Low-degree gravity change from GPS data of COSMIC and GRACE satellite missions

ContentslistsavailableatSciVerseScienceDirect

Journal

of

Geodynamics

jo u r n al h om ep a ge :h t t p : / / w w w . e l s e v i e r . c o m / l o c a t e / j o g

Low-degree

gravity

change

from

GPS

data

of

COSMIC

and

GRACE

satellite

missions

Tingjung

Lin

a

_,

_Cheinway

_Hwang

a,∗

_,

_Tzu-Pang

_Tseng

a,c

_,

_B.F.

_Chao

b a_{DeptofCivilEngineering,NationalChiaoTungUniversity,1001TaHsuehRoad,Hsinchu300,Taiwan} b_{InstituteofEarthSciences,AcademiaSinica,128,Sec.2,AcademiaRoad,Nangang,Taipei115,Taiwan}

c_{SPACEResearchCentre,SchoolofMathematicalandGeoSpatialSciences,RMITUniversity,394-412SwansonStreet,Melbourne3001,Australia}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received7March2011

Receivedinrevisedform12August2011 Accepted12August2011

Available online 19 August 2011 Keywords: FORMOSAT-3/COSMIC GPS GRACE Geoidchange Zonalcoefficient

a

b

s

t

r

a

c

t

Thispaperdemonstratesestimationoftime-varyinggravityharmoniccoefficientsfromGPSdataof COS-MICandGRACEsatellitemissions.ThekinematicorbitsofCOSMICandGRACEaredeterminedtothe cm-levelaccuracy.TheNASAGoddard’sGEODYNIIsoftwareisusedtomodeltheorbitdynamicsof COSMICandGRACE,includingtheeffectofastaticgravityfield.Thesurfaceforcesareestimatedper oneorbitalperiod.Residualorbitsgeneratedfromkinematicandreferenceorbitsserveasobservables todeterminetheharmoniccoefficientsintheweighted-constraintleast-squares.ThemonthlyCOSMIC andGRACEGPSdatafromSeptember2006toDecember2007(16months)areprocessedtoestimate harmoniccoefficientstodegree5.ThegeoidvariationsfromtheGPSandCSRRL04(GRACE)solutions showconsistentpatternsoverspaceandtime,especiallyinregionsofactivehydrologicalchanges.The monthlyGPS-derivedsecondzonalcoefficientcloselyresemblestheSLR-derivedandCSRRL04values, andthirdandfourthzonalcoefficientsresembletheCSRRL04values.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Lowearthorbitsatellites(LEOs)havebecomeabasicand effi-cienttoolfordeterminingglobalgravityfieldanditstimevariation. Anumberof satellitemissionswerelaunchedforthatpurpose, includingtheCHAllengingMinisatellitePayload(CHAMP;Reigber etal.,1996),thedual-satelliteGravityRecoveryandClimate Exper-iment(GRACE;Tapley,1997),andGravityFieldandSteady-State OceanCirculationExplorer(GOCE;ESA,1999).Atthesametime, theConstellationObservingSystemforMeteorology,Ionosphere andClimate(COSMIC)mission,alsoknownasFORMOSAT-3(Chao etal.,2000;Hwangetal.,2008)isalsoabletoyieldtime-varying gravitysignals.

Despitedifferentmeasurementtechniques,onecommon fea-tureof the missions is to useGPS (Global Positioning System) observations for precise orbit determination. Compared to the satellitelaserranging(SLR)techniquethatcanonlyobtain one-dimensional(scalar)distances,GPScoordinatemeasurementsare fully3-dimensionalandalsobeusedforgravityrecovery(Hwang etal.,2008).GPS-determinedprecisekinematicorbitscontainall informationoforbitalperturbationforces,includingthosedueto time-varyinggravitychanges,whichcanbeestimatedifother per-turbationforcesareproperlymodeled.BeforethelaunchofCHAMP

∗ Correspondingauthor.Fax:+88635716257.

E-mailaddress:cheinway@mail.nctu.edu.tw(C.Hwang).

andGRACE,time-varyinggravityfieldsaremainlydeterminedby SLR.Han(2003)usedabout2yearsofCHAMPorbitdatatorecover thetemporalvariationoftheEarthgravityfieldsuptodegreeand order3.FourreleasesofmonthlyGRACEgravityfieldsolutionsup todegreeandorder60orhighersolelyfromGRACEK-bandranging (KBR)andGPSmeasurementshavebeenpublishedbyCSR,GFZand JPL(seehttp://www.csr.utexas.edu/grace;Bettadpur,2007).

Therelease4(RL04)ofGRACEsolutionsisbasedonthe one-stepapproachtomodelthegravityfield,i.e.,tousetherawGPS measurementsdirectlyintheequationsofmotionforestimation ofharmoniccoefficients.TheGGMandEIGENseriesofstatic grav-ityfieldmodels(Tapleyetal.,2004;Reigberetal.,2005;Förste etal.,2006)basedonGRACEsatellite-to-satelliteKBRandGPS mea-surementsarecomputedinthisway.Thetwo-stepapproach,i.e., computingthekinematicand reference(i.e.,dynamic)orbits of LEOsfirstandestimatinggravityfieldsusingsuchorbits,is com-monlyusedforgravityfieldmodeling.Inthesecondstepofthis approachwherethegravityrecoveryiscarriedout,fourmethods maybeemployedbycombiningGPS-derivedorbitsofLEOswith differenttypesofspacemeasurements:(i)Kaula’slinear perturba-tiontheory(Kaula,1966);(ii)directnumericalintegration(Hwang, 2001;Visseretal.,2001;Rowlandsetal.,2002);(iii)energybalance approach(Wolff,1969;Wagner,1983;Jekeli,1999;Visseretal., 2003;Visser,2005)(iv)accelerationapproach(Ditmaretal.,2006). Inthispaper,wewillexperimentwith16monthsofCOSMIC andGRACE GPSdata,fromSeptember2006toDecember 2007, todemonstratethefeasibilityofgravityrecoverysolelyfromGPS 0264-3707/$–seefrontmatter © 2011 Elsevier Ltd. All rights reserved.

datafromthesixCOSMICsatellitesandtwoGRACEsatellites.The solutionwillbebasedonthetwo-stepapproachdescribedabove. GravitychangesfromGPSwillbecomparedwiththosefromtheSLR technique(onlyfordegree2zonalgeopotentialcoefficient)andthe latestGRACEsolutions.

2. Dataprocessing

2.1. COSMICandGRACEkinematicorbitdetermination

In this paper, the kinematic orbits were treated as three-dimensionalrangeobservationsandwereusedforgravityrecovery. With zero-differenced, ionosphere-free GPS phase observables, the precise kinematic orbits of LEOs were determined by the BerneseVersion5.0GPSsoftware(Dachetal.,2007).Thedetail ofGPS-determinedorbitsofGRACEandCOSMICsatellitesusing the kinematic approaches by Bernese have been documented by ˇSvehlaand Rothacher (2004), Jäggietal. (2007)andHwang et al. (2009). In the kinematic approach, epoch-wise parame-ters suchas coordinate components, GPS receiver clock errors and phase ambiguities weredetermined simultaneouslyin one orbit-arc solution. Both high-precision GPS satellite orbits and clock errors were used in kinematic orbit determinations, and were made available by theCenter for Orbit Determination in Europe(CODE,http://www.aiub.unibe.ch/igs.html).The reduced (orsimplified)dynamic orbitwasalsocomputed priorto kine-maticorbitanditservedasaprioriorbitforkinematicorbitand forremoving anomalouskinematicorbitvalues. In thereduced dynamicorbitdetermination,arc-dependentparameterssuchas theinitialstatevector(6Keplerianelements), 9solarradiation coefficientsand3stochasticpulsesintheradial,along-trackand cross-trackdirectionswereestimatedandnumericalintegrations werecarriedoutsubsequentlytodeterminethereduceddynamic orbits.

Ingeneral,theaccuracyofaLEO’skinematicorbitisgoverned bytheGPS-LEOgeometry,thenumberofvisibleGPSsatellites,the attitudeofLEOs,antennaphasevariation,GPSsatelliteorbit accu-racyandGPSclockerror(Byunand Schutz,2001;Hwangetal., 2009).Tsengetal.(2011)showthattheaccuraciesofCOSMICand GRACEkinematicorbitsare3and1cm,respectively,basedonan overlappinganalysis.

TherawGPSkinematicorbitsofCOSMICandGRACEsatellites werecomputedat5-sand10-sintervals,respectively.Becausethe primaryobjective ofthis paperis toestimatelow-degree grav-itychanges, suchhighsamplingrates arenot needed.Thus we compressedtherawkinematicorbitstoone-minutenormal-point

Table1

StandardsfortheorbitdynamicsofCOSMICandGRACEsatellites. Model/parameter Standard

N-body JPLDE-405

Earthgravitymodel GGM03S(70× 70)

Oceantides GOT00.2

Solidearthtides IERSstandard2000

Atmospheredensity Massspectrometerincoherentscatter(MSIS) empiricaldragmodel

Earthradiationpressure Second-degreezonalsphericalharmonicmodel Solarradiationpressure Onecoefficientpercycle

Atmospheredrag Onecoefficientpercycle Generalaccelerations 9parameterspercycle

orbitsusingthemethodofHwangetal.(2008),andtheone-minute

normal-pointkinematicorbitswereactuallyusedforgravity recov-ery.Also, outliersarepresentintherawGPSdataand mustbe removed bya properfilter.As defined in Ditmar et al. (2006), anoutlierhereisakinematicorbitcomponentwhosedifference withthereduced dynamic orbitexceeds 20cm, which is about 2.5timesoftheRMSorbitdifferencebetweentherawkinematic andreduceddynamicorbits.Inmostcases,rawkinematicorbits wererejecteddue tobadattitudedata andpoorreceiver clock resolution.

2.2. ReferencedynamicorbitsforCOSMICandGRACE

Thepurposeofobtainingthedynamicorbits ofCOSMICand GRACEisgeneratingresidualorbits(kinematicminusdynamic), which are then usedfor gravity recovery(Hwanget al.,2008). The reference orbitmodels theeffects of a static gravity field and allother satelliteperturbingforces, excluding theeffectof temporal gravity. In this paper,thereference orbits of COSMIC and GRACEweredetermined bytheNASAsoftware GEODYN II (Pavlisetal.,1996);thestandardsfortheorbitdynamicsaregiven in Table1. Thestatic gravityfield is described bytheGGM03S model (Tapley et al., 2007) based onfour years (January 2003 through December 2006) of GRACE KBR and GPS data.For the non-gravitational perturbing forces, we solved for coefficients of atmosphericdrag,radiation andgeneralaccelerations (Pavlis etal.,1996)alongtheradial,along-trackandcross-trackdirections perorbitalperiodusingtheCOSMICandGRACEkinematicorbits. The reference dynamic orbit is critical tothe determination of time-varyingcoefficientsfromtheresidualorbit.Agoodreference orbitdepends ongood model ofthe staticgravity field and all otherperturbing forcesactingonCOSMICandGRACEsatellites. Table2

NumbersofdailyobservationfilesanddailyusablekinematicorbitfilesfromSeptember2006toDecember2007.

Month FM1 FM2 FM3 FM4 FM5 FM6 GRA GRB 2006.9 26a_/26b _{15/14 26/26 27/27 29/29 23/23 30/30 30/30} 2006.10 27/24 30/27 27/27 28/25 28/28 25/24 31/31 31/31 2006.11 28/28 16/16 30/29 29/29 28/27 29/25 30/30 30/30 2006.12 27/27 26/26 26/26 29/29 29/29 22/21 31/31 28/28 2007.1 29/29 30/29 27/27 29/29 29/28 20/20 31/31 31/31 2007.2 26/26 27/27 28/27 28/28 28/28 16/14 28/28 28/28 2007.3 29/29 6/6 31/31 28/23 30/30 30/30 31/31 31/31 2007.4 30/29 13/13 30/29 23/18 29/29 20/20 30/30 30/30 2007.5 31/30 12/10 30/28 23/21 30/29 31/31 31/31 31/31 2007.6 30/30 22/21 25/25 30/30 30/30 26/26 30/30 30/30 2007.7 30/30 29/29 16/14 30/30 31/27 31/31 31/31 30/30 2007.8 31/31 18/18 17/17 30/29 31/30 29/28 31/31 31/31 2007.9 28/27 8/8 7/7 30/30 29/28 7/7 30/30 30/30 2007.10 28/15 27/27 21/21 31/31 31/31 0/0 31/31 31/31 2007.11 29/28 13/13 7/4 30/30 28/26 12/12 30/30 30/30 2007.12 27/27 27/27 23/23 31/29 29/27 28/27 31/31 30/30

a_{Numberofdailyobservationfiles.} b_{Numberofdailyusablekinematicorbitfiles.}

Table3

MonthlyRMSdifferencesbetweenreferenceandkinematicorbitsfromSeptember 2006toDecember2007(unit:cm).

Satellite Radial Alone-track Cross-track

FM1 7.24 6.96 6.66 FM2 7.02 6.76 6.46 FM3 7.30 7.00 6.78 FM4 7.25 6.95 6.68 FM5 7.00 6.73 6.31 FM6 6.88 6.59 6.33 GRA 6.28 6.26 5.01 GRB 6.38 6.38 5.42

Several experiments have been made in several previous

publications,e.g.,Hwang(2001)andHwangetal.(2008),basedon

simulateddatawithknowntime-varyingcoefficients.Thecurrent procedureusedinthispaperisoptimizedbasedontheresultsof thesesimulations.

Table2showsthenumberofGPS dailyfilesforthesix COS-MICsatellitesand theGRACEA andBsatellitesfor each ofthe

monthsfromSeptember2006toDecember2007.Thenumbers ofdailyusablekinematicorbitfilesarealsogiveninTable2.In general,COSMICcannot deliverfull-monthdataand full-month usable kinematic orbits. By contrast, the monthly GRACE GPS dataarealmostcomplete.Theunusablekinematicorbitdataare mostlydue topoorattitudecontrol orGPSobservationquality, or simplymissing observations.Fig.1 shows themonthlyRMS differencesbetweenthereferenceandkinematicorbits of COS-MICandGRACEsatellitesintheradial,along-trackandcross-track directions. The monthly RMS differences in these three direc-tionsfor theCOSMICandGRACEsatellites arelistedinTable3. Table4showsstatisticsofmonthlystandarderrorsofnormalpoint orbits.

2.3. Recoveringtemporalgravityfromresidualorbit

Theresidualorbitisa functionalofthetemporalgravityand isregardedasobservabletoestimatethelatter.Hereweusethe procedureandmethoddescribedinHwangetal.(2008)toestimate

Fig.1.MonthlyRMSdifferencesbetweendynamicandkinematicorbitsofCOSMICandGRACEsatellitesinradial(top),along-trackandcross-track(bottom)directionsfrom September2006toDecember2007.

Table4

Statisticsofmonthlystandarderrorsofnormalpointorbits(unit:cm).

Satellite Max. Mean Min.

FM1 2.00 1.81 1.51 FM2 1.94 1.75 1.46 FM3 2.02 1.82 1.55 FM4 2.00 1.84 1.51 FM5 1.94 1.74 1.42 FM6 1.90 1.70 1.43 GRA 1.81 1.51 1.15 GRB 1.84 1.55 1.24

thetime-variablegravity.First,thethreecomponentsofaresidual

orbitxiareexpressedas(Hwang,2001):

xi= 6



k=1 ci ksk( ¯Cnm,¯Snm)+ri+εi, i=1,2,3 (1) where n,m: spherical harmonicdegreeand order. i:variations correspondingtothethreecomponentsintheradial,along-track andcross-track(RTN)directions,respectively.sk:Keplerian vari-ations, or the perturbations in the six Keplerian elements as functionalsofchangesofharmoniccoefficients ¯Cnm,¯Snm.c_ki: coefficientfortransformingKeplerianvariationstoRTNvariations (Hwang,2001).εi:noiseofGPS-determinedorbit.ri:expression tocompensatethedeficiencyoftheKeplerianvariationsin model-ingthetemporalgravity.

Inthispaper,weusedthefollowingmodelforri(Colombo, 1984;Engelis,1987):

ri=a0+a1 cosu+a2sinu+a3cos2u+a4sin2u +a5tcosu+a6tsinu+a7tsin2u+a8tcos2u

+a9t+a10t2 (2)

whereuisargumentoflatitude,aktheempiricalcoefficients,andt thetimeelapsedwithrespecttoareferenceepoch.

Thecoefficients ¯Cnm,¯Snmaresolvedforbyweighted least-squareswithaprioriconstraints.Theconstraintsarebasedona modelofdegreevarianceofthetemporalgravitycomputedusing theGRACEmonthlygravitysolutionsofCSRRL04andGGM03S. Specifically,theharmoniccoefficientsofGGM03Sweresubtracted fromthemonthlycoefficientsofCSRRL04fromSeptember2006 toDecember2007toobtainmonthlyresidualgravitycoefficients. Thefollowingaverageddegreevariancesofthemonthlyresidual gravitycoefficientswerethencomputed:

¯ 2 n=_2n1₊₁ n



m=0 ( ¯C2 nm+¯S2nm) (3)

Fig.2showsaverageddegreevariances.Thesesdegreevariances werethenfittedbytheKaularule˛n−ˇ_{,where˛,ˇaretwo} param-etersdescribingthedecayoftemporalgravityfieldwithrespect toharmonicdegree.Theaverageddegreevarianceswereinversely weightedtothecorrespondingdiagonalelements(seeEq.(4))of thenormalequationsformedbytheobservationequationsandthe residualorbitsinEq.(1).

Pcnm=Psnm= 1 2 n (4) 3. Results

WeprocessedtheCOSMICandGRACEGPStrackingdatafrom September2006 toDecember2007 atone monthinterval. The resultistheNCTUgravitysolutioncontainingmonthlyestimates ofthetemporalvariationofthegravityfieldwithrespecttothe

Fig.2. ObservedandmodeleddegreevariancesofCSRRL04harmoniccoefficients. (static)GGM03Smodel.Itisbelievedthat,withGPSdataonly low-degreecoefficientscanbeestimatedwithsufficientconfidence(Xu etal.,2006;Hwangetal.,2008),andingeneralthesignal-to-noise ratiosofGPS-derivedharmoniccoefficientsarelargerthan1only fordegreesbelow10.Therefore,inthispaperweadopteddegree 5asthemaximumdegreeofharmonicexpansioninthegravity solution.

Fig.3shows selectedmonthsofgeoidvariationsconstructed fromCSRRL04uptodegree5(leftcolumn),ascomparedtothose fromtheNCTUsolution(rightcolumn).Ingeneral,theNCTUgeoid variationsshowhighsandlowssimilartotheCSRRL04results.Both resultsshowcleargravityvariationsoverareasoflargehydrological variationssuchastheAmazon,northernIndia,andcentralAfrica. Herethemaximumvariationsoccurinspring(April)andautumn (SeptembertoOctober)andthispatternisconsistentfromoneyear toanother.

However,thereexistdeviationsbetweenthetwosolutionsin thegeoid variations in certain months.For example, theNCTU monthlygeoid variationsin January,April and Octoberof 2007 containsomeartifactsatlatitudeshigherthan72◦,whichisthe inclinationangleofCOSMIC,orthemaximumlatitudecoveredby COSMIC(seebelow).Otherwise,themagnitudesofgeoidvariation signalsofNCTUsolutionsforFebruary,AprilandMayof2007 dis-agreewithCSRRL04solutionsduetolargedifferenceinthesecond zonalcoefficients.

WemakefurtherevaluationsparticularlyfortheGPS-derived zonalharmoniccoefficients.TheconventionalJnandthefully nor-malizedzonalharmoniccoefficient ¯Cn0arerelatedby

Jn=−Cn0=−



2n+1 ¯Cn0. (5)

Ries et al. (2008) has shown that the GRACE data are not conducive to estimation of the second zonal temporal coeffi-cient  ¯C20, mainly because of the polar orbit design and the presence of several long-period tidal aliases. The combination of satellite data of different inclinations such as COSMIC and GRACE will not only improve the accuracy of zonal harmonic coefficients,butalsothetesseralcoefficients(Zhengetal.,2008). Fig.4showsthemonthly ¯C20valuesandtheirstandarderrors from the NCTU solution in comparison to the CSR RL04 and SLR results (the monthly SLR  ¯C20 are from the Jet Propul-sion Laboratory GRACE ftp://podaac.jpl.nasa.gov/grace/doc/TN-05C20SLR.txt)(ChengandTapley,2004),over theperiod from September2006toDecember2007.Wefoundrelativelylarge dif-ferencesof ¯C20inApril,SeptemberandOctoberof2007.Fig.5 showsthecorrespondingrelativedifferencesof_{ ¯C}20oftheNCTU andtheCSRRL04coefficientswithrespecttotheSLR-derived ¯C20, showingthebetteragreementoftheNCTUsolution,thandoesthe

Fig.4. Timeseriesof ¯C20(changeofsecondzonalcoefficient)fromNCTU,SLR,and CSRRL04fromSeptember2006toDecember2007.

Fig.5. Relativedifferencesof ¯C20oftheNCTUandCSRRL04coefficientswith respecttotheSLR-derivedcoefficientsfromSeptember2006toDecember2007.

CSRRL04solution,totheSLRsolutionagree.Forthe ¯C30and ¯C40 values,theNCTUandCSRRL04solutionsshowsimilarmagnitudes ofvariationandalmostthesamephases(seeFigs.6and7).

Table5showsthecorrelationcoefficientsamongvarious har-moniccoefficients. For ¯C20,theSLR andNCTU solutionsshow astrongercorrelationthanthat betweentheSLR andCSRRL04

Table5

Correlationcoefficientsbetweenzonalharmoniccoefficientsfromtwosolutions.

Coefficient NCTU-GRACE NCTU-SLR SLR-GRACE

 ¯C20 0.64 0.82 0.76

 ¯C30 0.81 N/A N/A

 ¯C40 0.82 N/A N/A

Fig.6.Timeseriesof ¯C30(changeofthirdzonalcoefficient)fromNCTUandCSR

RL04fromSeptember2006toDecember2007.

Fig.7. Timeseriesof ¯C40(changeoffourthzonalcoefficient)fromNCTUandCSR

RL04fromSeptember2006toDecember2007.

solutions.For ¯C30and ¯C40,theCSRRL04andNCTUsolutions

againshowstrongcorrelationswithSLR.Thelinearratesof ¯C20,

 ¯C30and ¯C40fromNCTU,SLRandCSRRL04arelistedinTable6.

Again,therateof_{ ¯C}20fromNCTUmatchestheSLRresultbetter thantheratefromGRACE.For ¯C30and ¯C40,theamplitudesof theannualvariationsfromtheNCTUandCSRRL04solutionsare (1.196×10−10,1.162×10−10)and(4.549×10−11,5.594×10−11), respectively.ThephaseoftheannualvariationsofNCTUandCSR RL04solutions are(113.41◦,120.22◦)and(152.61◦,144.09◦)for  ¯C30and ¯C40,respectively.ThemagnitudesfromGPSappearto belargerthantheonesfromKBR,andthephasedifferencescan beupto8◦ (for ¯C40).Fig.8 showsthecorrelationcoefficients betweenharmoniccoefficientsfromNCTUandCSRRL04solutions. Thisdeviationispartlycausedbytheshortdatarecordsweusedin thispaper.

Table6

LinearratesofzonalcoefficientsfromNCTU,GRACEandSLRsolutions.

Coefficient NCTU GRACE SLR

 ¯C20 (−1.06±0.86)×10−10 (−1.98±0.86)×10−10 (−0.94±0.45)×10−10

 ¯C30 (−5.13±7.09)×10−11 (−1.58±6.07)×10−11 N/A

Fig.8.Correlationcoefficientsbetweenharmoniccoefficientstodegree5fromNCTUandCSRRL04solutions.

4. Conclusions

Thispaperdemonstrates experimentalmonthlygravity

solu-tionsproducedonthebasis of GPStracking datafromCOSMIC

and GRACE. Due to combining data fromsatellites of different

orbitalinclinations,theNCTU solutionsshowa higheraccuracy

oflow-degree zonalcoefficients thantheGRACE solutions.Due

tomainlymissingGPSdata,deviationsbetweenGPS(NCTU)and

GRACE-derivedmonthlygeoidchanges exist,and inmostcases

theyarelargelyduetothedifferencesinthezonalterms,

espe-ciallythesecondzonalcoefficient.TheGPS-derivedsecond,third

andfourthzonalharmoniccoefficientsareconsistentwiththeCSR

results,andtheircorrelationcoefficientswithGRACEresultsare

0.64,0.81and0.82,respectively.Forthesecondzonalcoefficient,

theGPS(NCTU)solutionshowsa highcorrelationcoefficientof

0.82withtheSLRsolution,andthiscorrelationisstrongerthan

thatbetweentheGRACEandSLR’ssecondzonalcoefficients.This

studyhighlightstheimportanceofusingGPSdatainrecovering

thelow-degreeharmoniccoefficients,especiallythesecondzonal

coefficient.Futureworkwillbetoextendthemonthlysolutions

fromGPStoalongerperiodtoimprovetheaccuracyoftheCOSMIC

kinematicorbitandtoincreasethepercentageofusableGPSdata

fromCOSMIC.

Acknowledgment

ThisresearchissupportedbytheNationalSpaceOrganizationof

Taiwan,undergrantNo.NSPO-S-099010(P).Wethankthereviewer

forhisconstructivereviewstoimprovethepaper.

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