6/29/2015 DifferentialCorrection(GPSandGIS)Part1

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DifferentialCorrection(GPSandGIS)Part1

INWHICHwetakeacloserlookatthesubjectofGPSaccuracyandexploretechniquesthatreduceerrors.

OVERVIEW

GPSAccuracyinGeneral

WhenyourecordasinglepositionwithagoodGPSreceiver,thepositionrecordedwillprobablybewithin5to15metershorizontallyofthetruelocationoftheantenna.

Whenasurveyorusesgood,surveygradeGPSequipmentheorshecanlocateapointtowithinacentimeterofitstruehorizontalposition.Whatarethefactorsthatallowthesurveyortobe1,000orsotimesmoreaccuratethanyouare?Thisisacomplicatedsubject.Theanswerincludes"verygoodequipment,""measuringtheactualnumberofwavesinthecarrier"(asdifferentiatedfrominterpretingthecodesimpressedonthecarrier),and"spendingalotoftime"ateachsite.1Wecancoveronlythebasicsinatopicofthisscope.Butyouwilllearnhowtoreduceerrorssothatyoucanrecordafixtowithinhalfametertothreemetersofitstruelocation.Oneprimarymethodofgainingsuchaccuracyiscalled"differentialcorrection."

DifferentialCorrectioninSummary

Inanutshell,thedifferentialcorrectionprocessconsistsofsettingaGPSreceiver(calledabasestation)atapreciselyknowngeographicpoint.Sincethebasestationknowsexactlywhereitsantennais,itcananalyzeandrecorderrorsintheGPSsignalsitreceivessignalsthattrytotellitthatitissomewhereelse.Thatis,thebasestationknowsthetruth,soitcanassesstheliesbeingtoldtoitbytheGPSsignals.ThesesignalerrorswillbealmostequivalenttothesignalerrorsaffectingotherGPSreceiversinthelocalarea,sotheaccuracyoflocationscalculatedbythoseotherreceiversmaybeimproved,dramatically,byinformationsuppliedbythebasestation.

ThinkingaboutError

Fortheloggingofagivenpoint,define"error"asthedistancebetweenwhatyourGPSreceiverrecordsasthepositionoftheantennaandthetruepositionoftheantenna.

Itisusefultodissecttheideaof"error."Wecanspeakoferrorinahorizontalplaneanddifferentiateitfromtheverticalerror.ThisisimportantinGPS,becausethegeometryofthesatellitesalmostalwaysdictatesthatno

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matterwhatwedo,verticalerrorwillalmostalwaysexceedhorizontalerroronornearthesurfaceoftheearth.Thefactthatallthesatellitesarenecessarilyabovethefixbeingtakengenerallymeansthatverticalerrorwillbe1.5to2.5asgreatashorizontalerror.

Anotherusefuldistinctionisbetweenwhatwemightcallrandomerrorandsystematicerror,orbias.Randomerrorsaredeviationsfroma"true"valuethatfollownopredictablepattern.Systematicerrorsdofollowapredictablepattern.Anexamplewillbeillustrative.Supposewehaveamachinedesignedtohurltennisballssothattheylandacertaindistanceawayonasmalltargetpaintedontheground.Ofcourse,noneoftheballswillhitthecenterofthetargetexactlytherewillalwaysbesomeerror.

Whatfactorsmightcauseerrors?Theballsareeachofslightlydifferentweighttheyarenotsymmetricalandwillbeloadedintothemachineindifferentorientations.Sinceitishardtodeterminetheeffectsofthesefactorsontheaccuracyoftheprocess,wesaythefactorsinducerandomerrors.Ifthereareonlyrandomerrorsintheprocess,someballswillhitshortofthecenterofthetarget,somebeyondit,someleft,someright,andsoon.Ifweshoot100ballsfromthemachinewewillseeapatternofstrikesintheareaoftargetwhichappearssomewhatrandom,butwhichclustersaroundthetarget.

Nowsupposethatwehadsetupourmachineanditstargetwhentherewasnowind,butthenaconstantbreezeof10milesperhourbeganblowingfromtherightacrossthepathofflightofthetennisballs.Thiswouldcreateasystematicerror:eachballwouldlandsomewhattotheleftofwhereitwouldhavelandedinthenowindcondition.Wewillstillseearandompatternofhits,buttheaverageofallhitswillbesomewhattotheleftofthetarget.Thisissystematicerrorthe"system,"includingthewind,causesit.Tocorrect,wecouldaimthemachinesomewhattotheright.

Otherexamplesoffactorsthatmightcontributetoerrorsare:asthetemperaturechanges,thecharacteristicsofthemachinemaychangetheatmosphericpressureandtherelativehumidityoftheairwillaffectthedragonaballandsoon.Whetherthesemightberandomerrorsorsystematiconesmightbehardtodetermine.

Generally,randomerrorsarethosecausedbyfactorswecannotmeasureorcontrolsystematicerrorsarethosewecanaccountfor,measure,and,perhaps,correctfor.

FirstLineofDefenseagainstError:Averaging

WhenIimpliedthatthesurveyorcouldbe1,000timesmoreaccuratethantheaveragepersonwithaGPSreceiver,Iwasbeingsomewhatdisingenuous,mostlyforeffect.Iwascomparingasinglereadingwithinexpensiveequipmentwiththeaverageofmanyreadingsfromexpensiveequipment.Thisisnotafaircontrast,sinceyoucanimprovetheaccuracyofthelessexpensiveequipmentbytakingmanyreadingsatafixedpoint.Yourecallthatthestrikesofthetennisballs,withnowind,tendedtoclusteraroundthetarget.GPSreadingstendtoclusteraroundthetruelocation.Wecanusethefactthatlargenumbersofrandomerrorstendtobeselfcanceling.Thatis,theaverageposition(ifyoutakethemeansofmanylatitudes,ofmanylongitudes,ofmanyaltitudes)willbemuchclosertothetruevaluethanthetypicalsinglemeasurement.

OnemeasureofaccuracyofGPSfixesiscalledCircularErrorProbable(CEP).Itistheradiusofacircleexpressedinalinearunit,suchasmeters.Foragivensituation,50%ofthefixeswillfallwithinthecircle,and50%outside.Anothermeasureofaccuracyisbasedontwostandarddeviationsofanormaldistributioncalled2dRMSwhereRMSmeansrootmeansquare.

Ninetyfivepercentofthefixeswillliewithinacirclewiththisradius.

ForNAVSTARGPS,anumberofexperimentssuggestthat50%ofthelatitudeandlongitudefixesyouobtainwithasinglereceiveroperatingbyitself(i.e.,autonomously)willliewithin12metersofthetruepoint.Fiftypercentofthealtitudefixeswillliewithin21meters.The2dRMSradiusis30metershorizontally,and70metersvertically.Thesenumbersassumethatselectiveavailabilityisoff.

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Ingeneral,themorefixesyoutakeandthemoretimeyouspend,thebetteryouraveragewillbe.Ifyouarepreparedtotakedataatonepointforseveralweekstoseveralmonthsyourerrorwillgetdowntoapproximatelyonetotwometers,basicallyduetothelawoflargenumbers.2Thismaynotbeapracticalwaytoreduceerrorinmostapplications.

Anotherapproach,whichisrelatedtoaveraginginadifferentway,istouse"overdetermined"positionfinding.Asyouknow,foursatellitesarerequiredfora3Dfix.Butsupposeyourreceiverhasaccesstofiveormoreatagiventime.Eachsetoffourofthesatellitesavailablewillprovideadifferentopiniononthepositionofthepointbeingsought.Acompromiseagreementbasedonallthesatellitesinputisprobablybetterthanthepositionindicatedbyanyonesetoffour.TheGeoExplorermaybesettocollectdatainthisway.

SourcesofGPSError

NowlookatthespecificsourcesoferrorsinGPSmeasurements.TypicalerrorsourcesandvaluesforreceiversofthePathfinderclassare:

satelliteclocks

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Thereceiverexpectseachsatellitetobeatacertainplaceataparticulargiventime.Everyhourorso,initsdatamessage,thesatellitetellsthereceiverwhereitispredictedtobeattime"t"hence.Ifthisephemerispredictionisincorrectthesatelliteisntwhereitisexpectedtobe,evenbyjustameterortwothenthemeasurementoftherangefromthereceiverantennatothesatellitewillbeincorrect.

ReceiverErrors

Theseerrorsresultfromanumberoffactorsrelatedtoreceiverdesign,cost,andquality.Somereceivers,forexample,cannotexactlymeasureandcomputethedistancetoeachsatellitesimultaneously.TheGeoExplorercantrackuptosixsatellitesatasinglemomentitisasixchannelreceiver.Inanyreceiverthecomputermustworkwithafixednumberofdigitsandisthereforesubjecttocalculationerrors.Thefactis,perfectioninpositioncalculationbycomputersimplyisnotpossible,becausecomputerscannotdoarithmeticonfractionsexactly.(Itistruethatothercomputeroperations,suchasadditionofintegers,areperfect.)

AtmosphericErrors

Formostofitstripfromthesatellitetothereceiverantenna,theGPSsignalenjoysatripthroughthevirtualvacuumof"emptyspace."Halfofthemassoftheearthsatmosphereiswithin3.5milesofsealevel.Virtuallyallofitiswithin100milesofthesurface.Sothesignalgetstogothespeedlimitforelectromagneticradiationformorethan12,000ofthemorethan12,600milesofthetrip.Whenitgetstotheearthsatmosphere,however,thespeeddropsveryslightlybyanamountthatvariessomewhatrandomly.Ofcourse,sincethecalculationoftherangetothesatellitedependsonthespeedofthesignal,achangeinsignalspeedimpliesanerrorindistance,whichproducesanerrorinpositionfinding.

Significantchangesinsignalspeedoccurthroughouttheatmosphere,buttheprimarycontributionstoerrorcomefromtheionosphere,whichcontainschargedparticlesundertheinfluenceoftheearthsmagneticfield,andfromthetropospherethatdensepartoftheatmospherethatwebreath,thatrainsonus,andthatdemonstrateslargevariationsinpressureanddepth.

MoresophisticatedGPSequipmentcan"calculateout"mostoftheionosphericerrorbecauseitconsidersboththefrequenciestransmittedbyeachsatellite.Sincetheionosphereaffectsthedifferentfrequenciesdifferentlyacorrectioncanbecalculated.Troposphericerrors,however,weseemtobeprettymuchstuckwith,especiallyusingthemoderatelyinexpensivecodebasedequipmentavailabletocivilians.Thiswillchangewhenasecondciviliansignalisaddedtothesystem.

MultipathErrors

Thiscanhappenifapartofthesignalisbouncedoffanobject,suchasabuilding.Thearrivaloftwoormorepartsofthesignalatdifferenttimescanconfusethereceiverandproduceafalsereading.Manyreceiversareprogrammedtodisregardthesecondsignal.Butaproblemcanoccurifthedirectsignalisblockedbuttherelatedbouncedsignalisseenbytheantennaandrecorded.

SelectiveAvailabilityAFormer(WeHope)SourceofError

UntilMay2,2000thelionsshareofthe"ErrorBudget"camefromdeliberatecorruptionofthesignalbytheU.S.DepartmentofDefense(DoD).AsIhavesaid,theydidntwantthecivilianGPSreceivertobeabletopinpointitsposition.Infact,intheearlydays,theydidntwantthecivilianworldtoknowGPSexisted.Selectiveavailabilitywasonetechniquethemilitarycouldusetokeepthesystemfrombeingtooaccurate.(Anotheriscalled"antispoofing.")Theerrorswereprobablyinducedbydithering(makinginaccurate)thesignalthattellsthegroundreceivertheexacttimeorbybroadcastingaslightlyfalseephemeris.Obviously,themilitarydidntwanttosaytoomuchaboutitstechniquesformakingthegoodGPSsignals"selectivelyavailable"i.e.,availableonlytomilitaryreceivers.

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TheDoDhad,onoccasion,turnedselectiveavailabilityoffbefore.(TwonotabletimeswereduringtheMideastWarwithIraqandduringthealmostinvasionofHaiti.)

WithSAoffanautonomousreceivercansupplymuchmoreaccuratepositioninformation.The50%measure(CEP)measureisabout12metershorizontallyand21metersvertically,duetotheerrorspreviouslydiscussed.Ninetyfivepercent(2dRMS)ofthefixesfallwithin30metershorizontally,and52metersvertically.

ReducingErrors

Asitturnsout,mostoftheseerrorscanbe"calculatedout"ofthemeasurementsbytheprocesscalleddifferentialcorrection.Tounderstandhowitworks,letmefirstshowyoutheresultsofanexperiment.

Figure413showstwosequencesofpoints(twodifferentfiles)takenwithaGeoExplorerreceiverplacedatasinglelocation,operatingduringtwodifferenttimeperiods.

Theneatstringatrightwasmadeoffixestakenabout6secondsapartthemessattheleftwasmeasuredbytakingpointsevery20secondsduringadifferenttimeperiod.(Thetickmarksare10metersapart.)Youshouldrealizethateachandeveryfixhereisattemptingtoapproximatethesametrueposition.Theyshowupindifferentpositionsbecauseoftheerrorsdiscussedabove.Wedontknowwhatthetruepositionoftheantennawas,butalmostcertainlyeachpointmissestheexact,truepositionbysomeamount.

Figure41.Twofilesofthesamepointdata(GeoExplorer).

Figure42showsdatacollectedatthesametimesasinthepreviousfigurethecomputationsoflocationwerebasedonthesamesetoffoursatellites.Theantennaforthisreceiver(aPathfinderBasic)wasinvirtuallythesamegeographiclocationastheother.

NowlookatacompositeofthetwosetsofdataasshowninFigure43.ThepointsoftheGeoExplorerdata

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aremarkedwithboxesthePathfinderdataaremarkedwithcircles.

Whatisinterestingisthatthetwopatternsoffixesshowremarkablesimilarity.Pairsoffixestakenatthesametimefromthesamesetofsatellitesappearinaboutthesameposition.Thissuggeststhattheerrorforeachassociatedpairofpointsisalmostidentical,eventhoughthefixeshavecomefromdifferentreceiversanddifferentantennas.Andthissuggestssomethingwhichturnsouttobetrue,thoughIhavenotprovedithere:thatreceiverscalculatingfromthesamesignalswillsufferfromalmostthesameerrors,providedthattheantennasare"close."Whatsclose?Tworeceiverswithin500kilometers(300miles)willtendtoshowthesamemagnitudeanddirectionoferrorswithrespecttothetruelocationsoftheirantennas,providedthepositionsarefoundusingthesamesetofsatellites.

Figure42.Samepointaspreviousfigure(PathfinderBasic).

MoreFormally

Theexperimentdescribedabovedemonstratesthatmuchoftheerrorisinherentinthesignalsthatis,theerrorsoccurbeforethesignalsreachthereceiverantenna.

ToseehowthathelpsusremovemostoftheerrorfromaGPSfix,letsfocusonbothasinglepointonEarthssurface(atruepoint,"T"),anditsrepresentationintheGPSreceiver(themeasured,orobserved,point,"O").

SupposewetakeaGPSreceiverantenna,andplaceitpreciselyatthatknownpoint"T"apointthathasbeensurveyedbyexactingmeansandwhosetruepositionisknowntowithinacentimeter.WecallsuchanantennareceiverconfigurationGPSbasestation.

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Figure43.Compositeoffourfilessameantennalocations.

Figure44.Errorvectorfromobservedpointtotruepoint.

Nowconsiderthatanobservation"O"istakenbyabasestationreceiver.Sowehavethreeentitiestoconsider,asshowninFigure44:

thepositionof"T,"

thereading"O,"and

the"difference"between"O"and"T."

Wevedrawnanarrowfromthemeasuredpointtothetruepoint.Thisarrow,whichisshownintwodimensionsbutwhichwouldreallyexistinthree,hasbothalength(calledamagnitude)andadirection.An

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entitythathasmagnitudeanddirectionisknownasavector.Welabelthevector"E,"for"error,"becauseitrepresentstheamountanddirectionbywhichthereadingmissedthetruepoint.UsuallywedontknowE,butherewecancalculateit.Thefollowingdiscussionindicateshow.

Ingeneral,whenwehaveusedGPS,wehaveusedthereportedcoordinates,"O,"asanapproximationof"T."Thevector"E"wasthe(unknown)amountbywhichwemisseddeterminingtheposition"T."Asanequationwecouldwrite:wherewerecord"O"andwedisregard"E"tofindanapproximationof"T."Thatis,thetruecoordinatesaretheobservedcoordinatesminustheerror.Atbest,wecouldestimatethemagnitudebutnotthedirectionof"E."(Itisimportanttorealizethatnoneofthesequantitiesarescalars[simplenumberslike23.5]butarethreedimensionalentities,sothe""signindicatesvectorsubtraction.Theconceptweareattemptingtocommunicatesurvivesthiscomplexity.)

Butifweknow"T"exactly,andofcoursewehavethemeasuredvalue"O,"thenwecanrewritetheaboveequationtofind"E":

Whatgoodisbeingabletocalculate"E"?ItallowsustocorrectthereadingsofotherGPSreceiversintheareathatarecollectingfixesatunknownpoints.

Wedemonstratedabove,withFigures41,42,and43,thatiftwoGPSreceiverantennasareclose,andusethesamesatellites,theywillperceivealmostthesameerrors.Thatis,foranygivenpointatanygivenmoment,"E"willbealmostexactlythesameforbothreceivers.Thus,foranynearbypointreportedbyaGPSreceiveras"o,"itstruevalue"t"canbecloselyapproximatedsimplybyapplyingtheequation:5

Sinceboth"o"and"E"areknown,theerroriseffectivelysubtractedout,resultinginanearlycorrectvaluefor"t,"asshownbyFigure45.

ThistechniqueprovidesanopportunityforcancelingoutmostoftheerrorinaGPSpositionfoundbyanantennathatisclosetoanotherantennawhichisoveraknownpoint.AsImentionedbefore,"close"isabout500kilometersor300miles.

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Figure45.Knowerrorvectorappliedtopointobservedbyrover.

Theformulafortheamountoferroryoumightexpectwithdifferentiallycorrecteddataisdependentonthedistancebetweenthebasestationantennaandtheroverantenna.Aruleofthumbisthatthefixwillbeinerrorbyoneadditionalcentimeterforeachthreekilometersbetweenthetwoantennas.Thisrelationshipisapproximatelylinear:300kilometerswouldproduceerrorofaboutameter.

Nextpost:DifferentialCorrection(GPSandGIS)Part2

Previouspost:ExaminingGPSData(GPSandGIS)Part4

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