GlobalPositioningSystem:whatitisandhowweuseitformeasuringthe

earth’smovement.May5,2009

References

•  LecturesfromK.Larson’s“IntroductiontoGNSS”http://www.colorado.edu/engineering/ASEN/asen5090/

•  Strang,G.andK.Borre“LinearAlgebra,Geodesy,andGPS”,Wellesley‐CambridgePress,1997

•  Blewitt,G.,“BasicsoftheGPSTechnique:ObservationEquations”,in“GeodeticApplicationsofGPS”

•  http://www.kowoma.de/en/gps/index.htm•  http://www.kemt.fei.tuke.sk/predmety/KEMT559_SK/GPS/GPS_Tutorial_2.pdf

•  LecturenotesfromG.Mattioli’(comp.uark.edu/~mattioli/geol_4733/GPS_signals.ppt)

Basicsofhowitworks

•  Trilateration•  GPSgivesdistanceto4satellites

‐  x,y,z,t‐  Earthcentered,EarthFixed

‐  Whyt?

‐  Whataresomeofreasonswhymeasuringdistanceisdifficult?

‐  Howdoweknowx,y,z,tofsatellites?

GPS:Spacesegment

•  SeveraldifferenttypesofGPSsatellites(BlockI,II,IIA,IIR)

•  Allhaveatomicclocks– Stabilityofatleast10‐13sec1secevery~300,000yrs

•  Dynamicsoforbit?

•  Referencepoint?

OrbitalPerturbations–(centralforceis0.5m/s2)Source Acceleration

m/s2

Perturbation

3 hrs

Type

Earth oblateness (J2 )

5 x 10-5 2 km @ 3 hrs secular + 6 hr

Sun & moon 5 x 10-6 5-150 m @ 3 hrs secular + 12hr

Higher Harmonics 3 x 10-7 5-80 m @ 3 hrs Various

Solar radiation pressure

1 x 10-7 100-800 m @2 days Secular + 3 hr

Ocean & earth tides

1 x 10-9 0-2m @2 days secular + 12hr

Earth albedo pressure

1 x 10-9 1-1.5m @2 days

FromK.Larson

GPS:SpaceSegment

•  24+satellitesinorbit– Cansee4atanytime,anypointonearth

– Satellitesneverdirectlyoverthepoles

– Formostmid‐latitudelocations,satellitestrackmainlynorth‐south

GPS:SatelliteGroundTrack

GPSSignal

•  Satellitetransmitsontwocarrierfrequencies:–  L1(wavelength=19cm)–  L2(wavelength=24.4cm)

•  Transmits3differentcodes/signals–  P(precise)code

•  Chiplength=29.3m–  C/A(courseacquisition)code

•  Chiplength=293m–  Navigationmessage

•  Broadcastephemeris(satelliteorbitalparameters),SVclockcorrections,ionoinfo,SVhealth

GPSSignal

•  Signalphasemodulated:

vs

Amplitudemodulation(AM) Frequencymodulation(FM)

C/AandPcode:PRNCodes

•  PRN=PseudoRandomNoise–  Codeshaverandomnoisecharacteristicsbutarepreciselydefined.

•  Asequenceofzerosandones,eachzerooronereferredtoasa“chip”.–  Calledachipbecausetheycarrynodata.

•  SelectedfromasetofGoldCodes.–  Goldcodesuse2generatorpolynomials.

•  ThreetypesareusedbyGPS–  C/A,PandY

PRNCodes:first100bits

PRNCodeproperties

•  HighAutocorrelationvalueonlyataphaseshiftofzero.

•  MinimalCrossCorrelationtootherPRNcodes,noiseandinterferers.

•  Allowsallsatellitestotransmitatthesamefrequency.

•  PRNCodescarrythenavigationmessageandareusedforacquisition,trackingandranging.

PRNCodeCorrelation

Non‐PRNCodeCorrelation

SchematicofC/A‐codeacquisition

SinceC/A‐codeis1023chipslongandrepeatsevery1/1000s,itisinherentlyambiguousby1msecor~300km.

BASICGPSMEASUREMENT:PSEUDORANGE

( )

= time of reception as observed by the receiver

= time of transmission as generated by the satellite

su

us

c t t

tt

ρ = −

•  Receivermeasuresdifferencebetweentimeoftransmissionandtimeofreceptionbasedoncorrelationofreceivedsignalwithalocalreplica

Themeasuredpseudorangeisnotthetruerangebetweenthesatelliteandreceiver.Thatiswhatweclarifywiththeobservableequation.

PSEUDORANGEOBSERVABLEMODEL

( )( )

1 1 1 1

2 2 2 2

1

2

= pseudorange measured on L1 frequency based on code = pseudorange measured on L2 frequency based on code

= geometrical range from satellit

su

su

R c t t T I M

R c t t T I M

R

ρ ρ ρ

ρ ρ ρ

ρ δ δ ε

ρ δ δ ε

ρ

ρ

= + − + + + +

= + − + + + +

1/ 2

1/ 2

1/ 2

e to user = user/receiver clock error

= satellite clock error = tropospheric delay

= ionospheric delay in code measurement on L1/2= multipath delay in code measurement on L1/2

=

us

s uttTIMρ

ρ

ρ

δ

δ

ε other delay/errors in code measurement on L1/2

CARRIERPHASEMODEL( )( )

1 1 1 1 1 1 1

2 2 2 2 2 2 2

1

2

= carrier phase measured on L1 frequency (C/A or P(Y) parts) = carrier phase measured on L2 frequency = geometrical range fr

su

su

R c t t T I M N

R c t t T I M N

R

ρ φ φ

ρ φ φ

φ λ δ δ λ ε

φ λ δ δ λ ε

φ

φ

= + − + − + + +

= + − + − + + +

1 2

1 2

om satellite to user = user/receiver clock error

= satellite clock error = tropospheric del

code measurementay

, = ionospheric delay in on L1/2, = multipath delay in carrier phase m

us

s uttTI IM Mρ ρ

φ φ

δ

δ

1 2

1 2

1 2

easurement on L1/2, = carrier phase bias or ambiguity, = carrier wavelength, = other delay/errors in carrier phase measurement on L1/2

N N

φ φ

λ λ

ε ε

COMPAREPSEUDORANGEandCARRIERPHASE

•  biastermNdoesnotappearinpseudorange•  ionosphericdelayisequalmagnitudebutoppositesign•  troposphere,geometricrange,clock,andtroposphereerrorsarethesameinboth

•  multipatherrorsaredifferent(phasemultipatherrormuchsmallerthanpseudorange)

•  noisetermsaredifferent(factorof100smallerinphasedata)

( )( )

1 1 1 1

1 1 1 1 1 1 1

su

su

R c t t T I M

R c t t T I M N

ρ ρ ρ

ρ φ φ

ρ δ δ ε

φ λ δ δ λ ε

= + − + + + +

= + − + − + + +

AtmosphericEffects

•  Ionosphere(50‐1000km)– Delayisproportionaltonumberofelectrons

•  Troposphere(~16kmatequator,wherethickest)– Delayisproportionaltotemp,pressure,humidity.

VerticalStructureofAtmosphere

Troposphericeffects•  Lowestregionoftheatmosphere–indexofrefraction=~1.0003at

sealevel

•  Neutralgasesandwatervapor–causesadelaywhichisnotafunctionoffrequencyforGPSsignal

•  Drycomponentcontributes90‐97%

•  Wetcomponentcontributes3‐10%

•  Totalisabout2.5mfor

zenithto25mfor5deg

Atlowerelevationangles,theGPSsignaltravelsthroughmoretroposphere.

Troposphericeffects

DryTroposphereDelay

Saastamoinenmodel:•  P0isthesurfacepressure(millibars)•  φisthelatitude•  histhereceiverheight(m)

Hopfieldmodel:•  hdis43km•  T0istemperature(K)

Mappingfunction:•  E–satelliteelevation

( )3, 02.277 10 1 0.0026cos 2 0.00028z dT h Pφ−= × + +

6 0,

0

77.6 105d

z dP h

TT

−= ×

10.00143sin

tan 0.0445

dmE

E

=+

+

~2.5matsealevel

1(zenith)–10(5deg)

WetTroposphereCorrection

Lesspredictablethandrypart,modeledby:

Saastamoinenmodel:

Hopfieldmodel:

•  hwis12km•  e0ispartialpressureofwatervaporinmbar

Mappingfunction:

3, 0

12552.277 10 0.05z wT eT

− = × +

0, 2

0

0.3735w

z we h

TT

=

10.00035sin

tan 0.017

dmE

E

=+

+

0–80cm

ExamplesofWetZenithDelay

Ionosphereeffects•Pseudorangeislonger–“groupdelay”

•CarrierPhaseisshorter–“phaseadvance”

( )( )

( )( )

( )

1 1 1 1

2 2 2 2

1 1 1 1 1 1 1

1 2 2 2 2 2 2

2

1 1 1 1 1 1 1

1

40.3

sL u L L L

sL u L L L

sL u L L L

sL u L L L

sL u L L L

R c t t I T MP

R c t t I T MP

R N c t t I T MP

R N c t t I T MP

TECI If

R N c t t I T MP

ρ ρ ρ

ρ ρ ρ

φ φ φ

φ φ φ

ρ φ

ρ φ φ

ρ δ δ ε

ρ δ δ ε

λ φ λ δ δ ε

λ φ λ δ δ ε

λ φ λ δ δ ε

λ

= + − + + + +

= + − + + + +

= − + − + + + +

= − + − + + + +

⋅≈ − ≈

= − + − − + + +

( )2 2 2 2 2 2s

L u L L LR N c t t I T MPρ φ φφ λ δ δ ε= − + − − + + +

TEC=TotalElectronContent

28

DeterminingIonosphericDelay

( )

( )

( )( )

22

1 2 12 21 2

21

2 2 12 21 2

2 21 2

2 12 21 2

Ionospheric delay on L1 pseudorange

Ionospheric delay on L2 pseudorange

40.3

L L L

L L L

L L

fIf ffI

f ff fTECf f

ρ

ρ

ρ ρ

ρ ρ

ρ ρ

= −−

= −−

= −−

WherefrequenciesareexpressedinGHz,pseudorangesareinmeters,andTECisinTECU’s(1016electrons/m2)

Ionospheremaps

30

Ionosphere‐freePseudorange

( )2

21 2 12 2

1 22 2

1 2" 3" 1 22 2 2 2

1 2 1 2

1 2

Ionospheric delay on L1 pseudorange

Ionosphere-free pseudorange

2.546 1.546

L L L

IF L L L

IF L L

fIf f

f ff f f f

ρ ρ ρ

ρ ρ ρ ρ

ρ ρ ρ

= −−

= = −− −

= −

Ionosphere‐freepseudorangesaremorenoisythanindividualpseudoranges.

Multipath

•  Reflectedsignals– Canbemitigatedbyantennadesign

– Multipathsignalrepeatswithsatelliteorbitsandsocanberemovedby“siderealfiltering”

StandardPositioningErrorBudgetSingleFrequency DoubleFrequency

EphemerisData 2m 2m

SatelliteClock 2m 2m

Ionosphere 4m 0.5–1m

Troposphere 0.5–1m 0.5–1m

Multipath 0‐2m 0‐2m

UERE 5m 2‐4m

UERE=UserEquivalentRangeError

IntentionalErrorsinGPS

•  S/A:Selectiveavailability–  Errorsinthesatelliteorbitorclock–  TurnedoffMay2,2000

WithSA–95%ofpointswithin45mradius.SAoff,95%ofpointswithin6.3m

•Didn’teffecttheprecisemeasurementsusedfortectonicsthatmuch.Whynot?

IntentionalErrorsinGPS

•  A/S:Anti‐spoofing– EncryptionofthePcode(Ycode)– DifferenttechniquesfordealingwithA/S

•  RecoverL1,L2phase•  Canrecoverpseudorange(rangeestimatedusingP‐code)

•  Generallyworsenssignaltonoiseratio

ASTechnologiesSummaryTable

Trimble4000SSi

AshtechZ‐12&µZ

FromAshjaee&Lorenz,1992

PSEUDORANGEOBSERVABLEMODEL

( )( )

1 1 1 1

2 2 2 2

1

2

= pseudorange measured on L1 frequency based on code = pseudorange measured on L2 frequency based on code

= geometrical range from satellit

su

su

R c t t T I M

R c t t T I M

R

ρ ρ ρ

ρ ρ ρ

ρ δ δ ε

ρ δ δ ε

ρ

ρ

= + − + + + +

= + − + + + +

1/ 2

1/ 2

1/ 2

e to user = user/receiver clock error

= satellite clock error = tropospheric delay

= ionospheric delay in code measurement on L1/2= multipath delay in code measurement on L1/2

=

us

s uttTIMρ

ρ

ρ

δ

δ

ε other delay/errors in code measurement on L1/2

EXAMPLEOFPSEUDORANGE(1)

( )1 1 1 1s

uR c t t T I Mρ ρ ρρ δ δ ε= + − + + + +

EXAMPLEOFPSEUDORANGE(2)

GEOMETRICRANGE

•  Distancebetweenpositionofsatelliteattimeoftransmissionandpositionofreceiverattimeofreception

( ) ( ) ( )2 2 2s s su u uR x x y y z z= − + − + −

PSEUDORANGEminusGEOMETRICRANGE

•  Differenceistypicallydominatedbyreceiverclockorsatelliteclock.

( )1 1 1 1s

uR c t t T I Mρ ρ ρρ δ δ ε− = − + + + +

L1PSEUDORANGE‐L2PSEUDORANGE

•  Differencingpseudorangesontwofrequenciesremovesgeometricaleffects,clocks,troposphere,andsomeionosphere

( )( )

1 1 1 1

2 2 2 2

1 2 1 2 1 2 1 2

su

su

R c t t T I M

R c t t T I M

I I M M

ρ ρ ρ

ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρ δ δ ε

ρ δ δ ε

ρ ρ ε ε

= + − + + + +

= + − + + + +

− = − + − + −

GeometryEffects:DilutionofPrecision(DOP)

GoodGeometry BadGeometry

DilutionofPrecision

€

VDOP =σ h

HDOP = σ n2 +σ e

2

PDOP = σ n2 +σ e

2 +σ h2

TDOP =σ t

GDOP = σ n2 +σ e

2 +σ h2 + c 2σ t

2

Covarianceispurelyafunctionofsatellitegeometry

DilutionofPrecision(VDOP)

Wuhan,China,30latCaseystation,Antarctica,66.3latitude

Positioning

•Mostbasic:solvesystemofrangeequationsfor4unknowns,receiverx,y,z,t

P1=((x1‐x)2+(y1‐y)2+(z1‐z)2)1/2+ct‐ct1

…P4=((x4‐x)2+(y4‐y)2+(z4‐z)2)1/2+ct‐ct4

•  Linearizeproblembyusingareference,orapriori,positionforthereceiver–  Eveninadvancedsoftware,needagoodaprioripositiontogetsolution.

Positioningvs.DifferentialGPS

•  Bydifferencingobservationsattwostationstogetrelativedistance,manycommonerrorssourcesdropout.

•  Thecloserthestations,thebetterthisworks•  Bringsprecisionuptomm,insteadofm.

SingleDifferencing

•Removessatelliteclockerrors•Reducestroposphereandionospheredelaystodifferentialbetweentwosites•Givesyourelativedistancebetweensites,notabsoluteposition

€

ΔLABj = ΔρAB

j + cΔτAB + ΔZABj −ΔIAB

j + ΔBABj

DoubleDifferencing

€

∇ΔLABjk =∇ΔρAB

jk +∇ΔZABjk −∇ΔIAB

jk + λ∇ΔNABjk

•Receiverclockerrorisgone•Randomerrorsareincreased(e.g.,multipath,measurementnoise)•Doubledifferencephaseambiguityisaninteger

€

ΔLABj = ΔρAB

j + cΔτAB + ΔZABj −ΔIAB

j + ΔBABj

ΔLABk = ΔρAB

k + cΔτAB + ΔZABk −ΔIAB

k + ΔBABk

HighprecisionGPSforGeodesy•  Usepreciseorbitproducts(e.g.,IGSorJPL)

•  Usespecializedmodelingsoftware

–  GAMIT/GLOBK

–  GIPSY‐OASIS–  BERNESE

•  Thesesoftwarepackageswill

–  Estimateintegerambiguities•  ReducesrmsofEastcomponentsignificantly

–  Modelphysicalprocessesthateffectprecisepositioning,suchasthosediscussedsofarplus

•  SolidEarthTides•  PolarMotion,LengthofDay

•  Oceanloading•  Relativisticeffects•  Antennaphasecentervariations

HighprecisionGPSforGeodesy•  Producedaily

stationpositionswith2‐3mmhorizontalrepeatability,10mmvertical.

•  Canimprovethesestatsbyremovingcommonmodeerror.