbits,which suggeststhat the modelingof thenonconservative forcesatGFZ is lessoptimal.Yet, unexpectedly, alsothechoiceof thegravitymodelappearsto affect thelong-wavelength fitsslightly(2.87 cm for DGM-E04 anda few millimetersmore for others).This indicatesthat a small partof the gravity-induced orbiterroris aliasedintothenonconservative force parameters(drag coefficientsand empiricalforces). When the aliasingwould be purely geographically correlatedor otherwise invariantfrom repeatcycle to repeatcycle, we would still not detectit in the collineartrack differences.Eliminationof a larger partof the gravityinducedorbit errorthusappearsto reducethe aliasingandmakesthe absorption of nonconservative forcesmore efficient.Again,Table7 demonstrates thesuperiority of DGM-E04 in thisrespectovertheothergravitymodels. When going from the shorttime intervalof 1 day of dualsatellite collinears to a sizeable interval of 35 days of singlesatellite collinears, the short-wavelengthheight differencesare significantly larger(around10.5cmrms)because of thedecorrela-
Table 7.
Orbit Error Differencesand Sea Height ResidualsBe-
tween Collinear
Pairs of ERS-1
Number of Orbit
Solution
and ERS-2
Number
Orbit
Height
Collinear
of Data
Error
Residuals
Pairs
Points
rms, cm
rms, cm
ERS-1 Minus ERS-2 (1-Day Interval) PGM055 JGM 3 EGM96 DGM-E04
10,103 10,193 10,214 10,220
14,699,314 14,826,690 14,844,032 14,861,823
3.69 3.09 3.03 2.87
5.62 5.62 5.62 5.62
contributions of the orbit error to XDs with a short time interval
(typicallyup to 1 day) and a long time interval(typically 2 weeks or more). Note that the fully correlatedpart doesnot contributeto the XDs, thatthe anticorrelated partaddsto the XDs at doubleefficiency,and that the nonconservative part hasa certaindecorrelation time.
After adding contributionsfor errorsin the various altimeter geophysicalcorrectionsand sealevel variability we obtainan overall budgetfor the rms XDs, rangingfrom the shortto long time intervals.
Plate 7 confirms these results and shows a dramatic re-
duction of the XDs from the PGM055
orbits to DGM-E04
orbits.
The latter even coincides with the level of TOPEX, which is shown
for referenceand indicatesthat at this point the altimetercorrection errorsand sea surfacevariabilityhave a significantlylarger contribution to the rms XDs than the orbits. In conclusion, for the
ERS TandemMissionthe radial rms accuracyof the GFZ PGM055 orbitsis 7.7 cm; the DUT third-generation preciseorbitshavea radial rms accuracyof 6.8 cm (JGM 3), 6.0 cm (EGM96), and 5.0 cm (DGM-E04). 7.
Conclusions
and Outreach
We have demonstratedthe limitations of currently available gravitymodelsfor the computation of ERS orbits.Focusingon the local average(time invariantpart) of XDs revealsstructuresthat can be linked to errorsin variousgravityfield coefficients,especially thosethat canbe associated with orbit errorsof frequencies closeto 1 cpr. The PGM055 gravity model appearsto be an unfortunatechoicefor the generationof the orbitson the OPR. This modelcausesgeographically anticorrelated orbiterrorsmuchlarger even than the older JGM 3 model.
ERS-1 Minus ERS-1 (35-Day Interval) 10,509 15,186,450 5.39
The time invariantpart of the XDs forms the basisfor the development of an ERS-tailored gravity model DGM-E04 within 10,516 15,186,699 5.12 the error marginsof JGM 3. This techniquehas demonstrated its 10,537 15,211,211 5.11 10,564 15,237,874 5.00 strength:XDs reduceexactly as predictedby the adjustmentsin the gravitycoefficients.Also, the acquiredgravitymodelperforms ERS-2 Minus ERS-2 (35-Day Interval) significantlybetterin termsof SLR trackingresiduals,improves PGM055 8,586 12,156,457 5.92 10.55 orbit determinationfor SPOT-2,andis competitivewith JGM 3 for JGM 3 8,825 12,485,460 5.31 10.56 TOPEX orbit determination.CrossingERS and TOPEX altimeEGM96 8,832 12,487,222 5.26 10.57 DGM-E04 8,862 12,522,016 5.11 10.57 try demonstrates a reductionof the geographicallyfully correlated radial orbit error on top of the forcedreductionof the anticorreRadial orbit error differencesarebasedon a five-parameterfit of the sea forcemodelingerrors surfaceheightdifferences;heightresidualsare indicativeof measurement latedpart. Remarkably,the nonconservative PGM055 JGM 3 EGM96 DGM-E04
10.44 10.44 10.44 10.44
andcorrectionerrorsand seasurfacevariability.Pairsare editedout when the orbit difference exceeds 3.5 times the nominal.
alsoappearto havediminishedin the DGM-E04 orbitscompared to their JGM 3 counterpart.The stepwiseimprovementof the orbit
8126
SCHARROO AND VISSER: PRECISE ORBIT DETERMINATION FOR ERS
Table 8. Contribution of OrbitErrors,AltimeterCorrections, andSeaSurfaceVariabilityto theErrorBudgetof AltimetricSeaSurface HeightsandCrossovers HeightBudget(CrossoverBudget)
Source
GFZ/PGM055
JGM 3
EGM96
DGM-E04
TOPEX/JGM3
Radial Orbit Errors
Geographically anticorrelated* Geographically fullycorrelatedt NongravitationalS Total
3.9(7.8-7.8) 5.4(0.0-0.0) 3.8(3.7-5.4)
3.1(6.2-6.2) 4.9(0.0-0.0) 3.6(3.1-5.1)
7.7 (8.6-9.5)
6.8 (6.9-8.0)
2.7(5.4-5.4) 4.0(0.0-0.0) 3.6(3.0-5.1) 6.0 (6.2-7.4)
1.5(3.1-3.1) 3.3(0.0-0.0) 3.5(2.9-5.0)
1.0(2.0-2.0) 1.0(0.0-0.0) 2.0(1.5-2.8)
7.5 (6.7-9.4)
6.1 (5.8-8.5)
5.0 (4.2-5.3)
2.4 (2.5-3.4)
MesoscaleFeatures(Common to All)
Drytropospheric correction Wettropospheric correction Ionospheric correction
1.0(1.0-1.4) 1.4(1.0-2.0) 1.0(1.0-1.4)
Oceantides Solid Earth tides Seastatebias Instrumenterrors
3.0 (4.2-4.2) 0.5 (0.7-0.7) 1.0 (1.0-1.4) 1.0 (1.0-1.4)
Seasurface variability
4.0(2.0-5.6)
Total
5.6 (5.2-7.8)
TotalError Budget
Total
9.5 (10.1-12.3)
8.8(8.6-11.2)
8.2 (8.1-10.8)
Theranges within brackets relate tocrossovers witha short timeinterval (1day)toa longtimeinterval (weeks). Values (incentimeters) aregiven for various ERSorbitsolutions, andTOPEXresultsaregivenforreference. ValuesforTOPEXarefromMarshallet al. [ 1995]. * On the basis of Plate 4.
t On the basisof Plate 6. SOnthebasisof Table7. determination for ERS seems now to have culminated in a radial
rmsaccuracy of 5.0 cm for theDGM-E04 orbits. 7.1. Room for Improvement
process.Likewise,we can add SLR and finallyPRARE dataas well andusean averagingprocesssimilarto whatis shownin section 4.2 to isolatethe gravity-induced part of the residuals.This pavesthe way to includealsothe zonalgravitycoefficients in the tuningprocess. With thelong-termoperation of ERS-1andERS-2,
When studyingERS/TOPEX dual-satelliteXO differences, the sufficientdata will be availablefor a further gravity model tuning geographically fully correlatedorbiterrorappears to belargerthan to thesesatellites,whichis thenequallysuitablefor theirsuccessor: the anticorrelated. It is not yet certainthattheobserved differences EnviSat. By far the largestcontributorto the orbiterror,however,is the arefully accountable to gravitymodelerrors.Whenthisisclarified, that thereis marginfor improvethisdatatypecaneasilybe includedin thegravitymodeltailoring time variantpart. This suggests mentof the surfaceforces(dragandsolarradiation).The time variant part,however,alsocomprises the gravitational effectof solid Earth and ocean tides, which so far have not been considered.Yet this does not affect our results, since all time variant orbit errors are
simplymingledinto the onethatwe labeled"nonconservative" or "nongravitational." 7.2. Remaining Considerations
===
ERS-1/2GFZ/PGM055
ERS-I• DUT
ERS-1/2 OUT/EGM96 ERS-1/2 DUT/DGM-E04
TOPEX NASA/JGM 3
A widelydistributed legendarguesthatwhenorbitcomputations use the samegravity field for all altimetersatellites,thereis no mismatchbetweenthe gravity-inducedorbit errorsandhencethey cancelwhendifferencingthe differentdatasets.This is, however, nottrue. Becauseof theirdistinctinclination,repeatcycle,altitude, andchoicefor orbitalarc length,gravitymodelerrorsimpactdifferentlyonthecomputed orbitalaltitude.Thebestchoiceof gravity modelis the onethat introducesthe leasterrorsfor eachparticular mission. Tailored modelswhich are tuned to a particularsatellite missionare thereforethe bestcandidatesfor adoptionin the POD,
aslongastheseact withinthe errormarginsof a general-purpose 0
2
½
6
8
10
12
14.
16
18
model.
time difference (dq/s) This is irrevocablydemonstrated when differencingERS and Plate 7. The rms and meanXD (ascending-descending) as a TOPEX altimetryin Plate6 of section4.4. Nearlythe worstof functionof timeinterval.Databeyond66ø latitudeandin high- all performance is givenby usingthesamegravitymodel(JGM3)
variability areasareexcluded. A 3.5-ceditingisapplied.
for bothsatellites.The bestmatchis obtained usingDGM-E04
SCHARROOAND VISSER:PRECISEORBITDETERMINATIONFORERS
for ERS andJGM 3 for TOPEX. In fact,JGM 3 is verymuchoptimized for TOPEX, which is also reflected in the error variance-
8127
Marsh,J. G., et al., The GEM-T2 gravitationalmodel,NASA Tech.Memo. 100746,91 pp., 1989. Marshall,J. A., N. P. Zelensky,S. M. Klosko,D. S. Chinn, S. B. Luthcke, and K. E. Rachlin, The temporaland spatialcharacteristics of TOPEX/POSEIDON radial orbit error,J. Geophys.Res., 100(C12), 25,331-
covariancematrix of JGM 3. The tailoringto ERS thusmainly concernsthe weakly determined(lumped)coefficients thatare best 25,352, 1995. observedin ERS XDs. Consequently,for TOPEX orbitdetermination, DGM-E04 and JGM 3 are very similar, as was also demon- Massmann,F.-H., C. Reigber,R. Ktnig, andJ. C. Raimondo,ERS-1 orbit strated in section 3.4.
informationprovidedby D-PAF, in Proceedings of the SecondERS-I Symposium, Eur. SpaceAgencySpec.Publ., ESASP-361, pp. 765-770, 1994.
Acknowledgments. The ERS orbit determination at DUT wouldnot havebeensucha success withoutthesupportfrommanyof ourcolleagues. In particular,we like to thankBob Cheneyof NOAA for supporting the operationalorbit determinationfinanciallyand to colleaguesat GSFC for providingnumerousgravitymodelsand our workhorse,the GEODYN orbit determinationsoftware. We also acknowledgethe supportof the oreciseorbitdetermination throughthe EC projectESAMCA, led by Duncan Winghamof UCL/MSSL. Furthermore,we are gratefulto thosewhoseinputsanddiscussions were veryvaluablethroughout the years:C. K. Shum, JohnRies, Pierre-YvesLe Traon, Carl Wagner,Michael Anzenhofer,and Arthur Smith. Specialthanksgo to JohnLillibridge for hostingthe DUT orbitson the NOAA ftp site and to FrankLemoine(GSFC) and Richard Rapp(OSU) for theirindependent assessments of theDGM-E04 model.
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