Lecture2:GPSPseudorange

GEOS655TectonicGeodesyJeffFreymueller

GPSDesignTimeline•  NAVSTAR=NAVigaEonSystem

withTimingandRanging•  Always-on,instantglobal

posiEoning•  Developmentbeganin1973•  Firstsatelliteslaunched1978•  Userequiptests1980•  1983:KoreanAir007shotdownby

SovietUnion–  Planehadstrayedaconsiderable

distanceintoSovietairspace–  LedUSPres.Reagantomandate

futurecivilianuseofGPS

GPSUserHardware–Old!

GPSUserHardware–Modern!

DifferentModesofUse

Naviga&on•  Instantaneous•  SinglestaEon•  Originalintendeduse•  Accuracy

–  Fewmeters–  Sub-meterw/differenEalcorrecEons

Surveying•  Usuallypost-process•  UsuallymulE-staEon•  Scienceorsurvey•  Accuracy

–  1-2cmatworst–  1-2mmatbest

•  Also“seismology”

BasicPrinciples:Surveying•  Requiresdatafromn≥4satellites,m≥2receivers

–  PointposiEoningapproachesworkwith1receiver•  RequiresconEnuoustrackingoverEme•  Post-processedbutreal-Emebeingdeveloped•  Usepseudorangeandcarrierphasemeasurementsfromeachsatellitetoreceiver

•  OrbitsofsatellitesfixedoresEmated•  ClockerroronsatellitesesEmatedordifferencedout•  EsEmatereceiverposiEon(X,Y,Z)andclockerror•  Modelawidevarietyofpathdelaysandothereffects

PosiEoningByRanging1

A 2D example: If you know you are a certain distance from Boise, your position could be anywhere on the circle.

PosiEoningByRanging2

With two distances, you know you are at one of two points.

PosiEoningByRanging3

With three distances, you know you are in Denver. In 3D the circles become spheres, and with three distances you still

have two possible locations – one on the surface and one out in space.

GPSPosiEoning•  MeasureposiEonbymeasuringrangestosatellites– Afewsatellitescanserveanunlimitednumberofusersontheground,anywhereintheworld

•  Howdoweknowwheresatellitesare?– TheybroadcasttheirposiEons(orbits)inanaviga&onmessage

–  (or)someonegivesuspreciseorbits

•  Measuredrangesarecalledpseudoranges•  High-precisionGPSusesthephaseoftheGPScarriersignaltomeasurechangesinrange

Whycallita“pseudorange”?

•  Rangeisthedistancefromsatellitetoreceiver,pluspathdelays.

•  Pseudorangeisdistancepluseffectsofclockerrors

•  TheterminologyhasoldrootsinnavigaEon.– VLBIandGPSarepseudoranging;SLRisranging

•  Geometricrangeρistruedistance.•  P=ρ+c*(clockerrors)+c*(pathdelays)

EvolvingSatellites

1970s-1980s: smaller, relatively simple 2000s: big, more complex

SatelliteConstellaEon

SatelliteConstellaEonFacts•  Nominally4satellites(SVs)ineachof6equallyspacedorbitalplanes(now5ineachplane).

•  Orbitalplanesinclined55°fromequator.•  NearlycircularorbitsR=26,600km~4RE•  Orbitalperiodis11h58m,twoorbitspersiderealday

•  Siderealdayislengthofdaydefinedbywhenstarsappearinsameplaceinsky– DiffersfromrotaEonaldaybecauseofmoEonofeartharoundthesun.

Orbits

•  CanesEmateorbitsorfixorbitstopre-determinedvalues•  RepresentaEonoforbit

–  Broadcast:Keplerianelements+Eme-dependentcorrecEons–  TabularfileofXYZsatelliteposiEons–  Trajectory:iniEalcondiEons+integrateequaEonsofmoEon

(neededtoesEmateorbits)

•  InpracEce,highlypreciseorbitsareavailablefromtheIGS–  Ultra-Rapid:includespredict-aheadforrealEmeuse–  Rapid:Availablenextday–  Final:Availablein<2weeks

KeplerianElements•  AnellipEcalorbitandtheposiEonofabodycanbedescribedby6parameters(Keplerianelements)–  Semi-majoraxis–  Eccentricity–  Fouranglesshownatles

•  Earthisnotapointmass,sosatelliteorbitsarenotexactlyellipEcal.–  Otherforcesalsoperturborbit

SatelliteGroundTracks

24hoursofGPSDataskytracks

Southern California Fairbanks

These are the paths you would see in the sky if you could see the satellites

GPSSignalStructure•  ThreefrequenciesatL-band,L1,L2,andL5

–  L1at154*10.23MHz(~19cm)–  L2at120*10.23MHz(~24cm)–  L5at115*10.23MHz(~25cm)

•  CodesModulated(phasemodulaEon)ontoeachcarrier–  P-codeat10.23MHzonL1+L2–  C/A(CoarseAcquisiEon)codeat1.023MHzonL1+L2(newL2C)

–  NavigaEonmessageat50bitspersecond

•  PandC/Acodesaretypesofpseudo-randomnoise(PRN)codes

TypesofsignalmodulaEon

Amplitude Modulation (AM)

Frequency Modulation (FM)

Phase Modulation (PM)

SatelliteSignals

PrecisionofObservaEons

•  Code“chiplength”isthedistanceassociatedwitheachbitofthecode.–  C/A:293m

•  Repeatsevery~300km

–  P:29.3m

•  Carrierwavelengthisanalogoustochiplength==2-3ordersofmagnitudemoreprecise

“chip length”

DenialofAccuracy•  USDoDcanreduceaccuracyforreal-Emecivilianusers•  (S/A)SelecEveAvailability–onfrom1990stolate1990s–  Epsilon(introduceerrorsinnavigaEonmessage)–  Dither(introducerapidvariaEoninSVclocks)– Militaryreceivershavespecialchipstoundothis

•  (A/S)AnE-Spoofing–onsince1994–  EncrypEonofP-code–  Prevents“theenemy”fromimitaEng(spoofing)GPSsignal– ModernreceiversgetaroundthisencrypEoninvariousways(includinganear-reverse-engineeringinonecase).

Pseudo-RandomNoise•  Computerscannotgeneratetruerandomnumbers,butcangenerateasequenceofnumberswithrandomstaEsEcalproperEes.– Butthesequencecanberepeatedexactly– BeginwithsomestarEngvalue,thenperformaseriesofoperaEons

•  C/Acodehas1023bits,repeats1000Emespersecond

•  Pcodehasalotofbits,repeatsevery266.4days;eachSVgetsa7-daypieceofcode

CodeCorrelaEonforRanging

Receiver generates a copy of the (known) code and correlates with the received code

PseudorangeObservaEonModel•  ThecorrelaEonEmeshisgivesanesEmateofthetravelEme,whichisthefundamentalpseudorangemeasurement.– TravelEme=(EmeofrecepEon)–(Emeoftransmission)

•  PS=(T–TS)c– T=receiverclockreadingatrecepEon– TS=satelliteclockreadingattransmission– c=speedoflight=299792458m/s

AccounEngforClockBiases•  Clockbiasorclockerror?

– Error==mistake– Error==bias,measurementerror– Error==esEmateofuncertaintyintheabove

•  VLBI“removed”clockerrorsbyusingultra-stablehydrogenmaserclocks– VLBIactuallymodelsclockbiasesasquadraEc

•  GPSmustesEmatereceiverclockbias(andsatelliteclockbiasforhighprecisionwork)

ObservaEonModelwithClocks•  PS=(T–TS)c

–  T=t+τ–  TS=tS+τS–  t,tSaretruereceive,transmitEmes,τareclockerrors

•  SubsEtuEng–  PS=[(t+τ)–(tS+τS)]c–  PS=(t-tS)c+(τ–τS)c–  PS=ρS(t,tS)+(τ–τS)c

•  ρS(t,tS)israngefromreceiveratreceiveEmetosatelliteattransmitEme:

€

ρS (t, t S ) = xS (t S ) − x(t)( )2

+ yS (t S ) − y(t)( )2

+ zS (t S ) − z(t)( )2

|τ|≤1millisecond

|τS|issmall(CesiumorRubidiumclocks)

LightTimeEquaEon•  ThetransmissionEmeis~0.07sec.ToevaluatethegeometricrangeweneedtomapthesatelliteposiEonbacktothetransmissionEme.ButwestartoutknowingonlythereceiveEme.WecansolvethisproblemiteraEvely:

€

t(0)S = t = (T − τ)

t(1)S = t −

ρS (t,t(0)S )

c

t(2)S = t −

ρS (t,t(1)S )

c!

First guess: Transmit time = receive time

Next iteration: Correct for satellite position based on the transmit time estimated from the previous iteration.

SetofSimplifiedObserv.EquaEons

•  Now,generalizetomulEplesatellites.WeuseasuperscripttoidenEfyeachsatellite(don’tconfusewithanexponent).LaterwewillhavetouseasubscripttokeeptrackofmulEplereceivers:– P(1)=[(x(1)–x)2+(y(1)–y)2+(z(1)–z)2]1/2+cτ–cτ(1)– P(2)=[(x(2)–x)2+(y(2)–y)2+(z(2)–z)2]1/2+cτ–cτ(2)– P(3)=[(x(3)–x)2+(y(3)–y)2+(z(3)–z)2]1/2+cτ–cτ(3)– P(4)=[(x(4)–x)2+(y(3)–y)2+(z(3)–z)2]1/2+cτ–cτ(4)

x, y, z, τ: receiver position and clock error

LinearizingNonlinearEquaEons•  Therearesimplewaysto

solvesystemsoflinearequaEons,likematrixinversionorleastsquares.Butwehaveanonlinearproblem.Oneapproachistolinearize,orconstructalinearapproximaEontothenon-linearproblem.WecandothatwithTaylor’stheorem(TaylorSeries)

f(x)=

LinearizingPart2

•  WeapproximatebytakingjustthelineartermsoftheTaylorSeries.

•  Welinearizeaboutapproximatevalues(a,b)

•  ParEalderivaEvesarecomputedat(a,b)

LinearizingOurEquaEons

•  WelinearizeourequaEonsaboutapproximatevalues(x0,y0,z0,τ0)

€

P(x,y,z,τ ) = P(x0,y0,z0,τ 0) +∂P∂x

x − x0( ) +∂P∂y

y − y0( ) +∂P∂z

z − z0( ) +∂P∂τ

τ − τ 0( )

P(x,y,z,τ ) = P(x0,y0,z0,τ 0) +∂P∂x

Δx +∂P∂y

Δy +∂P∂zΔz +

∂P∂τ

Δτ

P(x,y,z,τ ) − P(x0,y0,z0,τ 0) =∂P∂x

Δx +∂P∂y

Δy +∂P∂zΔz +

∂P∂τ

Δτ

Pobserved − Pcomputed =∂P∂x

Δx +∂P∂y

Δy +∂P∂zΔz +

∂P∂τ

Δτ

ObservaEonModel•  TosolvetheseequaEons,weneedtowritethemintheformofanobservaEonmodel.– ObservaEons=Model+measurementnoise

•  Pobserved=P(x,y,z,τ)+v

€

Pobserved = Pcomputed +∂P∂x

Δx +∂P∂y

Δy +∂P∂zΔz +

∂P∂τ

Δτ + v

Pobserved − Pcomputed =∂P∂x

Δx +∂P∂y

Δy +∂P∂zΔz +

∂P∂τ

Δτ + v

ΔP =∂P∂x

∂P∂y

∂P∂z

∂P∂τ

⎛

⎝ ⎜

⎞

⎠ ⎟

ΔxΔyΔzΔτ

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

+ v

MatrixEquaEon

•  ItiseasiertodealwiththisequaEonifwewriteitasamatrixequaEon:

€

ΔP (1)

ΔP (2)

ΔP (3)

ΔP (4 )

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

=

∂P(1)

∂x∂P(1)

∂y∂P(1)

∂z∂P(1)

∂τ∂P(2)

∂x∂P(2)

∂y∂P(2)

∂z∂P(2)

∂τ∂P(3)

∂x∂P(3)

∂y∂P(3)

∂z∂P(3)

∂τ∂P(4 )

∂x∂P(4 )

∂y∂P(4 )

∂z∂P(4 )

∂τ

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

ΔxΔyΔzΔτ

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

+

v(1)

v(2)

v(3)

v(4 )

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

EvaluatetheParEalDerivaEves

•  Thisisosenwriyeninmatrixformlike– b=Ax+v Aiscalledthe“Designmatrix”–  Ifρ(i)=[(x0–x(i))2+(y0–y(i))2+(z0–z(i))2]1/2

€

A =

x0 − x(1)

ρ(1)y0 − y

(1)

ρ(1)z0 − z

(1)

ρ(1)c

x0 − x(2)

ρ(2)y0 − y

(2)

ρ(2)z0 − z

(2)

ρ(2)c

x0 − x(3)

ρ(3)y0 − y

(3)

ρ(3)z0 − z

(3)

ρ(3)c

x0 − x(4 )

ρ(4 )y0 − y

(4 )

ρ(4 )z0 − z

(4 )

ρ(4 )c

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

These have the form of trig functions, and can also be written in terms of the azimuth to the satellite and

the inclination of the satellite above the

horizon.

SolvingtheEquaEons

•  Ifthereare4observaEonsexactly,thenthesystemofequaEonscanbesolvedexactly:– b=Ax+v èx=A-1b(noisev=0isassumed)

•  Ingeneral,wewillhavemorethan4satellitesobservedataEme.Sohowdowefindthe“best”soluEon.Leastsquares!– LeastsquaressoluEonminimizesthesumofsquaresofresiduals,thatis

– FindthexthatgivestheminimumvTv.

LeastSquaresSoluEon

•  TheleastsquaressoluEon,forequallyweighteddata,is– x’=(ATA)-1ATb

•  Thisassumesthat(ATA)-1exists.Itwillexistaslongasthereare4ormoresatelliteslocatedindisEnctdirecEonsinthesky.Twosatelliteslocatedinexactlythesameplacewouldcountasone.

BiasesandErrors•  Supposeweknowwhatthe

measurementerrorsare.HowwouldtheseknownerrorsbiasouresEmatesofposiEon(andclockbias)?

–  vx=(ATA)-1ATv–  Youmightusethistodetermine

whetheranewlydiscoverederrorhasabigimpactonyouresEmatedparameters,butingeneralyouwouldsimplywanttocorrectthedata!

•  Inreality,youdon’tknowthemeasurementerrors,butyoumayknowtheirstaEsEcalproperEes.Forexample,youmayknowthatthemeanmeasurementerroris0,withuncertaintyσ,andthatthemeasurementerrorsfollowaGaussianornormaldistribu&on.

In this case: Expectation: E(v) = 0 Covariance: C = E(vvT) = σ2I

CovarianceMatrix•  Thecovarianceisgivenasa(symmetric)matrix.Fortheproblemwehavejustsolved,– Cx=σ2(ATA)-1

– σ2isthedatanoise.(ATA)-1relatesonlytothegeometry.

•  Orintermsofthecomponents:

€

Cx =σ 2

σ x2 σ xy σ xz σ xτ

σ yx σ y2 σ yz σ yτ

σ zx σ zy σ z2 σ zτ

στx στy στz στ2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

LocalCoordinates

•  YoucanalsotransformtheXYZcoordinatestolocalcoordinates(east,north,height).We’llleavetheequaEonsforthatforlater.Butifyoutakejustthecoordinatespartofthecovariance,youget:

€

CL =σ 2

σ e2 σ en σ eh

σ ne σ n2 σ nh

σ he σ hn σ h2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

DOPs–DiluEonofPrecision•  YourhandheldGPSprobablyreportsanumbercalled“PDOP”,whichstandsfor“PosiEonDiluEonofPrecision”.TheseareotherDOPsaswell,whichallgivemeasuresofhowthesatellitegeometrymapsintoposiEonorEmeprecision.–  VDOP=σh

–  HDOP=(σe2 + σn

2)1/2 –  PDOP=(σe

2 + σn2 + σh

2)1/2 –  GDOP=(σe

2 + σn2 + σh

2 + c2στ2)1/2

–  TDOP=στ•  MulEplyPDOPbymeasurementprecisiontogetuncertaintyin3DposiEon.

PDOP > 5 considered poor