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