Lecture 3: GPS Carrier · PDF fileLecture 3: GPS Carrier Phase ... Measurement Trick: Beat...

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Transcript of Lecture 3: GPS Carrier · PDF fileLecture 3: GPS Carrier Phase ... Measurement Trick: Beat...

  • Lecture3:GPSCarrierPhaseGEOS655TectonicGeodesy

    JeffFreymueller

  • ThinkingAboutClocks AllofourclocksarereallyperiodicmoFondevices:rotaFonofearth,orbitofearth,pendulum,quartzoscillator, ThereforeFmeisproporFonaltophase T(t)=k((t)0)

    EveryclockkeepsperfectFmefromitsownperspecFve KeepinourmindthedisFncFonbetweentrueFmeandFmekeptbysomeclock(moreimportantfortheultra-preciseGPSphasemeasurementsthanforpseudorange).

    WecandefinetheclockFmeintermsofthenominalfrequencyoftheoscillator,f0: T(t)=((t)0)/f0

  • PhaseTracking ReceivermeasureschangesinphaseofcarriersignaloverFmeFirstmustremovecodestorecoverrawphaseThentrackconFnuousphase,keepingrecordofthenumberofwholecycles

    Phasehasanintegerambiguity(iniFalvalue) Problemsoccurifreceiverlosesphaselock

    0 1 cycle 2 cycles

  • MeasurementTrick:BeatPhase RemovePRNcode

    modulaFonbymulFplyingsignalbycoderemovesphaseshi\sandrecoversoriginalcarriersignal

    Mixreceivedphasewithreferencephasesignal

    Filterhighfrequencybeatsandmeasurephaseoflowfrequencybeat

  • MeasuringBeatPhase

    DopplerShi\shi\seachSVsfrequencyslightly ReceivergeneratesreferencesignalatnominalGPSfrequency(fGwithphaseG)

    BeatphaseBandbeatfrequencyfBare B(t)=R(t)-G(t) fB=fR-fG

    Beatfrequencyismuchlowerthannominal,easiertomeasurebeatphase,butwecanrecoverallvariaFonsinphaseofthetransmi`edcarriersignalfromthebeatphase

  • PhaseAmbiguity Onedrawbackofthephaseis

    thatwecanaddanarbitraryconstantnumberofcyclestothetransmi`edcarriersignal,andwewouldgetexactlythesamebeatphase: +N=RG Actualrecordedphaseis

    Also,wemusttrackthephaseconFnuously.IfwelosetrackofthephaseoverFme,andstartover,wegetadifferentN. Losingtrackiscalledacycleslip

  • LossofLock(CycleSlips) Ifreceiverlosesphaselock,therewillbeajumpofanintegernumberofcyclesinthephasedata

    Thismustbedetectedandrepairedbytheanalysisso\ware

    SlightlydifferentproceduresareusuallyappliedmulFpleFmestofindallofthecycleslips

    t

    t

    2n

    Before After

  • CarrierBeatPhaseModel ObservaFonofsatelliteSproducesthephaseobservableS S(T)=(T)S(T)-NS RememberTisthereceiveFmeaccordingtothereceiverclock,andtrueFmereceiveFmet.

    WethenusethefactthatthesignalatthereceiveratreceiveFmeTisthesame(samephase)asthesignalatthetransmi`erattransmitFmeTS S(x,y,z,T)=S(xS,yS,zS,TS)

    Likewithpseudorange,welleventuallyhavetodealwithhowTrelatestoTS

  • CarrierBeatPhaseModel2 Wecanusetheclock-Fmeequivalency,T(t)=((t)0)/f0,towritetheobservableintermsofFmes: S(T)=f0T+0f0TS0S-NS S(T)=f0(TTS)+00S-NS

    Thelastthreetermsareallarbitraryconstantsandcanbecombinedintoasinglebiasterm

    Wellputthisintermsofdistance(range)insteadofphase,andaddsubscriptstoidenFfyeachstaFon LAj(TA)=0S(TA)=0f0(TATj)+0(0A0jNaj) CarrierphasebiasBAj=0(0A0jNaj) Alsonotethat0f0=c

  • ObservaFonEquaFons

    ComparethephaseobservaFonmodelwiththepseudorangemodel: LAj(TA)=c(TATj)+Baj PAj(TA)=c(TATj) Exactlythesameexceptforthephasebias!

    Weneedtoaddmoretermstodealwiththeclockerrorsandwithpathdelayterms LAj(TA)=Aj(tA,tj)+cAcj+ZajIAj+BAj PAj(TA)=Aj(tA,tj)+cAcj+Zaj+IAj PathdelaytermsareZforthetroposphere,andIforionosphere.Wellcomebacktotheselateron.

  • TimeTagBiasandLightTimeEquaFon WesFllneedtosolvetheLightTimeEquaFontocomputegeometricrangeAj(tA,tj)correctly.

    WhenwedealtwiththisinthepseudorangeequaFons,IsaidwecouldignorethedifferencebetweenthetruereceiveFmetAandthereceiverclocksreceiveFmeTA tA=TAA

    Togoto1mmprecision,weneedtouseanesFmateofAaccurateto1microsecond.ThatismoredemandingthanGPSreceiverclockscanactuallybesynchronized. Thisisonereasonforthetemp.stabilizedclocksintheoldTI-4100receiver

  • SeveralSoluFonstoThisProblem

    1. EsFmatethereceiverclockbiasfirstusingapseudorangesoluFon

    2. IteratetheleastsquaressoluFonusingbothphaseandpseudorangedata Startoutwithreceiverclockbias=0,esFmate RepeatesFmaFonassuminglastclockbias

    3. UseanesFmateoftruetransmitFmegivensatelliteclockerrorandapproximatestaFonposiFon RequiresstaFonposiFonto~300m

    4. ExpandrangemodelinaTaylorseries,addingrange-rateterm

    Any of these would work: in general software usually uses #1 or #3

  • DifferencingTechniques Receiverandsatelliteclock

    biasescanberemovedbydifferencingdatafrommulFplesatellitesand/orreceivers. Differencebetweenreceivers

    (singledifference)removessatelliteclock

    Differencebetweensatellites(singledifference)removesreceiverclock

    Differenceofdifferences(doubledifference)removesbothclocks

  • Advantages/DisadvantagesofDifferencing

    Advantage Removesclockerrors,whichareapainMakesforfasteresFmaFon(fewerparametersifyoudonotneedtoesFmateclockerrorateveryobservaFonFme)

    Phasebiasparametersreducetointegervalues Disadvantages

    Requiresamethodtoselectdifferences RequiresaddiFonalbookkeeping NotaFongetsmessy

  • SingleDifference ReceiversAandBobserve:

    LAj=Aj+cAcj+BAj LBj=Bj+cBcj+BBj

    FormadifferencebetweenreceiversAandB LABj=LAjLBj LABj=(AjBj)+(cAcB)+(BAjBBj) LABj=ABj+cAB+BABj

    Weusetoindicateadifferencebetweengroundreceivers.

    Aj Bj

    satellite j

    receiver A records LAj

    receiver B records LBj

  • DoubleDifference

    ReceiversAandBobserve: LAj=Aj+cAcj+BAj LAk=Ak+cAck+BAk LBj=Bj+cBcj+BBj LBk=Bk+cBck+BBk

    FormthesingledifferencebetweenreceiversAandB,andthendifferencebetweensatellitesjandk: LABjk=LABjLABk LABjk=(ABjABk)+(BABjBABk) LABjk=ABjk+BABjk

    Weusetoindicateadoubledifference.

    Aj

    Bj

    satellite j

    receiver A records LAj records LAk

    receiver B records LBj records LBk

    Ak

    Bk

    satellite k

  • Double-DifferencedAmbiguity Thedouble-differencedphaseambiguiFesbecomeexactlyintegers: BABjk=BABjBABk=0NABjk

    EachBhasthreeparts: BAj=0(NAj+0A0j) Thereceiverbias0Aiscommontoallsatellites,anddifferencesoutlikethereceiverclock.

    Thesatellitebias0jiscommontoallreceivers,anddifferencesoutlikethesatelliteclock.

    Therearesomeclevertechniquestoremovethephaseambiguitycompletely,ifyoucanresolveittothecorrectinteger.

  • TripleDifference Thetripledifferenceaddsa

    differenceinFme.Ifyoudifferencethedouble-differencedobservaFonsfromoneepochinFmetothoseofthepreviousepoch,yougetatripledifference: LABjk(i+I,i)=ABjk(i+I,i)

    Weusetoindicateadoubledifference.ThetripledifferencealsoremovesmostofthegeometricstrengthfromGPS,soitproducesonlyweaklydeterminedposiFons.ButthetripledifferencecanbeappliedtokinemaFcproblems.

    Aj

    Bj

    satellite j

    receiver A records LAj records LAk

    receiver B records LBj records LBk

    Ak

    Bk

    satellite k

  • FinalNotesonDifferencing Whenyoudifferencebetweenreceivers,thenineffectyouarenowesFmaFngthebaselinevectorbetweenthetworeceivers,ratherthanthetwoposiFons.

    Youhavetotakesomecareinchoosingwhichdifferencestouse CannotuselinearlydependentobservaFonsMustbecarefulinchoosingbaselinestodifference,satellitestodifferencebetween.

    Eachso\waredoesitdifferently Someso\waresdonotdifferenceatall,butesFmateclockerrorsinstead.

  • DesignMatrixforPhase Letslookatonerowofthedesignmatrixforacaseof2staFons,4satellites.WewillconsiderstaFonAtobefixed,andesFmatestaFonB.3doubledifferencedobservaFons3doubledifferenceambiguiFes6parameters!Need2epochsofdata!

    AAB(24 )(i) = LAB

    ( 24 )

    (i)xB

    LAB( 24 )

    (i)yB

    LAB( 24 )

    (i)zB

    LAB( 24 )

    (i)NAB

    21LAB

    ( 24 )

    (i)NAB

    23LAB

    ( 24 )

    (i)NAB

    24

    xyzNAB21

    NAB23

    NAB24

    AAB(24 )(i) = AB

    ( 24 )

    (i)xB

    AB( 24 )

    (i)yB

    AB( 24 )

    (i)zB

    0 0 0

  • DesignMatrixforPhase2 Thephasedesignmatrixisthesameasthepseudorange

    version,exceptforthephaseambiguiFes.Thedouble-differencedversioninthelastslideresolvesto:

    Ifwecomparethedouble-differencedversiontotheundifferencedversion:

    AB( 24 )

    (i)xB

    =xB

    A(2)(i) B

    (2)(i) A(4 )(i) + B

    (4 )(i)( )

    AB( 24 )

    (i)xB

    =B

    (4 )(i)xB

    B

    (2)(i)xB

    AB( 24 )

    (i)xB

    =xB 0 x

    (4 )(i)B(4 )(i)

    xB 0 x

    (2)(i)B(2)(i)

    (undifferenced) (double differenced)xB 0 x

    (4 )(i)B(4 )(i)

    xB 0 x

    (4 )(i)B(4 )(i)

    xB 0 x

    (2)(i)B(2)(i)

  • WeightedLeastSquares ToproperlyesFmatetheposiFonwithdifferencing,youhavetoaccountforthefactthatthemeasurementsarenowcorrelated BecausetheobservaFonfromstaFonA,satellitejmaybeusedinmorethanonedifferencedobservaFon.

    Thefirststepistodefineaweightmatrix.Theweightmatrixmustbesymmetric(wji=wij).Thematrixatrightshowsaweightmatrixforequally-weightedanduncorrelateddata

    W =

    w11 w12 w1nw21 w22 ! w2n" " # "wn1 wn2 ! wnn

    W =2

    1 0 ! 00 1 0 0" " # "0 0 ! 1

  • WeightedLeastSquaresSoluFon

    TheweightmatrixWisgoingtobetheinverseofthedatacovariancematrixW=C-1

    TheweightedleastsquaressoluFonlooksverysimilartotheunweightedversionUnweighted: x=(ATA)-1ATbWeighted:x=(ATWA)-1ATWb

    ThemodelcovarianceisalsoverysimilarUnweighted: Cx=(ATA)-1Weighted:Cx=(ATWA)-1

  • ProductisDifferenFalPosiFon

    Or a set of relative positions (all sites in network relative to each other)

  • XYZtoLLH

  • IonosphericCalibraFon

    ToaverygoodapproximaFon,thepathdelayduetoionosphericrefracFonisproporFonalto1/f2

    Thephaseisadvanced,whilethepseudorangedataaredelayed InformaFontravelsatgroupvelocity

    Specifically,thepathdelayis(40.3/f2)TEC,whereTECisthetotalelectroncontent.Thispathdelaycanbeaslargeasmeters. ThedelaytermIaj=40.3TEC/f2;f=f1forL1,f2forL2

  • Ionosphere-freeCombinaFon Wecanremovetheeffectsoftheionospherebyforminga