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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