lecture 2: gps pseudorangewhy call it a “pseudorange”? • range is the distance from satellite...
TRANSCRIPT
![Page 1: Lecture 2: GPS PseudorangeWhy call it a “pseudorange”? • Range is the distance from satellite to receiver, plus path delays. • Pseudorange is distance plus effects of clock](https://reader034.vdocuments.us/reader034/viewer/2022042318/5f07a2b47e708231d41df8a8/html5/thumbnails/1.jpg)
Lecture2:GPSPseudorange
GEOS655TectonicGeodesyJeffFreymueller
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GPSDesignTimeline• NAVSTAR=NAVigaEonSystem
withTimingandRanging• Always-on,instantglobal
posiEoning• Developmentbeganin1973• Firstsatelliteslaunched1978• Userequiptests1980• 1983:KoreanAir007shotdownby
SovietUnion– Planehadstrayedaconsiderable
distanceintoSovietairspace– LedUSPres.Reagantomandate
futurecivilianuseofGPS
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GPSUserHardware–Old!
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GPSUserHardware–Modern!
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DifferentModesofUse
Naviga&on• Instantaneous• SinglestaEon• Originalintendeduse• Accuracy
– Fewmeters– Sub-meterw/differenEalcorrecEons
Surveying• Usuallypost-process• UsuallymulE-staEon• Scienceorsurvey• Accuracy
– 1-2cmatworst– 1-2mmatbest
• Also“seismology”
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BasicPrinciples:Surveying• Requiresdatafromn≥4satellites,m≥2receivers
– PointposiEoningapproachesworkwith1receiver• RequiresconEnuoustrackingoverEme• Post-processedbutreal-Emebeingdeveloped• Usepseudorangeandcarrierphasemeasurementsfromeachsatellitetoreceiver
• OrbitsofsatellitesfixedoresEmated• ClockerroronsatellitesesEmatedordifferencedout• EsEmatereceiverposiEon(X,Y,Z)andclockerror• Modelawidevarietyofpathdelaysandothereffects
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PosiEoningByRanging1
A 2D example: If you know you are a certain distance from Boise, your position could be anywhere on the circle.
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PosiEoningByRanging2
With two distances, you know you are at one of two points.
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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.
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GPSPosiEoning• MeasureposiEonbymeasuringrangestosatellites– Afewsatellitescanserveanunlimitednumberofusersontheground,anywhereintheworld
• Howdoweknowwheresatellitesare?– TheybroadcasttheirposiEons(orbits)inanaviga&onmessage
– (or)someonegivesuspreciseorbits
• Measuredrangesarecalledpseudoranges• High-precisionGPSusesthephaseoftheGPScarriersignaltomeasurechangesinrange
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Whycallita“pseudorange”?
• Rangeisthedistancefromsatellitetoreceiver,pluspathdelays.
• Pseudorangeisdistancepluseffectsofclockerrors
• TheterminologyhasoldrootsinnavigaEon.– VLBIandGPSarepseudoranging;SLRisranging
• Geometricrangeρistruedistance.• P=ρ+c*(clockerrors)+c*(pathdelays)
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EvolvingSatellites
1970s-1980s: smaller, relatively simple 2000s: big, more complex
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SatelliteConstellaEon
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SatelliteConstellaEonFacts• Nominally4satellites(SVs)ineachof6equallyspacedorbitalplanes(now5ineachplane).
• Orbitalplanesinclined55°fromequator.• NearlycircularorbitsR=26,600km~4RE• Orbitalperiodis11h58m,twoorbitspersiderealday
• Siderealdayislengthofdaydefinedbywhenstarsappearinsameplaceinsky– DiffersfromrotaEonaldaybecauseofmoEonofeartharoundthesun.
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Orbits
• CanesEmateorbitsorfixorbitstopre-determinedvalues• RepresentaEonoforbit
– Broadcast:Keplerianelements+Eme-dependentcorrecEons– TabularfileofXYZsatelliteposiEons– Trajectory:iniEalcondiEons+integrateequaEonsofmoEon
(neededtoesEmateorbits)
• InpracEce,highlypreciseorbitsareavailablefromtheIGS– Ultra-Rapid:includespredict-aheadforrealEmeuse– Rapid:Availablenextday– Final:Availablein<2weeks
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KeplerianElements• AnellipEcalorbitandtheposiEonofabodycanbedescribedby6parameters(Keplerianelements)– Semi-majoraxis– Eccentricity– Fouranglesshownatles
• Earthisnotapointmass,sosatelliteorbitsarenotexactlyellipEcal.– Otherforcesalsoperturborbit
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SatelliteGroundTracks
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24hoursofGPSDataskytracks
Southern California Fairbanks
These are the paths you would see in the sky if you could see the satellites
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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
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TypesofsignalmodulaEon
Amplitude Modulation (AM)
Frequency Modulation (FM)
Phase Modulation (PM)
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SatelliteSignals
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PrecisionofObservaEons
• Code“chiplength”isthedistanceassociatedwitheachbitofthecode.– C/A:293m
• Repeatsevery~300km
– P:29.3m
• Carrierwavelengthisanalogoustochiplength==2-3ordersofmagnitudemoreprecise
“chip length”
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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).
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Pseudo-RandomNoise• Computerscannotgeneratetruerandomnumbers,butcangenerateasequenceofnumberswithrandomstaEsEcalproperEes.– Butthesequencecanberepeatedexactly– BeginwithsomestarEngvalue,thenperformaseriesofoperaEons
• C/Acodehas1023bits,repeats1000Emespersecond
• Pcodehasalotofbits,repeatsevery266.4days;eachSVgetsa7-daypieceofcode
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CodeCorrelaEonforRanging
Receiver generates a copy of the (known) code and correlates with the received code
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PseudorangeObservaEonModel• ThecorrelaEonEmeshisgivesanesEmateofthetravelEme,whichisthefundamentalpseudorangemeasurement.– TravelEme=(EmeofrecepEon)–(Emeoftransmission)
• PS=(T–TS)c– T=receiverclockreadingatrecepEon– TS=satelliteclockreadingattransmission– c=speedoflight=299792458m/s
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AccounEngforClockBiases• Clockbiasorclockerror?
– Error==mistake– Error==bias,measurementerror– Error==esEmateofuncertaintyintheabove
• VLBI“removed”clockerrorsbyusingultra-stablehydrogenmaserclocks– VLBIactuallymodelsclockbiasesasquadraEc
• GPSmustesEmatereceiverclockbias(andsatelliteclockbiasforhighprecisionwork)
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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)
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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.
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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
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LinearizingNonlinearEquaEons• Therearesimplewaysto
solvesystemsoflinearequaEons,likematrixinversionorleastsquares.Butwehaveanonlinearproblem.Oneapproachistolinearize,orconstructalinearapproximaEontothenon-linearproblem.WecandothatwithTaylor’stheorem(TaylorSeries)
f(x)=
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LinearizingPart2
• WeapproximatebytakingjustthelineartermsoftheTaylorSeries.
• Welinearizeaboutapproximatevalues(a,b)
• ParEalderivaEvesarecomputedat(a,b)
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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
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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
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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 )
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
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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.
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SolvingtheEquaEons
• Ifthereare4observaEonsexactly,thenthesystemofequaEonscanbesolvedexactly:– b=Ax+v èx=A-1b(noisev=0isassumed)
• Ingeneral,wewillhavemorethan4satellitesobservedataEme.Sohowdowefindthe“best”soluEon.Leastsquares!– LeastsquaressoluEonminimizesthesumofsquaresofresiduals,thatis
– FindthexthatgivestheminimumvTv.
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LeastSquaresSoluEon
• TheleastsquaressoluEon,forequallyweighteddata,is– x’=(ATA)-1ATb
• Thisassumesthat(ATA)-1exists.Itwillexistaslongasthereare4ormoresatelliteslocatedindisEnctdirecEonsinthesky.Twosatelliteslocatedinexactlythesameplacewouldcountasone.
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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
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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
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
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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
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
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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