time series and error analysis - geowebgeoweb.mit.edu/.../pdf/33-error_analysis.pdf · tools for...
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Timeseriesanderroranalysis
M.A.FloydMassachusettsInstituteofTechnology,Cambridge,MA,USA
SchoolofEarthSciences,UniversityofBristolUnitedKingdom2–5May2017
MaterialfromT.A.Herring,R.W.King,M.A.Floyd(MIT)andS.C.McClusky (nowANU)
http://web.mit.edu/mfloyd/www/courses/gg/201705_Bristol/
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IssuesinGPSErrorAnalysis
• Whatarethesourcesoftheerrors?
• Howmuchoftheerrorcanweremovebybettermodeling?
• Dowehaveenoughinformationtoinfertheuncertaintiesfromthedata?
• Whatmathematicaltoolscanweusetorepresenttheerrorsanduncertainties?
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DeterminingtheUncertaintiesofGPSParameterEstimates
• Rigorousestimateofuncertaintiesrequiresfullknowledgeoftheerrorspectrum—bothtemporalandspatialcorrelations(neverpossible)
• Sufficientapproximationsareoftenavailablebyexaminingtimeseries(phaseand/orposition)andreweightingdata
• Whatevertheassumederrormodelandtoolsusedtoimplementit,externalvalidationisimportant
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ToolsforErrorAnalysisinGAMIT/GLOBK
• GAMIT:AUTCLNreweight=Y (default)usesphaserms frompostfit edittoreweightdatawithconstant+elevation-dependentterms
• GLOBK– rename(eq_file)_XPSor_XCLtoremoveoutliers– sig_neu addswhitenoisebystationandspan;bestwayto“rescale”therandomnoise
component;alargevaluecanalsosubstitutefor_XPS/_XCLrenamesforremovingoutliers
– mar_neu addsrandom-walknoise:principalmethodforcontrollingvelocityuncertainties– Inthegdl files,canrescalevariancesofanentireh-file:usefulwhencombiningsolutions
fromwithdifferentsamplingratesorfromdifferentprograms(Bernese,GIPSY)• Utilities
– tsview andtsfit cangenerate_XPScommandsgraphicallyorautomatically– grw andvrw cangeneratesig_neu commandswithafewkeystrokes– FOGMEx (“realisticsigma”)algorithmimplementedintsview (MATLAB)andtsfit/ensum;
sh_gen_stats generatesmar_neu commandsforglobk basedonthenoiseestimates– sh_plotvel (GMT)allowssettingofconfidenceleveloferrorellipses– sh_tshist andsh_velhist (GMT)canbeusedtogeneratehistogramsoftimeseriesand
velocities.
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SourcesofError
• Signalpropagationeffects– Receivernoise– Ionospheric effects– Signalscattering(antennaphasecenter/multipath)– Atmosphericdelay(mainlywatervapor)
• Unmodeled motionsofthestation– Monumentinstability– Loadingofthecrustbyatmosphere,oceans,andsurfacewater
• Unmodeled motionsofthesatellites
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Epochs
12345Hours
20
0mm
-20
Elevationangleandphaseresidualsforsinglesatellite
CharacterizingPhaseNoise
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Fixedantennas
Walls
Poles
Reinforcedconcretepillars
Deep-bracing
http://pbo.unavco.org/instruments/gps/monumentation
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Timeseriescharacteristics
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Timeseriescomponentsobservedposition
(linear)velocityterm
initialposition
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observedposition
(linear)velocityterm
annualperiodsinusoid
initialposition
Timeseriescomponents
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observedposition
(linear)velocityterm
annualperiodsinusoid
semi-annualperiodsinusoid
initialposition
seasonalterm
Timeseriescomponents
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observedposition
(linear)velocityterm
annualperiodsinusoid
semi-annualperiodsinusoid
initialposition
seasonaltermε=3mmwhitenoise
Timeseriescomponents
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“White”noise• Time-independent(uncorrelated)• Magnitudehascontinuousprobabilityfunction,e.g.Gaussian
distribution• Directionisuniformlyrandom
“True”displacementpertimestepIndependent(“white”)noiseerrorObserveddisplacementaftertimestept(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”displacementpertimestepCorrelated(“colored”)noiseerror*Observeddisplacementaftertimestept(v=d/t)*exampleis“randomwalk”(time-integratedwhitenoise)
• Mustbetakenintoaccounttoproducemore“realistic”velocities
Thisisstatisticalandstilldoesnotaccountforallother(unmodeled)errorselsewhereintheGPSsystem
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Annualsignalsfromatmosphericandhydrologicalloading,monumenttranslationandtilt,andantennatemperaturesensitivityarecommoninGPStimeseries
VelocityErrorsduetoSeasonalSignalsinContinuousTimeSeries
TheoreticalanalysisofacontinuoustimeseriesbyBlewitt andLavallee [2002,2003]
Top: Biasinvelocityfroma1mmsinusoidalsignalin-phaseandwitha90-degreelagwithrespecttothestartofthedataspan
Bottom:Maximumandrms velocitybiasoverallphaseangles– TheminimumbiasisNOTobtainedwith
continuousdataspanninganevennumberofyears
– Thebiasbecomessmallafter3.5yearsofobservation
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CharacterizingtheNoiseinDailyPositionEstimates
Notetemporalcorrelationsof30-100daysandseasonalterms
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Figure5fromWilliamsetal [2004]:Powerspectrumforcommon-modeerrorintheSOPACregionalSCIGNanalysis.Linesarebest-fitWN+FNmodels(solid=meanampl;dashed=MLE)
Notelackoftaperandmisfitforperiods>1yr
SpectralAnalysisoftheTimeSeriestoEstimateanErrorModel
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SummaryofSpectralAnalysisApproach
• Powerlaw:slopeoflinefittospectrum– 0=whitenoise– -1=flickernoise– -2=randomwalk
• Non-integerspectralindex(e.g.“fractionwhitenoise”à 1>k>-1)
• GooddiscussioninWilliams[2003]
• Problems:– Computationallyintensive– Nomodelcapturesreliablythelowest-frequencypartofthespectrum
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CATS(Williams,2008)
• CreateandAnalyzeTimeSeries• Maximumlikelihoodestimatorforchosenmodel– Initialpositionandvelocity– Seasonalcycles(sumofperiodicterms)[optional]– Exponentofpowerlawnoisemodel
• Requiressomelinearalgebralibraries(BLASandLAPACK)tobeinstalledoncomputer(commonnowadays,butcheck!)
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Hector(Bos etal.,2013)• MuchthesameasCATSbutfasteralgorithm• Maximumlikelihoodestimatorforchosenmodel– Initialpositionandvelocity– Seasonalcycles(sumofperiodicterms)[optional]– Exponentofpowerlawnoisemodel– Also
• RequiresATLASlinearalgebralibrariestobeinstalledoncomputer
• LinuxpackageavailablebuttrickytoinstallfromsourceduetoATLASrequirement
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sh_cats/sh_hector
• ScriptstoaidbatchprocessingoftimeserieswithCATSorHector
• RequiresCATSand/orHectortobepre-installed
• Outputs– Velocitiesin“.vel”-fileformat– Equivalentrandomwalkmagnitudesin“mar_neu”commandsforsourcinginglobk commandfile
• Cantakealong time!
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Whitenoisevs flickernoisefromMaoetal. [1999]spectralanalysisof23globalstations
Short-cut(Maoetal,1998):Usewhitenoisestatistics(wrms)topredicttheflickernoise
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“RealisticSigma”AlgorithmforVelocityUncertainties
• Motivation:computationalefficiency,handletimeserieswithvaryinglengthsanddatagaps;obtainamodelthatcanbeusedinglobk
• Concept:Thedeparturefromawhite-noise(sqrt n)reductioninnoisewithaveragingprovidesameasureofcorrelatednoise.
• Implementation:– Fitthevaluesofchi2vs averagingtimetotheexponentialfunction
expectedforafirst-orderGauss-Markov(FOGM)process(amplitude,correlationtime)
– Usethechi2valueforinfiniteaveragingtimepredictedfromthismodeltoscalethewhite-noisesigmaestimatesfromtheoriginalfit
– and/or– FitthevaluestoaFOGMwithinfiniteaveragingtime(i.e.,random
walk)andusetheseestimatesasinputtoglobk (mar_neu command)
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Extrapolatedvariance(FOGMEx)• Forindependentnoise,variance∝ 1/√Ndata
• Fortemporallycorrelatednoise,variance(or𝜒2/d.o.f.)ofdataincreaseswithincreasingwindowsize
• Extrapolationto“infinitetime”canbeachievedbyfittinganasymptoticfunctiontoRMSasafunctionoftimewindow– 𝜒2/d.o.f.∝ e−𝜎𝜏
• Asymptoticvalueisgoodestimateoflong-termvariancefactor
• Use“real_sigma”optionintsfit
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Yellow:Daily(raw)Blue:7-dayaverages
UnderstandingtheFOGMEx algorithm:Effectofaveragingontime-seriesnoise
Notethedominanceofcorrelatederrorsandunrealisticrateuncertaintieswithawhitenoiseassumption:.01mm/yrN,E.04mm/yrU
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Samesite,Eastcomponent(dailywrms 0.9mmnrms 0.5)
64-davgwrms 0.7mmnrms 2.0
100-davgwrms 0.6mmnrms 3.4
400-davgwrms 0.3mmnrms 3.1
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Redlinesshowthe68%probabilityboundsofthevelocitybasedontheresultsofapplyingthealgorithm.
UsingTSVIEW tocomputeanddisplaythe“realistic-sigma”results
Noterateuncertaintieswiththe“realistic-sigma”algorithm:
0.09mm/yrN0.13mm/yrE0.13mm/yrU
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Comparisonofestimatedvelocityuncertaintiesusingspectralanalysis(CATS)andGauss-Markovfittingofaverages(FOGMEx)
PlotcourtesyE.Calais
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SummaryofPracticalApproaches
• Whitenoise+flickernoise(+randomwalk)tomodelthespectrum[Williamsetal.,2004]
• Whitenoiseasaproxyforflickernoise[Maoetal.,1999]• Randomwalktomodeltomodelanexponentialspectrum[Herring“FOGMEx”
algorithmforvelocities]• “Eyeball”whitenoise+randomwalkfornon-continuousdata______________________________________• OnlythelasttwocanbeappliedinGLOBKforvelocityestimation• Allapproachesrequirecommonsenseandverification
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Externalvalidationofvelocityuncertaintiesbycomparingwithamodel- Simplecase:assumenostrainwithinageologicallyrigidblock
GMTplotat70%confidence
17sitesincentralMacedonia:4-5velocitiespierceerrorellipses
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..samesolutionplottedwith95%confidenceellipses
1-2of17velocitiespierceerrorellipses
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McCaffreyetal.2007
Externalvalidationofvelocityuncertaintiesbycomparingwithamodel- amorecomplexcaseofalargenetworkintheCascadiasubduction zone
Colorsshowslippingandlockedportionsofthesubductingslabwherethesurfacevelocitiesarehighlysensitivetothemodel;areatotheeastisslowlydeformingandinsensitivetothedetailsofthemodel
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Velocitiesand70%errorellipsesfor300sitesobservedbycontinuousandsurvey-modeGPS1991-2004
Testarea(nextslide)iseastof238E
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Residualstoelasticblockmodelfor73sitesinslowlydeformingregion
Errorellipsesarefor70%confidence:13-17velocitiespiercetheirellipse
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CumulativehistogramofnormalizedvelocityresidualsforEasternOregon&Washington(70sites)
Noiseaddedtopositionforeachsurvey:0.5mmrandom1.0mm/sqrt(yr))randomwalk
Solidlineistheoreticalforachidistribution
PercentWithinRatio
Ratio(velocitymagnitude/uncertainty)
StatisticsofVelocityResiduals
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Ratio(velocitymagnitude/uncertainty)
PercentWithinRatio
Sameaslastslidebutwithasmallerrandom-walknoiseadded:
0.5mmrandom0.5mm/yrrandomwalk
(vs 1.0mm/sqrt(yr))RWfor‘best’noisemodel)
Notegreaternumberofresidualsinrangeof1.5-2.0sigma
StatisticsofVelocityResiduals
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PercentWithinRatio
Sameaslastslidebutwithlargerrandomandrandom-walknoiseadded:
2.0mmwhitenoise1.5mm/sqrt(yr))randomwalk
(vs 0.5mmWNand1.0mm/sqrt(yr))RWfor‘best’noisemodel)
Notesmallernumberofresidualsinallrangesabove0.1-sigma
Ratio(velocitymagnitude/uncertainty)
StatisticsofVelocityResiduals
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Summary
• Allalgorithmsforcomputingestimatesofstandarddeviationshavevariousproblems:Fundamentally,ratestandarddeviationsaredependentonlowfrequencypartofnoisespectrumwhichispoorlydetermined.
• Assumptionsofstationarityareoftennotvalid
• FOGMEx (“realisticsigma”)algorithmisaconvenientandreliableapproachtogettingvelocityuncertaintiesinglobk
• Velocityresidualsfromaphysicalmodel,togetherwiththeiruncertainties,canbeusedtovalidatetheerrormodel
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References
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EffectofseasonaltermsonvelocityestimatesBlewitt andLavallee [J.Geophys.Res.107,2001JB000570,2002]
RealisticSigmaAlgorithmHerring[GPSSolutions,7,194,2003]Reilinger etal.[J.Geophys.Res.,111,B5,2006]
ValidationinvelocityfieldsMcClusky etal.[J.Geophys.Res.105,5695,2000]McClusky etal.[Geophys.Res.Lett.,28,3369,2000]Davisetal.[J.Geophys.Res.Lett.2003GL016961,2003]McCaffreyetal.,[Geophys J.Int.,2007.03371,2007]
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