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Subscribe whatwhenhow In Depth Tutorials and Information Differential Correction (GPS and GIS) Part 1 IN WHICH we take a closer look at the subject of GPS accuracy and explore techniques that reduce errors. OVERVIEW GPS Accuracy in General When you record a single position with a good GPS receiver, the position recorded will probably be within 5 to 15 meters horizontally of the true location of the antenna. When a surveyor uses good, surveygrade GPS equipment he or she can locate a point to within a centimeter of its true horizontal position. What are the factors that allow the surveyor to be 1,000 or so times more accurate than you are? This is a complicated subject. The answer includes "very good equipment," "measuring the actual number of waves in the carrier" (as differentiated from interpreting the codes impressed on the carrier), and "spending a lot of time" at each site.1 We can cover only the basics in a topic of this scope. But you will learn how to reduce errors so that you can record a fix to within half a meter to three meters of its true location. One primary method of gaining such accuracy is called "differential correction." Differential Correction in Summary In a nutshell, the differential correction process consists of setting a GPS receiver (called a base station) at a precisely known geographic point. Since the base station knows exactly where its antenna is, it can analyze and record errors in the GPS signals it receives—signals that try to tell it that it is somewhere else. That is, the base station knows the truth, so it can assess the lies being told to it by the GPS signals. These signal errors will be almost equivalent to the signal errors affecting other GPS receivers in the local area, so the accuracy of locations calculated by those other receivers may be improved, dramatically, by information supplied by the base station. Thinking about Error For the logging of a given point, define "error" as the distance between what your GPS receiver records as the position of the antenna and the true position of the antenna. It is useful to dissect the idea of "error." We can speak of error in a horizontal plane and differentiate it from the vertical error. This is important in GPS, because the geometry of the satellites almost always dictates that no Explore New Zealand Temukan Keindahan Alam Kami dan Nikmati Liburan Impian Anda di Sini newzealand.com/NewZealand

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Differential Correction Part 1

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  • 6/29/2015 DifferentialCorrection(GPSandGIS)Part1

    http://whatwhenhow.com/globalpositioningsystemandgis/differentialcorrectiongpsandgispart1/ 1/9

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    whatwhenhow

    InDepthTutorialsandInformation

    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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