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Erroranalysis
T.A.HerringM.A.FloydMassachuse(sIns,tuteofTechnology
GAMIT/GLOBK/TRACKShortCourseforGPSDataAnalysisKoreaInsDtuteofGeoscienceandMineralResources(KIGAM)
Daejeon,RepublicofKorea23–27May2016
MaterialfromT.A.Herring,R.W.King,M.A.Floyd(MIT)andS.C.McClusky(nowANU)
IssuesinGPSErrorAnalysis
• Whatarethesourcesoftheerrors?
• Howmuchoftheerrorcanweremovebybe[ermodeling?
• DowehaveenoughinformaDontoinfertheuncertainDesfromthedata?
• WhatmathemaDcaltoolscanweusetorepresenttheerrorsanduncertainDes?
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DeterminingtheUncertainDesofGPSParameterEsDmates
• RigorousesDmateofuncertainDesrequiresfullknowledgeoftheerrorspectrum—bothtemporalandspaDalcorrelaDons(neverpossible)
• SufficientapproximaDonsareoaenavailablebyexaminingDmeseries(phaseand/orposiDon)andreweighDngdata
• Whatevertheassumederrormodelandtoolsusedtoimplementit,externalvalidaDonisimportant
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ToolsforErrorAnalysisinGAMIT/GLOBK
• GAMIT:AUTCLNreweight=Y(default)usesphasermsfromposeitedittoreweightdatawithconstant+elevaDon-dependentterms
• GLOBK– rename(eq_file)_XPSor_XCLtoremoveoutliers– sig_neuaddswhitenoisebystaDonandspan;bestwayto“rescale”therandomnoise
component;alargevaluecanalsosubsDtutefor_XPS/_XCLrenamesforremovingoutliers
– mar_neuaddsrandom-walknoise:principalmethodforcontrollingvelocityuncertainDes– Inthegdlfiles,canrescalevariancesofanenDreh-file:usefulwhencombiningsoluDons
fromwithdifferentsamplingratesorfromdifferentprograms(Bernese,GIPSY)• UDliDes
– tsviewandtsfitcangenerate_XPScommandsgraphicallyorautomaDcally– grwandvrwcangeneratesig_neucommandswithafewkeystrokes– FOGMEx(“realisDcsigma”)algorithmimplementedintsview(MATLAB)andtsfit/ensum;
sh_gen_statsgeneratesmar_neucommandsforglobkbasedonthenoiseesDmates– sh_plotvel(GMT)allowsseongofconfidenceleveloferrorellipses– sh_tshistandsh_velhist(GMT)canbeusedtogeneratehistogramsofDmeseriesand
velociDes.
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SourcesofError
• SignalpropagaDoneffects– Receivernoise– Ionosphericeffects– Signalsca[ering(antennaphasecenter/mulDpath)– Atmosphericdelay(mainlywatervapor)
• UnmodeledmoDonsofthestaDon– Monumentinstability– Loadingofthecrustbyatmosphere,oceans,andsurfacewater
• UnmodeledmoDonsofthesatellites
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Epochs
12345Hours
20
0mm
-20
ElevaDonangleandphaseresidualsforsinglesatellite
CharacterizingPhaseNoise
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Fixedantennas
Walls
Poles
Reinforcedconcretepillars
Deep-bracing
h[p://pbo.unavco.org/instruments/gps/monumentaDon
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TimeseriescharacterisDcs
TimeseriescomponentsobservedposiDon
(linear)velocityterm
iniDalposiDon
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observedposiDon
(linear)velocityterm
annualperiodsinusoid
iniDalposiDon
Timeseriescomponents
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observedposiDon
(linear)velocityterm
annualperiodsinusoid
semi-annualperiodsinusoid
iniDalposiDon
seasonalterm
Timeseriescomponents
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observedposiDon
(linear)velocityterm
annualperiodsinusoid
semi-annualperiodsinusoid
iniDalposiDon
seasonaltermε=3mmwhitenoise
Timeseriescomponents
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“White”noise• Time-independent(uncorrelated)• MagnitudehasconDnuousprobabilityfuncDon,e.g.Gaussian
distribuDon• DirecDonisuniformlyrandom
“True”displacementperDmestepIndependent(“white”)noiseerrorObserveddisplacementaaerDmestept(v=d/t)
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“Colored”noise• Time-dependent(correlated):power-law,first-orderGauss-
Markov,etc• Convergenceto“true”velocityisslowerthanwithwhite
noise,i.e.velocityuncertaintyislarger
“True”displacementperDmestepCorrelated(“colored”)noiseerror*ObserveddisplacementaaerDmestept(v=d/t)*exampleis“randomwalk”(Dme-integratedwhitenoise)
• Mustbetakenintoaccounttoproducemore“realisDc”velociDesThisisstaDsDcalandsDlldoesnotaccountforallother(unmodeled)errorselsewhereintheGPSsystem
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Annualsignalsfromatmosphericandhydrologicalloading,monumenttransla,onand,lt,andantennatemperaturesensi,vityarecommoninGPS,meseries
VelocityErrorsduetoSeasonalSignalsinConDnuousTimeSeries
TheoreDcalanalysisofaconDnuousDmeseriesbyBlewi(andLavallee[2002,2003]
Top:Biasinvelocityfroma1mmsinusoidal
signalin-phaseandwitha90-degreelagwithrespecttothestartofthedataspan
Bo(om:Maximumandrmsvelocitybiasover
allphaseangles– TheminimumbiasisNOTobtainedwith
conDnuousdataspanninganevennumberofyears
– Thebiasbecomessmallaaer3.5yearsofobservaDon
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CharacterizingtheNoiseinDailyPosiDonEsDmates
NotetemporalcorrelaDonsof30-100daysandseasonalterms
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Figure5fromWilliamsetal[2004]:Powerspectrumforcommon-modeerrorintheSOPACregionalSCIGNanalysis.Linesarebest-fitWN+FNmodels(solid=meanampl;dashed=MLE)Notelackoftaperandmisfitforperiods>1yr
SpectralAnalysisoftheTimeSeriestoEsDmateanErrorModel
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SummaryofSpectralAnalysisApproach
• Powerlaw:slopeoflinefittospectrum– 0=whitenoise– -1=flickernoise– -2=randomwalk
• Non-integerspectralindex(e.g.“fracDonwhitenoise”à1>k>-1)
• GooddiscussioninWilliams[2003]
• Problems:– ComputaDonallyintensive– Nomodelcapturesreliablythelowest-frequencypartofthespectrum
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CATS(Williams,2008)
• CreateandAnalyzeTimeSeries• MaximumlikelihoodesDmatorforchosenmodel– IniDalposiDonandvelocity– Seasonalcycles(sumofperiodicterms)[opDonal]– Exponentofpowerlawnoisemodel
• Requiressomelinearalgebralibraries(BLASandLAPACK)tobeinstalledoncomputer(commonnowadays,butcheck!)
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Hector(Bosetal.,2013)• MuchthesameasCATSbutfasteralgorithm• MaximumlikelihoodesDmatorforchosenmodel– IniDalposiDonandvelocity– Seasonalcycles(sumofperiodicterms)[opDonal]– Exponentofpowerlawnoisemodel– Also
• RequiresATLASlinearalgebralibrariestobeinstalledoncomputer
• LinuxpackageavailablebuttrickytoinstallfromsourceduetoATLASrequirement
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sh_cats/sh_hector
• ScriptstoaidbatchprocessingofDmeserieswithCATSorHector
• RequiresCATSand/orHectortobepre-installed
• Outputs– VelociDesin“.vel”-fileformat– Equivalentrandomwalkmagnitudesin“mar_neu”commandsforsourcinginglobkcommandfile
• CantakealongDme!
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WhitenoisevsflickernoisefromMaoetal.[1999]spectralanalysisof23globalstaDons
Short-cut(Maoetal,1998):UsewhitenoisestaDsDcs(wrms)topredicttheflickernoise
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“RealisDcSigma”AlgorithmforVelocityUncertainDes
• MoDvaDon:computaDonalefficiency,handleDmeserieswithvaryinglengthsanddatagaps;obtainamodelthatcanbeusedinglobk
• Concept:Thedeparturefromawhite-noise(sqrtn)reducDoninnoisewithaveragingprovidesameasureofcorrelatednoise.
• ImplementaDon:– Fitthevaluesofchi2vsaveragingDmetotheexponenDalfuncDon
expectedforafirst-orderGauss-Markov(FOGM)process(amplitude,correlaDonDme)
– Usethechi2valueforinfiniteaveragingDmepredictedfromthismodeltoscalethewhite-noisesigmaesDmatesfromtheoriginalfit
– and/or– FitthevaluestoaFOGMwithinfiniteaveragingDme(i.e.,random
walk)andusetheseesDmatesasinputtoglobk(mar_neucommand)
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Extrapolatedvariance(FOGMEx)• Forindependentnoise,variance∝1/√Ndata
• Fortemporallycorrelatednoise,variance(or𝜒2/d.o.f.)ofdataincreaseswithincreasingwindowsize
• ExtrapolaDonto“infiniteDme”canbeachievedbyfionganasymptoDcfuncDontoRMSasafuncDonofDmewindow– 𝜒2/d.o.f.∝e−𝜎𝜏
• AsymptoDcvalueisgoodesDmateoflong-termvariancefactor
• Use“real_sigma”opDonintsfit
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Yellow:Daily(raw)Blue:7-dayaverages
UnderstandingtheFOGMExalgorithm:EffectofaveragingonDme-seriesnoise
NotethedominanceofcorrelatederrorsandunrealisDcrateuncertainDeswithawhitenoiseassumpDon:.01mm/yrN,E.04mm/yrU
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Samesite,Eastcomponent(dailywrms0.9mmnrms0.5)
64-davgwrms0.7mmnrms2.0
100-davgwrms0.6mmnrms3.4
400-davgwrms0.3mmnrms3.1
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Redlinesshowthe68%probabilityboundsofthevelocitybasedontheresultsofapplyingthealgorithm.
UsingTSVIEWtocomputeanddisplaythe“realisDc-sigma”results
NoterateuncertainDeswiththe“realisDc-sigma”algorithm:0.09mm/yrN0.13mm/yrE0.13mm/yrU
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ComparisonofesDmatedvelocityuncertainDesusingspectralanalysis(CATS)andGauss-Markovfiongofaverages(FOGMEx)
PlotcourtesyE.Calais
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SummaryofPracDcalApproaches
• Whitenoise+flickernoise(+randomwalk)tomodelthespectrum[Williamsetal.,2004]
• Whitenoiseasaproxyforflickernoise[Maoetal.,1999]• RandomwalktomodeltomodelanexponenDalspectrum[Herring“FOGMEx”
algorithmforvelociDes]• “Eyeball”whitenoise+randomwalkfornon-conDnuousdata______________________________________• OnlythelasttwocanbeappliedinGLOBKforvelocityesDmaDon• AllapproachesrequirecommonsenseandverificaDon
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ExternalvalidaDonofvelocityuncertainDesbycomparingwithamodel-Simplecase:assumenostrainwithinageologicallyrigidblock
GMTplotat70%confidence
17sitesincentralMacedonia:4-5velociDespierceerrorellipses
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..samesoluDonplo[edwith95%confidenceellipses
1-2of17velociDespierceerrorellipses
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McCaffreyetal.2007
ExternalvalidaDonofvelocityuncertainDesbycomparingwithamodel-amorecomplexcaseofalargenetworkintheCascadiasubducDonzone
ColorsshowslippingandlockedporDonsofthesubducDngslabwherethesurfacevelociDesarehighlysensiDvetothemodel;areatotheeastisslowlydeformingandinsensiDvetothedetailsofthemodel
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VelociDesand70%errorellipsesfor300sitesobservedbyconDnuousandsurvey-modeGPS1991-2004Testarea(nextslide)iseastof238E
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ResidualstoelasDcblockmodelfor73sitesinslowlydeformingregionErrorellipsesarefor70%confidence:13-17velociDespiercetheirellipse
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CumulaDvehistogramofnormalizedvelocityresidualsforEasternOregon&Washington(70sites)NoiseaddedtoposiDonforeachsurvey:0.5mmrandom1.0mm/sqrt(yr))randomwalkSolidlineistheoreDcalforachidistribuDon
PercentWithinRaDo
RaDo(velocitymagnitude/uncertainty)
StaDsDcsofVelocityResiduals
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RaDo(velocitymagnitude/uncertainty)
PercentWithinRaDo
Sameaslastslidebutwithasmallerrandom-walknoiseadded:0.5mmrandom0.5mm/yrrandomwalk(vs1.0mm/sqrt(yr))RWfor‘best’noisemodel)Notegreaternumberofresidualsinrangeof1.5-2.0sigma
StaDsDcsofVelocityResiduals
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PercentWithinRaDo
Sameaslastslidebutwithlargerrandomandrandom-walknoiseadded:2.0mmwhitenoise1.5mm/sqrt(yr))randomwalk(vs0.5mmWNand1.0mm/sqrt(yr))RWfor‘best’noisemodel)Notesmallernumberofresidualsinallrangesabove0.1-sigma
RaDo(velocitymagnitude/uncertainty)
StaDsDcsofVelocityResiduals
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Summary
• AllalgorithmsforcompuDngesDmatesofstandarddeviaDonshavevariousproblems:Fundamentally,ratestandarddeviaDonsaredependentonlowfrequencypartofnoisespectrumwhichispoorlydetermined.
• AssumpDonsofstaDonarityareoaennotvalid
• FOGMEx(“realisDcsigma”)algorithmisaconvenientandreliableapproachtogeongvelocityuncertainDesinglobk
• Velocityresidualsfromaphysicalmodel,togetherwiththeiruncertainDes,canbeusedtovalidatetheerrormodel
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