Lecture3:GPSCarrierPhaseGEOS655TectonicGeodesy

JeffFreymueller

ThinkingAboutClocks•  AllofourclocksarereallyperiodicmoFondevices:rotaFonofearth,orbitofearth,pendulum,quartzoscillator,…•  ThereforeFmeisproporFonaltophase•  T(t)=k(φ(t)–φ0)

•  EveryclockkeepsperfectFmefromitsownperspecFve•  KeepinourmindthedisFncFonbetween“true”FmeandFmekeptbysomeclock(moreimportantfortheultra-preciseGPSphasemeasurementsthanforpseudorange).

•  WecandefinetheclockFmeintermsofthenominalfrequencyoftheoscillator,f0:•  T(t)=(φ(t)–φ0)/f0

PhaseTracking•  ReceivermeasureschangesinphaseofcarriersignaloverFme– Firstmustremovecodestorecoverrawphase– ThentrackconFnuousphase,keepingrecordofthenumberofwholecycles

– Phasehasanintegerambiguity(iniFalvalue)

•  Problemsoccurifreceiverlosesphaselockφ

φ

0 1 cycle 2 cycles

MeasurementTrick:BeatPhase•  RemovePRNcode

modulaFonbymulFplyingsignalbycode–removesphaseshi\sandrecoversoriginalcarriersignal

•  Mixreceivedphasewithreferencephasesignal

•  Filterhighfrequencybeatsandmeasurephaseoflowfrequencybeat

MeasuringBeatPhase

•  DopplerShi\shi\seachSVsfrequencyslightly•  ReceivergeneratesreferencesignalatnominalGPSfrequency(fGwithphaseφG)

•  BeatphaseφBandbeatfrequencyfBare–  φB(t)=φR(t)-φG(t)–  fB=fR-fG

•  Beatfrequencyismuchlowerthannominal,easiertomeasurebeatphase,butwecanrecoverallvariaFonsinphaseofthetransmi`edcarriersignalfromthebeatphase

PhaseAmbiguity•  Onedrawbackofthephaseis

thatwecanaddanarbitraryconstantnumberofcyclestothetransmi`edcarriersignal,andwewouldgetexactlythesamebeatphase:•  Φ+N=φR–φG•  ActualrecordedphaseisΦ

•  Also,wemusttrackthephaseconFnuously.IfwelosetrackofthephaseoverFme,andstartover,wegetadifferentN.•  Losingtrackiscalleda“cycleslip”

LossofLock(CycleSlips)•  Ifreceiverlosesphaselock,therewillbeajumpofanintegernumberofcyclesinthephasedata

•  Thismustbedetectedandrepairedbytheanalysisso\ware

•  SlightlydifferentproceduresareusuallyappliedmulFpleFmestofindallofthecycleslips

t

φ

t

φ2πn

Before After

CarrierBeatPhaseModel•  ObservaFonofsatelliteSproducesthephaseobservableΦS

–  ΦS(T)=φ(T)–φS(T)-NS

–  RememberTisthereceiveFmeaccordingtothereceiverclock,andtrueFmereceiveFmet.

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

•  Likewithpseudorange,we’lleventuallyhavetodealwithhowTrelatestoTS

CarrierBeatPhaseModel2•  Wecanusetheclock-Fmeequivalency,T(t)=(φ(t)–φ0)/f0,towritetheobservableintermsofFmes:–  ΦS(T)=f0T+φ0–f0TS–φ0S-NS

–  ΦS(T)=f0(T–TS)+φ0–φ0S-NS

•  Thelastthreetermsareallarbitraryconstantsandcanbecombinedintoasinglebiasterm

•  We’llputthisintermsofdistance(range)insteadofphase,andaddsubscriptstoidenFfyeachstaFon–  LAj(TA)=λ0ΦS(TA)=λ0f0(TA–Tj)+λ0(φ0A–φ0j–Na

j)–  CarrierphasebiasBAj=λ0(φ0A–φ0j–Na

j)–  Alsonotethatλ0f0=c

ObservaFonEquaFons

•  ComparethephaseobservaFonmodelwiththepseudorangemodel:•  LAj(TA)=c(TA–Tj)+Baj•  PAj(TA)=c(TA–Tj)•  Exactlythesameexceptforthephasebias!

•  Weneedtoaddmoretermstodealwiththeclockerrorsandwithpathdelayterms•  LAj(TA)=ρAj(tA,tj)+cτA–cτj+Zaj–IAj+BAj•  PAj(TA)=ρAj(tA,tj)+cτA–cτj+Zaj+IAj•  PathdelaytermsareZforthetroposphere,andIforionosphere.We’llcomebacktotheselateron.

TimeTagBiasandLightTimeEquaFon•  WesFllneedtosolvetheLightTimeEquaFontocomputegeometricrangeρAj(tA,tj)correctly.

•  WhenwedealtwiththisinthepseudorangeequaFons,IsaidwecouldignorethedifferencebetweenthetruereceiveFmetAandthereceiverclock’sreceiveFmeTA–  tA=TA–τA

•  Togoto1mmprecision,weneedtouseanesFmateofτAaccurateto1microsecond.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,whichareapain– MakesforfasteresFmaFon(fewerparametersifyoudonotneedtoesFmateclockerrorateveryobservaFonFme)

–  Phasebiasparametersreducetointegervalues

•  Disadvantages–  Requiresamethodtoselectdifferences–  RequiresaddiFonalbookkeeping–  NotaFongetsmessy

SingleDifference•  ReceiversAandBobserve:

–  LAj=ρAj+cτA–cτj+BAj–  LBj=ρBj+cτB–cτj+BBj

•  FormadifferencebetweenreceiversAandB–  ΔLABj=LAj–LBj–  ΔLABj=(ρAj–ρBj)+(cτA–cτB)+(BAj–BBj)–  ΔLABj=ΔρABj+cΔτAB+ΔBABj

•  WeuseΔtoindicateadifferencebetweengroundreceivers.

ρAj

ρBj

satellite j

receiver A records LA

j receiver B records LB

j

DoubleDifference

•  ReceiversAandBobserve:•  LAj=ρAj+cτA–cτj+BAj•  LAk=ρAk+cτA–cτk+BAk•  LBj=ρBj+cτB–cτj+BBj•  LBk=ρBk+cτB–cτk+BBk

•  FormthesingledifferencebetweenreceiversAandB,andthendifferencebetweensatellitesjandk:•  ΔΔLABjk=ΔLABj–ΔLABk•  ΔΔLABjk=(ΔρABj–ΔρABk)+(ΔBABj–ΔBABk)•  ΔΔLABjk=ΔΔρABjk+ΔΔBABjk

•  WeuseΔΔtoindicateadoubledifference.

ρAj

ρBj

satellite j

receiver A records LA

j records LA

k

receiver B records LB

j records LB

k

ρAk

ρBk

satellite k

Double-DifferencedAmbiguity•  Thedouble-differencedphaseambiguiFesbecomeexactlyintegers:–  ΔΔBABjk=ΔBABj–ΔBABk=λ0ΔNAB

jk

•  EachBhasthreeparts:–  BAj=λ0(NA

j+φ0A–φ0j)–  Thereceiverbiasφ0Aiscommontoallsatellites,anddifferencesoutlikethereceiverclock.

–  Thesatellitebiasφ0jiscommontoallreceivers,anddifferencesoutlikethesatelliteclock.

•  Therearesomeclevertechniquestoremovethephaseambiguitycompletely,ifyoucanresolveittothecorrectinteger.

TripleDifference•  Thetripledifferenceaddsa

differenceinFme.Ifyoudifferencethedouble-differencedobservaFonsfromoneepochinFmetothoseofthepreviousepoch,yougetatripledifference:•  ΔΔΔLABjk(i+I,i)=ΔΔΔρABjk(i+I,i)

•  WeuseΔΔΔtoindicateadoubledifference.ThetripledifferencealsoremovesmostofthegeometricstrengthfromGPS,soitproducesonlyweaklydeterminedposiFons.ButthetripledifferencecanbeappliedtokinemaFcproblems.

ρAj

ρBj

satellite j

receiver A records LA

j records LA

k

receiver B records LB

j records LB

k

ρAk

ρBk

satellite k

FinalNotesonDifferencing•  Whenyoudifferencebetweenreceivers,thenineffectyouarenowesFmaFngthebaselinevectorbetweenthetworeceivers,ratherthanthetwoposiFons.

•  Youhavetotakesomecareinchoosingwhichdifferencestouse–  CannotuselinearlydependentobservaFons– Mustbecarefulinchoosingbaselinestodifference,satellitestodifferencebetween.

•  Eachso\waredoesitdifferently•  Someso\waresdonotdifferenceatall,butesFmateclockerrorsinstead.

DesignMatrixforPhase•  Let’slookatonerowofthedesignmatrixforacaseof2staFons,4satellites.WewillconsiderstaFonAtobefixed,andesFmatestaFonB.– 3doubledifferencedobservaFons– 3doubledifferenceambiguiFes– 6parameters!Need≥2epochsofdata!

€

AAB(24 )(i) =

∂LAB( 24 )

(i)∂xB

∂LAB( 24 )

(i)∂yB

∂LAB( 24 )

(i)∂zB

∂LAB( 24 )

(i)∂NAB

21∂LAB

( 24 )

(i)∂NAB

23∂LAB

( 24 )

(i)∂NAB

24

⎛

⎝ ⎜

⎞

⎠ ⎟

ΔxΔyΔzNAB21

NAB23

NAB24

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

AAB(24 )(i) =

∂ρAB( 24 )

(i)∂xB

∂ρAB( 24 )

(i)∂yB

∂ρAB( 24 )

(i)∂zB

0 0 −λ0⎛

⎝ ⎜

⎞

⎠ ⎟

DesignMatrixforPhase2•  Thephasedesignmatrixisthesameasthepseudorange

version,exceptforthephaseambiguiFes.Thedouble-differencedversioninthelastslideresolvesto:

•  Ifwecomparethedouble-differencedversiontotheundifferencedversion:€

∂ρAB( 24 )

(i)∂xB

=∂∂xB

ρA(2)(i) − ρB

(2)(i) − ρA(4 )(i) + ρB

(4 )(i)( )

∂ρAB( 24 )

(i)∂xB

=∂ρB

(4 )(i)∂xB

−∂ρB

(2)(i)∂xB

∂ρAB( 24 )

(i)∂xB

=xB 0 − x

(4 )(i)ρB(4 )(i)

−xB 0 − x

(2)(i)ρB(2)(i)

€

(undifferenced) (double − differenced)xB 0 − x

(4 )(i)ρB(4 )(i)

⇒xB 0 − x

(4 )(i)ρB(4 )(i)

−xB 0 − x

(2)(i)ρB(2)(i)

WeightedLeastSquares•  ToproperlyesFmatetheposiFonwithdifferencing,youhavetoaccountforthefactthatthemeasurementsarenowcorrelated–  BecausetheobservaFonfromstaFonA,satellitejmaybeusedinmorethanonedifferencedobservaFon.

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

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W =

w11 w12 … w1nw21 w22 ! w2n

" " # "wn1 wn2 ! wnn

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

€

W =σ−2

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

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

WeightedLeastSquaresSoluFon

•  TheweightmatrixWisgoingtobetheinverseofthedatacovariancematrixW=C-1

•  TheweightedleastsquaressoluFonlooksverysimilartotheunweightedversion– Unweighted: x’=(ATA)-1ATb– Weighted:x’=(ATWA)-1ATWb

•  Themodelcovarianceisalsoverysimilar– Unweighted: Cx=(ATA)-1– Weighted: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

linearcombinaFonofthedataatthetwofrequencies–  LC=-f12/(f22–f12)L1+f22/(f22–f12)L2–  PC=-f12/(f22–f12)P1+f22/(f22–f12)P2

•  Tryit:ForL1andL2,thebiasesare(40.3TEC/f12,40.3TEC/f22)•  Notethatthetwocoefficientssumto1.Theyhavevaluesof

approximately(-1.54,2.54)•  Thisremovesallionosphericeffectsexceptfora1/f4

dependence.Therearenow“second-orderionosphere”modelscomingintouse,whichhaveanimpactonposiFonsatthefewmmlevelorless.

Ionosphere-OnlyCombinaFon

•  IfonelinearcombinaFonofthedatacanremovetheionosphere,thentheremustbeanorthogonalcombinaFonthathasnothingbuttheionosphere:

•  L1–L2=–40.3TEC(1/f12–1/f22)+B1–B2•  P1–P2=–40.3TEC(1/f12–1/f22)

–  HereB1andB2arethephaseambiguiFes–  Allothertermscancel:LAj(TA)=ρAj(tA,tj)+cτA–cτj+Zaj–IAj+BAj

•  ThiscombinaFonmaybeusedforionosphericstudies,butalsotostudywavespropagaFnginionopsherefromearthquakesorvolcanicexplosions.

IonosphericWavesfromanExplosionatMontserrat

Dautermann et al., 2009, JGR (doi:10.1029/2008JB005722)

PhysicalMechanism•  Volcanicexplosionsorshallowearthquakes

triggeracousFcandgravitywavesthatpropagateintheatmosphereatinfrasonicspeeds.

•  Ationosphericheights,couplingbetweenneutralparFclesandfreeelectronsinducesvariaFonsofelectrondensitydetectablewithdual-frequencyGlobalPosiFoningSystem(GPS)measurements.

•  Raytracingofneutralatmopressurewave•  TotalacousFcenergyreleaseof1.53×1010Jfor

theprimaryexplosioneventatSoufrièreHillsVolcanoassociatedwiththepeakdomecollapse.

•  Thismethodcanbeappliedtoanyexplosionofsufficientmagnitude,providedGPSdataareavailableatneartomediumrangefromthesource.

SomeOtherImportantModels

•  Troposphericdelay(esFmated)–  Hopfield,Saastamoinen,Lanyi,Niell,GMF,VMF1

•  EarthFdes(wellknown)–  Upto~70cmpeaktopeak

•  OceanTidalLoading–  ResponseofsolidearthtochangingloadofoceanFdes

•  AntennaPhaseCentervariaFonswithelevaFon–  Phasecenteristhepointontheantennathatweactuallymeasuredistancesto

–  Itisanimaginarypointinspace,notaphysicalpoint

Vertical Solid Earth Tide - over one month

K. Larson, University of Colorado

Troposphere•  TroposphericpathdelayaffectsbothfrequenciesidenFcally.

Ithastwocomponents.–  “Dry”delay:duetoairmass(~proporFonaltopressure)–  “Wet”delay:duetointegratedwatervaporalongpath

•  DelayinbothcasesislargestatlowelevaFonsabovethehorizon,becausethepathlengththroughatmosphereislongerthere.

•  InpracFce,weesFmateazenithdelay,andusea“mappingfuncFon”ofelevaFonangletomapthistolowerelevaFons–  MappingfuncFonis~1/sin(e)

e

To SV

•  DetailedformofmappingfuncFonisaconFnuedfracFon:

–  Approx.forlayeredatmosphere

•  PathdelayatsomeelevaFonangleeisZTD*mf(e)

•  VMFmappingfuncFonsvarywithspaceandFmebasedondistribuFonofwatervapor.

•  GMFisseasonalaverageofVMF

WetTroposphericMappingFuncFonFortaleza, Brazil, 5° elevation

MapofPrecipitableWaterVapor

PWV is proportional to the path delay, given pressure, temperature

OceanTidalLoading•  SolidearthrespondstochangingloadofoceanFdes

•  Displacementslargenearcoast,whereFdalrangeislarge

•  DetailsdependonoceanFdes,coastline

•  AccurateremovaldependsongoodFdalmodels

ComparetoChina

Computed in Center of Mass of Solid Earth frame

FrameUsedforOTLModelsisCriFcal

•  ModernFdalmodelsareverysimilar

•  Cancomputedisplacementsincenterofmassofsolidearth(CE)orcenterofmassofearth+fluids(CM)

•  FrameusedforOTLcomputaFonmakesabigdifference.

ComparisonofFramesCEframe CMframe

AntennaPhaseCenterModels•  Ideally,phasecenterisa

pointinspace.•  Differentforeverytypeof

antenna•  Inreality,thephase

centerdependsontheazimuthandelevaFonofincomingsignal.

•  ModelsassumeazimuthalsymmetryandfitelevaFon-dependence

AmbiguityResoluFon

•  AmbiguityresoluFonisatrickthatcandramaFcallyimproveposiFonqualityforshortsurveysorkinemaFcposiFoning.

•  Ifyouknowtheambiguityisaninteger,andcandeterminewhichinteger,thenyoucanfixtheambiguitytothatintegervalue.–  RemovingtheambiguityparameterdramaFcallyimprovesthestrengthofthedatatobeusedfordeterminingtheposiFon

•  We’lltalkaboutthismoreinthekinemaFcdiscussion