Bollettino di Geofisica Teorica ed Applicata Vol. 54, n. 4, pp. 367-384; December 2013

DOI 10.4430/bgta0108

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Fully dynamic approach for GOCE precise orbit determination

S. CaSotto1,2, F. Gini2, F. Panzetta2 and M. Bardella1

1 Dipartimento di Fisica e Astronomia, Università di Padova, Italy2 Centro Interdipartimentale di Studi e Attività Spaziali (CISAS), Università di Padova, Italy

(Received: January 15, 2013; accepted: July 10, 2013)

ABSTRACT Gravityfieldandsteady-stateOceanCirculationExplorer(GOCE)waslaunchedin2009at250kmaltitudetorecovertheEarth’sstaticgravityfield.AspartoftheGOCE-Italyproject,wecarriedoutGPS-based,fullydynamicPreciseOrbitDetermination(POD)ofGOCEfordailyarcscoveringabout500days(November1,2009-May31,2011).Threesequencesweredefinedand implementedwith thesoftwareNAPEOS(ESA/ESOC).Afirstsequenceusestheorbitpropagatedfromthepreviousdayasana-priori orbit but, to avoid one-day failures compromising all the subsequent PODprocessingchain,othertwosequenceswerebuiltusingtheofficialkinematicPreciseScienceOrbits (PSO) as a-priori orbits.For thosedayswhere the sequencesbasedon thePSOgave lessaccurateresults,orevenfailure, thea-prioriorbitpropagatedfrom the previous day was employed. Results show an average post-fit RMS ofzero-differencephasemeasurementsbelow10mmforabout90%ofthedailyarcs.Mostorbitscomparetolessthan6cm3DRMSwithrespecttotheofficialkinematicand reduced-dynamicsPSOorbits.Toevaluate thequalityof thePODresults,366overlapping arcs of 5 hourswere compared, showing an averagedistancebelow1cm.

Key words:GOCE;PreciseOrbitDetermination(POD);GPS;FullyDynamicPOD.

© 2013 – OGS

1. Introduction

TheGravity field and steady-stateOceanCirculationExplorer (GOCE) (ESA, 1999) isthe firstEarth explorer coremission of theEuropeanSpaceAgency (ESA). Itwas launchedonMarch17,2009fromPlesetsk,Russia.Themission isdedicated tohigh-resolutiongravityfield extraction and carries, as primary instrument, a three-axis gradiometer for determiningthegravityfieldwithanunprecedentedaccuracyof1mGalandthegeoidwithanaccuracyof1cm,bothataspatialresolutionof100km(Drinkwateret al.,2007).Inadditionthemissionis equippedwith two 12-channel dual-frequency LAGRANGEGPS/GLONASS Satellite-to-SatelliteTracking Instruments (SSTIs) consisting of an independent receiver and antennaeach.Themeasurementsof themainonboardGPS receiver (SSTI-A)allow forPreciseOrbitDetermination(POD)ofthesatelliteandforgravityfieldrecoveryofthelowdegreeandorderterms.

POD for the GOCE satellite is one task of the GOCE-Italy group, an ESA projectendorsementfundedbytheAgenziaSpazialeItaliana(ASI).BasedontheGPSdataavailability

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GOCEorbitswereestimatedattheUniversityofPaduaandthiswasperformedwithadensely-parameterized,fullydynamicPODapproach.

The results discussed subsequentlywere all obtained from 1Hz data of the SSTI-A(RINEXdata)inthetimeintervalfromNovember1,2009toMay31,2011(496daysofdataavailability).

GeneralGOCEdatainformationemployedtoperformthePODisprovidedinSection2,thedynamicalmodelandtheestimatedandfixedparametersadoptedduringthePODprocessaredescribedinSection3,thesoftwareandthePODchainsimplementedarepresentedinSection4.InSection5theGOCEorbitresultsforthe496-dayperiodarethenillustratedanddiscussed.ConclusionsarereportedinSection6.

2. GOCE useful data for POD processing

GOCEPODwas performedusing as themain input dataGOCEGPSphase observables,InternationalGNSSService (IGS) (Dow et al., 2009) finalGPSorbits and clock solutions,GOCEGPS antenna information inANTEX format,GOCE radiation cross-section area,aerodynamiccross-sectionarea,andattitudeinformation.

GOCEGPSrawobservationsarecollectedanddeliveredat1HzbyGOCESSTI-A,whichconsists of a 12-channel dual-frequencyLAGRANGE (LabenGNSSReceiver forAdvancedNavigation,GeodesyandExperiments)GPS/GLONASSreceiver(theGOCEflightmodelonlytracksGPSmeasurements)andanL-bandantenna;SSTI-Bisafullyredundantbackupreceiver.The internal clockof theGOCELAGRANGEreceiver isnot steered to integer secondswith

Table1-GOCESSTIdataavailability:dailyreportsovertheperiodNovember1,2009-May31,2011.

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somejumpsof20ms.ForthePODprocess,GOCEGPSdatacoveringthetimeintervalfromNovember 1, 2009 (beginning of the operational phase) toMay 31, 2011were obtained inRINEX2.20format(filetypeSST_RIN_1b)throughEOLi-SA(EarthObservationLink-StandAlone) (EOLI-SA, 2012), an interactive tool to view and order products fromESA’sEarthobservationcatalogues.

SSTIdataavailabilityover theconsideredperiod isshown inmoredetail inTable1.Notethatthedatasetisnotcontinuousandseveralintervalsorsingleepochsaremissing,generallydue to onboard system failures or anomalies, as reported inmonthly reports available on theGOCEMonthlyStatisticswebsite(GOCE,2013).

OfficialGOCEkinematic PSO solutionswere used as a-priori orbits for the smoothingof rawobservations during thePODprocess and as a termof comparison for the estimatedorbits.Theseofficial orbits includepositions at 1 s sampling (file typesSST_PSO_2) for thekinematicorbitsandpositionsandvelocitiesat10ssamplingforthereduced-dynamicsorbits.They are generated at theAstronomical Institute of theUniversity ofBern (AIUB,Bern,Switzerland)with the support of the Institute ofAstronomical andPhysicalGeodesy (IAPG,TechnischeUniversitätMünchen,München,Germany)(Bocket al.,2007;Visseret al.,2009).TheofficialPSOsolutionsareprovidedwithanaccuracyof2cm(Visseret al.,2010;Bocket al.,2011b)andarebasedonundifferencedGPSobservationsprocessedwiththeBerneseGPSsoftwarepackage (Dachet al.,2007).KinematicorbitsmaycontaindatagapssincepositionscanbeestimatedonlyatepochsforwhichasufficientnumberofGPSsatellitesisvisible.PSOavailabilityovertheconsideredperiodisshowninmoredetailinTable2.

PreciseGPSorbits,deliveredwitha15minsample intervaland30sclocksolutionswereobtainedfromIGSFinalProductsftpwebsite(IGS,2013b)foreachdayofeachGPSweek.Theaccuracyisabout2.5cmforGPSorbitsandabout75psforclocksolutionsandisreportedontheIGSProductswebsite(IGS,2013a).

GOCEorbits are estimatedwith respect to the centre ofmass (CoM) of the satellite, so

Table2-GOCEPSOavailability:dailyreportsovertheperiodNovember1,2009-May31,2011.

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modelingGPSobservations requires precise coordinates of the centre of themountingplane(CMP)of themainGOCEGPSantenna (SSTI-A)with respect to the satelliteCoM.Table 3shows coordinates of theGOCECoMandSSTI-A antennaCMP in the SatelliteReferenceFrame (SRF).No change in theCoM location has beenmodelled, keeping its coordinatesconstantandequaltotheBeginningOfLife(BOL)conditions.Fig.1showsaschematicviewoftheGOCEsatellite,withthelocationofthetwoSSTIantennasandthelayoutoftheright-handedSRFandtheSSTI-AAntennaReferenceFrame(ARF).

XSRF (m) YSRF (m) ZSRF (m)

CoM 2.5000 0.0036 0.0011

CMP 3.1930 0.0000 -1.0922

Table3-CoordinatesofthesatelliteCoMandSSTI-AantennaCMPintheSRF(BigazziandFrommknecht,2010).

Moreover,because theantennaphasecentre locationdiffers from theCMP, it isnecessaryto determine two sets of phase centre corrections:PhaseCentreOffsets (PCOs), defining thepositionofthemeanantennaphasecentrewithrespecttotheCMP(Table4),andPhaseCentreVariations (PCVs), accounting for the azimuth and zenith/nadir dependence,which are bothprovidedinANTEX(ANTennaEXchange)formatfortheL1andL2frequenciesinanantenna-fixedreferenceframeandareavailableontheGOCESSTI-AANTEXwebsite(GOCESSTI-AANTEX,2013).TheimpactofPCVsonthePODprocessisimportant,reachingmagnitudeupto3-4cm.Infact,whilePCOvectorsarefixed,PCVswereestimatedfromin-flightcalibrationusing154daysofGOCEGPSdata(Bocket al.,2011a).Ingeneral,bothPCOsandPCVsaresourcesofsignificantsystematicerrorsinGPSdataprocessingifnotcorrectlymodeled(Jäggiet al.,2009).

Fig.1-SchematicviewoftheSSTIantennalocationsandtheSRFandARFreferencesystems.

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Other auxiliary parameters forGOCEPODprocessingwere retrieved from theEuropeanSpaceOperationsCentre (ESA/ESOC) ftpwebsite (ESA/ESOC,2013), inparticular aGOCEradiationareaof11.2m2,adragareaof1.035m2,andEuleranglesdescribinga3-2-1rotationfromtheSRFtotheARF(seeTable5).Aconstantsatellitemassof1050kgwasused.AccordingtoDrinkwateret al.(2007)theyaw-steeringmodewasusedfortheattitudemodeling.

Table4-SSTI-AantennaPCOsintheARFforbothfrequenciesL1andL2(Bocket al.,2011a).

XARF (mm) YARF(mm) ZARF (mm)

L1 -0.18 3.51 -81.11

L2 -1.22 -1.00 -84.18

Table5-EulerrotationanglesfromtheSRFtotheARF(ESA/ESOC,2013).

XSRF Axis (deg) YSRF Axis (deg) ZSRF Axis (deg)

Euler Rotation Angle 180.0 0.0 90.0

3. Dynamical model

TheEarth’sstaticgravityfieldusedintheGOCEPODprocessingistheEIGEN-6Cmodel(Försteet al., 2011) up to degree and order 200, combiningLAGEOS,GRACEandGOCEsatellitemeasurements,gravityandaltimetricdata,available fromtheInternationalCentreforGlobalEarthModels(ICGEM)website(ICGEM,2013).

The third-bodygravitational forces of theSun, theMoon and the planets are considered,togetherwith the indirectoblatenessperturbation,which isa feedbackeffectoriginating fromtheinteractionofthelunarmotionperturbedbytheEarthsecondzonalharmonic.

SolidEarth andocean tide perturbations are included in the dynamicalmodel.Earth tidesaremodeledaccording to theIERS-TN32solid tidemodel (McCarthyandPetit,2003),whiletheFES2004model(Lyardet al.,2006)uptodegreeandorder50for106constituentshasbeenadoptedforoceantides.

Thesatelliteaccelerationisalsocorrectedforgeneralrelativisticeffectsduetothecurvatureofspace-timegeneratedbyarotatingEarth(McCarthyandPetit,2003).

Three analyticalmodels are implemented inNAvigation Package forEarthObservationSatellites (NAPEOS) in order to compute the non-gravitational perturbations due to SolarRadiation Pressure (SRP) (McCarthy and Petit, 2003), Earth albedo and infrared radiation(ArnoldandDow,1984).Inparticular,SRPcoefficientsareestimatedduringthePODprocessconsidering a constantGOCE radiation cross-section area,while the albedo and infraredcoefficientsarefixed.

TheaerodynamicforcesarenotconsideredduringthePODprocessbecauseGOCEisflyinginaquasidrag-freemodeviaanelectricpropulsionsystemusedtocontinuouslycounteracttheatmosphericdraginthedirectionofthemotion.

Finally,empiricalaccelerationsintheradial(r),along-track(a)andcross-track(c)directionscan be included in the forcemodel to compensatemodel omission errors.They consist of acombinationoftwoperiodicterms,eachafunctionofthesatelliteargumentoflatitudeu,andaconstantone,asfollows:

∆➝ar=(ar0+arccosu+arssinu)➝ur

∆➝aa=(aa0+aaccosu+aassinu)➝ua (1)

∆➝ac=(ac0+acccosu+acssinu)➝uc.

ThenineparametersappearingintheseequationsarereferredtoasCPRsbecausetheperiodoftheseaccelerationsisonecycleperrevolution.ThethreeradialCPRsarenotconsideredinthe adopted forcemodel because there is a direct correlation between these and along-trackCPRs.Thus,onlythealong-trackandcross-trackparametersareestimatedoverhourlyintervals,whichestablishesadensely-parameterized,fullydynamicPODapproach.

Tables6and7,respectively,exhibittheforcemodelsandtheestimatedandfixedparametersadoptedfortheGOCEPODprocess.

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Table7-EstimatedandfixedparametersduringtheGOCEPODprocess.

Parameters Description Type Reference

State vector estimated

Radiation pressure Solar radiation pressure estimated (a) Albedo radiation pressure fixed, equal to 1 (b) Infrared radiation pressure fixed, equal to 1 (b)

Empirical accelerations CPR constant along-track estimated (1 per hour) (c) CPR constant cross- track estimated (1 per hour) (c) CPR cosine along- track estimated (1 per hour) (c) CPR cosine cross- track estimated (1 per hour) (c) CPR sine along- track estimated (1 per hour) (c) CPR sine cross- track estimated (1 per hour) (c)

(a) McCarthy and Petit (2003); (b) Arnold and Dow (1984); (c) ESA/ESOC (2009).

Table6-Gravitational,non-gravitationalandempiricalforcesadoptedfortheGOCEPODprocess.

Dynamical models Description Reference

Static gravity field EIGEN-6C 200x200 (a)

Solid Earth tides IERS-TN32 (b)

Ocean tides FES2004 106 constituents, 50x50 (c)

Third body perturbation Lunar gravity Solar gravity Planetary gravity Indirect oblateness perturbation

Relativistic correction Correction according to general relativity (b)

Aerodynamic forces not considered

Radiation Pressure Solar radiation pressure (b) Albedo radiation pressure (d) Infrared radiation pressure (d)

Empirical accelerations CPR along-, cross-track parameters (e)

(a) Förste et al. (2011); (b) McCarthy and Petit (2003); (c) Lyard et al. (2006); (d) Arnold and Dow (1984); (e) ESA/ESOC (2009).

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4. Fully dynamic orbit generation

NAPEOS,developedandmaintainedbyESA,wasused toperform theGOCEPOD.Thissoftwaresystemiscomposedofseveralprograms thatperformdifferent tasks.APODtask isimplementedthroughascript,whichisusedtosetupasequencewhichcallsdifferentprogramsand organizes the I/O channels between them under specific processing procedures, calledmodes.Threesequencesweredefined toprocessGOCEdata (seeFig.2).Theyaredescribedin the following, andwhere appropriate the specificprogramand itsmode isprovided in theformProgram/Modes.ThePODproductscontaintheGPSobservationresiduals,theestimatedparametersandtheestimationcovariancematrices.

4.1. The sequence GOCE_PSO_aThesequencestartswiththeIGSclockfilesthatareconvertedtoNAPEOSclockfilesformat

(.tcb)forthe24hoftheanalyzedday(ClockUpd/GOCE).ThesatellitepositionsfromthePSOorbits are converted toNAPEOSTrackingDataFormat (NTDF) (Tracksim/ORBIT-FIT) thatarethenusedtogeneratethebest-fittingorbitforthedaybeingprocessed(Bahn/ORBIT-FIT).As a result ofthis step theparameters shown inTable7 are estimated and an a-priori orbit isgenerated.TheIGSsp3orbitsaremergedwiththecomputeda-prioriorbittoobtainacombinedsp3(OrbUpd/ORBIT-FIT).TheRINEXfilesavailableforthedaybeingprocessedarecollectedin a catalogue file (BuildCat/GOCE), and pre-processed to initialize the receiver clocks areestimatedatthesametime(GnssObs/GOCE-RAW).AfirstraworbitisthencomputedatmetrelevelusingonlyGPSundifferencedpseudorangeobservations,andGOCEclocks(Bahn/GOCE-RAW).Inthisstepthepre-computedsatellitestatevector,CRandCPRsareusedtoinitializetheestimationprocess.The resulting raworbit is thenmergedwith the IGS sp3orbits (OrbUpd/GOCE-RAW).The available observations are then pre-processed (GnssObs/GOCE) and theeffective orbit estimation is performed atmillimetre level using both theGPSundifferencedcarrier phase and pseudorange observations, taking as input the a-priori raw orbit and theinitializedparametersandclocks(Bahn/GOCE).

4.2. The sequence GOCE_PSO_bItwasobservedthatGnssObswasnotutilizingalltheavailableobservationsforsomedaily

arcs.Thiswas traced to a large disalignment of the observation time tagwith respect to anintegersecond.TheproblemwasfixedbytweakingstepsGnssObs/GOCE-RAWandGnssObs/GOCE of sequenceGOCE_PSO_a to instruct the program to correct bothmeasurements andtime tags tobring themeasurement to an integer second, andby setting the interval betweenobservations, for them to be considered simultaneous, to 1.0 and 0.9 s respectively for theGnssObsGOCE-RAWandGOCEmodes.

4.3. The sequence GOCE_cThissequenceisidenticaltotheGOCE_PSO_asequence,exceptthatitusesasthea-priori

orbittheonepropagatedfromthepreviousdayratherthanthePSOsolution.Thiswasnecessarydue to the lackof convergenceof thePOD taskbasedon the kinematicPSO solutions.ThisbehaviorisprobablyduetothepresenceofgapsinthekinematicPSOephemeris,butwarrantsfurtherinvestigation.

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Fig.2-GeneralsequenceofNAPEOSprogramsusedforGOCEPODprocess.

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5. POD results

The three sequences described in Section 4were executed for each 24-hour observationperiodtocoverallGOCEdataavailablefromNovember1,2009toMay31,2011foratotalof496days.TheGOCE_PSO_asequencewasappliedinthefirstplace, thentheGOCE_PSO_bsequencewas selectively applied to those dayswhereGOCE_PSO_a sequence failed toconverge,orwheretheresultingpost-fitRMSwasgreaterthan1cm.TheGOCE_csequence,whichisbasedontheorbitpropagatedfromthepreviousday’ssolution,wasrunforthosedayswherethePSOwasnotavailable,orwheretheprevioussequenceshadfailedorgiveninaccurateresults.ThestatisticsofsuccessfulprocessingforthedifferentsequencesareshowninTable8.Acomprehensiveviewofthedataavailability,post-fitRMSlevel,andthesuccessfulsequenceappliedoneachofthe496daysofthedataintervalconsideredisshowninTable8.Thethreesequencesweresuccessfullyrunfor487daysofthetotalavailabledaysofdata(98.2%)andthePODtaskwasperformedwithaRMSbelow10mmfor448days(90.3%).

Table8-StatisticsoftheGOCEPODprocessforthethreesequences.

Run sequences Days successfully Days with Percentage of days Percentage of days processed RMS ≤ 1 cm processed RMS ≤ 1 cm

GOCE _PSO_a 375 339 75.6 68.3

GOCE _PSO_a,_b 426 377 85.9 76.0

GOCE_PSO_a,_b,_c 487 448 98.2 90.3

AsTable8shows,thePODwascarriedoutwithloweraccuracy(avalueoftheRMSgreaterthan10mm)fortheperiodsimmediatelyafterthedatagaps(i.e.,FebruaryandSeptember2010)andseveraldays in2011.For themajorityof thedailydataavailablefor2011, thesequencesbasedonthePSOorbitsfailedduetodivergenceintheestimationprocess.Thethirdsequencewashencerun,andthisallowedtoperformthePODsuccessfullyforthemajorityofthemissingdays. InFig. 3 (in blue),where the dailyRMSof the residuals for the undifferenced carrierphasearereported, it ispossible toobserve thatmostof theresultsexhibitvaluebetween6.7mmand6.8mm,andthatonlyafewspikesexceedthe1.5cmlevel.

Fig. 4 shows the total numberof observations available forBahn processing for eachdayofdata.Theaveragenumberofobservationsisabout28000perday,whiletheaverageof therejectedobservationsisabout515perday.Forthemajorityofthedaysprocessedthenumberof rejected observations is below300 (about 1%of the total available observations), but forsome days (i.e.,DayOfYear (DOY) 295, 296, 297, 310, 314, 315, 318 of year 2010) thenumberof rejections ismuchhigher (evenmore than50%of theobservationsaresometimesrejected). For those dayswhere a huge number of observationswas rejected, the flag thatcorrectsmeasurementsand time tags tobring themeasurements to the integersecondwassetin the data pre-processing program (GnssObs).This flag allowed to utilize all the availableobservationsbut, in turn,sometimesreduced theaccuracyof theestimatedorbits leading toahighobservationrejectionrate.

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The parameters thus estimatedwere then analyzed.The solar radiation coefficientCR,estimateddaily,isshown(inblue)inFig. 5withalargescatterof3.97aboutameanvalueof2.48.This behavior points to somedeficiencies in ourmodeling, since negative values alsoappear and some formof signal seems to be present, notwithstanding the discontinuities dueto data outages.The hourly estimatedCPRs for the along-track and cross-track components

Table9-StatisticsofGOCEPODprocessingovertheperiodfromNovember1,2009toMay31,2011.

Fig.3-Post-fitRMSofundifferencedcarrierphases(inmillimetres)forthedataprocessed.Drag-freearcs(noCDestimated)areshowninblue,non-drag-freearcs(CDestimated)areshowninred.

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Fig.4-Numberofundifferencedcarrierphaseobservationsprocessed(blue)orrejected(red)eachdayfromNovember1,2009toMay31,2011.

arerespectivelyshowninblueinFigs.6and7.TheCPRvaluesaregenerallyconstant,exceptforlargevariationsof2ordersofmagnitude(fromabout10-11to10-09km/s2)inthealong-trackcomponentnear thedatagapsofFebruary,JulyandSeptember,2010.Thesameeffect isalsoobserved,buttoalesserextent,inthecross-trackCPRs.NoCRorCPRcoefficientsareplottedfromDOY19 to 40 of 2011, because they are associatedwith an unmodeled switch of theGOCEGPSantenna,asexplainedlater.

The unexpected pattern of the non-gravitational parameters solution prompted us toinvestigatewhether thesatellitehadsufferedanyanomalousbehaviornear thosedates. Itwasfound (ESA, 2012a, 2012b, 2012c) that in February, July andSeptember, 2010GOCEhadnolongerbeenorbitingin thenominaldrag-freemode,sincetheionthrusterhadbeenbiasedin order to raise the orbit for altitude recovery following spacecraft anomalies.Clearly, theestimationofthealong-trackCPRcoefficientsisnotcapableofaccuratelymodelingthesatellitethrust,asshownbytheassociatedhigherpost-fitRMSvaluesshowninblueinFig.3.

ThechangeinthesatellitedynamicsduetotheactionoftheionthrusterwasthenintroducedintheNAPEOSforcemodelthroughtheactivationoftheatmosphericdragmodel.Preciseorbitdeterminationrunswereperformedonlyforthedayswheredrag-freemodewasnotoperating.Threepiecewise-lineardragcoefficients(CDs)perarcwereestimated,oneevery12hours.TheirvaluesareshowninFig.8andaremostly-andparadoxically-positive,rangingbetween3and5.Theresultingpost-fitRMS,showninredinFig.3,issignificantlyreducedwithrespecttothepreviouslyused forcemodel,wherenoCDswere estimated.The computedRMS is generallybelow10mm, except for a fewdaysduringFebruary, 2010.Thenewly estimatedvalues forthe solar radiation pressure coefficients and for the along- and cross-trackCPRs are shownrespectively in red inFigs. 5, 6 and7.While the behavior of the newly estimated values oftheCDsisnotsodifferentfromthatofthepreviouslyestimatedvalues,theCPRs-mainlythealong-trackcomponent,showasmallerexcursion,exceptfortheconstanttermoverafewdaysinFebruary,2010.

The precise orbits determined usingNAPEOSwere then comparedwith the officialkinematic and reduced-dynamicsPSOs for all the available days of data.The comparison isshown inFig.9 for thekinematicPSOsandFig.10 for the reduced-dynamicsPSOs.The3DRMS - later referred toas thedistancebetweenorbits - isbelow10cmformostof theorbitcomparisons(76%and83%of thetotalavailabledaysofdata,respectively,for thekinematicandreduced-dynamicsPSOs).BeforeDOY200of2010boththePODRMS andthePOD-PSO

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Fig. 5 -Behaviour of the solar radiation pressure coefficientsCRs, estimated once per day.Drag-free arcs (noCD estimated)areshowninblue,non-drag-freearcs(CDestimated)areshowninred(DOYstartsin2009).

Fig.6-Behaviourofthehourlyestimatedconstant,sineandcosinecomponentsofthealong-trackCPRcoefficients.Drag-freearcs(noCDestimated)areshowninblue,non-drag-freearcs(CDestimated)areshowninred(DOYstartsin2009).

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Fig.7-Behaviourofthehourlyestimatedconstant,sineandcosinecomponentsofthecross-trackCPRcoefficients.Drag-freearcs(noCDestimated)areshowninblue,non-drag-freearcs(CDestimated)areshowninred(DOYstartsin2009).

Fig.8-Behaviourofthepiecewise-lineardragcoefficientsCDs,estimatedonceperdayovertheperiodofnon-drag-freemotion(DOYstartsin2009).

orbitdistancearegenerally low,whileafterDOY240of2010, theRMSshavehighervalues,especiallyduring2011.

Thereisahighdegreeofcorrelationbetweendaysshowingapost-fitRMS higherthan3cmandtheRMS ofthedistanceoftheresultingephemerisfromthePSOorbits.Amongtheorbitsestimatedwithapost-fitRMS lessthan10mm,about40%and20%significantly(3DRMS >6cm)differrespectivelyfromthekinematicandreduced-dynamicsPSOsolutions,sometimes

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Fig.9-Comparisonbetween theestimatedand theofficialkinematicPSOorbits in theEarth-CentredEarth-Fixed(ECEF)referenceframe.The3DorbitdifferenceRMSwithrespecttothekinematicPSOsisshownincentimetres.

Fig.10-Comparisonbetweentheestimatedandtheofficialreduced-dynamicsPSOorbitsintheEarth-CentredEarth-Fixed(ECEF)referenceframe.The3DorbitdifferenceRMSwithrespecttothereduced-dynamicsPSOsisshownincentimetres.

reachingapositiondifferenceRMS uptoseveralmetres.SomeofthesearcsarecharacterizedbynotalltheobservationsbeingusedintheGOCEPODprocess,thusleadingtolowaccuracyorbitsevenifalowvalueoftheresidualsRMS wasachieved.

ThecorrelationbetweentheRMS oftheresidualsoftheestimatedorbitsandthe3DRMS ofthedifferencebetweentheseorbitsandthePSOorbitsismoreevidentinbothFigs.11and12.Themajorityofthedataanalyzed(about70%and75%oftheorbitscompared,respectivelywith the kinematic and reduced-dynamicsPSO solutions) is located at the lower left corner

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Fig.11-ComparisonbetweentheestimatedandtheofficialkinematicPSOorbits.The3DorbitdifferenceRMSwithrespecttothekinematicPSO’sisplottedagainstthepost-fitRMS.

Fig.12-Comparisonbetweentheestimatedandtheofficialreduced-dynamicsPSOorbits.The3DorbitdifferenceRMSwithrespecttothereduced-dynamicsPSOsisplottedagainstthepost-fitRMS,showingalinearcorrelation.

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of theplot,where theRMSof thefit isbelow10mmandthe3DRMS is lower than10cm.Increasingvaluesof thepost-fitRMS correspondtohighervaluesof thePOD-PSO3DRMS.These data are distributed linearly on the semi-logplot.ThePOD-PSO3DRMS is between10 cm and 12m even if thePODRMS is below10mm for about 60 and 40 days of datarespectivelycomparedwiththekinematicandreduced-dynamicsPSOsolutions.Thisispartlydue to sparseness of the observations used and partly to the unmodeled switch of theGPSantennafortheperiodimmediatelyaftertheSSTIanomalyoccurredonJanuary2,2011(ESA,2011a,2011b).DuringtheperiodbetweenJanuary2,2011andFebruary10,2011theredundantGPSantenna(SSTI-B)wasusedtotrackthesatellite,butthiswasnottakenintoaccountintheNAPEOSprocessingscheme.ThisisclearlyobservableinFigs.9and10,wherethePOD-PSO3DRMS shows an almost constant bias of about 186 cm (the distancebetweenSSTI-AandSSTI-Bisabout183cm),aswellasinFig.3,whichshowsananomalouslyhighandconstantpost-ftRMS for these dates, and inFigs. 11 and 12,where two flat series of constant orbitdistancevaluesappearindependentofthevalueofthepost-fitRMS.

Inorder toevaluate the internalqualityof thePODresults, the regular24-hourarc lengthused for orbit determinationwas extended for 366 arcs to 30 hours, thus ensuring 6-houroverlapscentredat24:00eachday.Toavoidboundaryeffects,thecomparisonoftheoverlapswascarriedoutconsideringonly thecentral5hours (21:30-02:30).Thisshowedanaveragedistanceoftheoverlappingarcsof9.6mm.

6. Conclusions

Theprecise orbit determination ofGOCEwas successfully carried out for a total of 487dailyarcsovertheperiodfromNovember1,2009toMay31,2011,whichincludesalltrackingmeasurements from the LAGRANGEGPS receiver onboardGOCEwhichwere publiclyavailable at the timeof processing.TheNAPEOS software systemwasused to carryout theorbitdeterminationtask.Resultsshowanaveragepost-fitRMS oftheundifferencedGPScarrierphaseobservationsbelow10mmfor448days,or90.3%ofthetotal.Thismaybepartlyduetodegradedsystemperformanceimmediatelyafterlossoftelemetry(February12-14,2010)andonboardcomputerfailures(July-August,2010).

The quality of the POD results obtained usingNAPEOSwas evaluated internally bycomputingthedifferencebetween5hour-longoverlappingarcs,whichshowsanaverageRMSof9.6mm.Anexternalevaluationofthequalitywasobtainedbycomparingourorbitswiththeofficialprojectkinematicandreduced-dynamicsPSOorbitsovertheentiredataspan.Notethatthereisafundamentaldifferencebetweentheorbitreductionapproachesusedfordeterminingthe kinematicPSOs and the orbits estimatedhere.The approach adoptedby theGOCE-Italyteam is a densely-parameterized, fully dynamicalmethod,while the official kinematic PSOorbitsarecomputedbasedonakinematicalapproach,whichissubjecttoleavingwidegapsinthereconstructedephemerisduetoitslackofcapabilitytopredictandinterpolateorbitalarcs.The 3DRMS difference between the orbits determined in this investigation and the officialPSO solutions is below 10 cm for about 70% and 75%of the daily arcswhen compared,respectively,withthekinematicandreduced-dynamicsversions.Inparticular,dayswithapost-fitRMS higherthan3cmalsoexhibitafairlyhighRMS ofthedistancefrombothkinematic

Fully dynamic approach for GOCE precise orbit determination Boll. Geof. Teor. Appl., 54, 367-384

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andreduced-dynamicsPSOorbits.Amongtheorbitsestimatedwithagoodpost-fitRMS(RMS< 10mm),about30%and15%respectivelydiffer(3DRMS> 6cm)fromthekinematicandreduced-dynamicsPSOsolutions,sometimesreachingapositiondifferenceRMSuptoseveralmetres.Thisresultismainlycausedbysparsenessintheobservationsavailableforprocessingandtoanon-modeledswitchoftheGPSreceiverantenna.

Theresultsobtainedareencouraging,butfurtherinvestigationisneededtoovercomeseveralweakpointsintheanalysis.Inparticular,thenextphaseoftheinvestigationwillmakeuseofthethrusthistoryprofileaswellasoftheattitudeandmasshistoryprofiles.Togethertheseadditionswill contribute to improving the forcemodeling in such away as to resolve the paradoxicalresultofpositivedragcoefficientsduringtheorbitraisingphaseandexplaintheapparentsignalintheradiationpressurecoefficienttimeseries.TheorbitalfitwillalsoimproveduetocorrectmodelingoftheGPSantennainuse.

Thepresentresults,andtheexpectedimprovementfromthenextstudyphase,formasolidbasisforaninvestigationoftherecoveryofoceantideparameterswhichisbeingcarriedoutasaparalleltask.Infact,bothPSOorbittypescanabsorbtidalperturbationsignalthattheNAPEOSforcemodel,howeverdenselyparameterized, cannot.Besides, thepartialswith respect to thetidalparametersarenotavailablefromtheofficialGOCEproducts.Thanksto its lowaltitudeofabout250km,GOCEiswellsuitedtotidalrecovery,exceptforaliasingproblemsstemmingfromSun-synchronicity,throughtheapplicationofclassicalorbitperturbationanalysismethods.ThisrequiresanextensionoftheNAPEOSsolutioncapabilitytoincludeoceantideparameters,whichhasrecentlybeenimplemented(BardellaandCasotto,2012).

Acknowledgements. TheauthorsgratefullyacknowledgetheGOCE-ItalyprojectfundedbytheASI.WethankH.BockandF.Vespefortheircriticalreviewofthemanuscript.AspecialthankgoestoT.SpringerandhisgroupforNAPEOSsoftwaresupport.TheESAisalsoacknowledgedforprovidingtheGOCEdata.

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Corresponding author: Stefano Casotto Dip. Fisica e Astronomia, Università di Padova Via dell’Osservatorio 3, 35122 Padova, Italy Phone: +39 049 8278224; fax +39 049 8278212; e-mail: stefano.casotto@unipd.it