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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/

IssuesinGPSErrorAnalysis

• Whatarethesourcesoftheerrors?

• Howmuchoftheerrorcanweremovebybettermodeling?

• Dowehaveenoughinformationtoinfertheuncertaintiesfromthedata?

• Whatmathematicaltoolscanweusetorepresenttheerrorsanduncertainties?

2017/05/04 Basicerroranalysis 1

DeterminingtheUncertaintiesofGPSParameterEstimates

• Rigorousestimateofuncertaintiesrequiresfullknowledgeoftheerrorspectrum—bothtemporalandspatialcorrelations(neverpossible)

• Sufficientapproximationsareoftenavailablebyexaminingtimeseries(phaseand/orposition)andreweightingdata

• Whatevertheassumederrormodelandtoolsusedtoimplementit,externalvalidationisimportant


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.


SourcesofError

• Signalpropagationeffects– Receivernoise– Ionospheric effects– Signalscattering(antennaphasecenter/multipath)– Atmosphericdelay(mainlywatervapor)

• Unmodeled motionsofthestation– Monumentinstability– Loadingofthecrustbyatmosphere,oceans,andsurfacewater

• Unmodeled motionsofthesatellites


Epochs

12345Hours

20

0mm

-20

Elevationangleandphaseresidualsforsinglesatellite

CharacterizingPhaseNoise


Fixedantennas

Walls

Poles

Reinforcedconcretepillars

Deep-bracing

http://pbo.unavco.org/instruments/gps/monumentation


Timeseriescharacteristics

Timeseriescomponentsobservedposition

(linear)velocityterm

initialposition


observedposition


annualperiodsinusoid

initialposition

Timeseriescomponents


observedposition



semi-annualperiodsinusoid

initialposition

seasonalterm



observedposition



semi-annualperiodsinusoid

initialposition

seasonaltermε=3mmwhitenoise



“White”noise• Time-independent(uncorrelated)• Magnitudehascontinuousprobabilityfunction,e.g.Gaussian

distribution• Directionisuniformlyrandom

“True”displacementpertimestepIndependent(“white”)noiseerrorObserveddisplacementaftertimestept(v=d/t)


“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


Annualsignalsfromatmosphericandhydrologicalloading,monumenttranslationandtilt,andantennatemperaturesensitivityarecommoninGPStimeseries

VelocityErrorsduetoSeasonalSignalsinContinuousTimeSeries

TheoreticalanalysisofacontinuoustimeseriesbyBlewitt andLavallee [2002,2003]

Top: Biasinvelocityfroma1mmsinusoidalsignalin-phaseandwitha90-degreelagwithrespecttothestartofthedataspan

Bottom:Maximumandrms velocitybiasoverallphaseangles– TheminimumbiasisNOTobtainedwith

continuousdataspanninganevennumberofyears

– Thebiasbecomessmallafter3.5yearsofobservation


CharacterizingtheNoiseinDailyPositionEstimates

Notetemporalcorrelationsof30-100daysandseasonalterms


Figure5fromWilliamsetal [2004]:Powerspectrumforcommon-modeerrorintheSOPACregionalSCIGNanalysis.Linesarebest-fitWN+FNmodels(solid=meanampl;dashed=MLE)

Notelackoftaperandmisfitforperiods>1yr

SpectralAnalysisoftheTimeSeriestoEstimateanErrorModel


SummaryofSpectralAnalysisApproach

• Powerlaw:slopeoflinefittospectrum– 0=whitenoise– -1=flickernoise– -2=randomwalk

• Non-integerspectralindex(e.g.“fractionwhitenoise”à 1>k>-1)

• GooddiscussioninWilliams[2003]

• Problems:– Computationallyintensive– Nomodelcapturesreliablythelowest-frequencypartofthespectrum


CATS(Williams,2008)

• CreateandAnalyzeTimeSeries• Maximumlikelihoodestimatorforchosenmodel– Initialpositionandvelocity– Seasonalcycles(sumofperiodicterms)[optional]– Exponentofpowerlawnoisemodel

• Requiressomelinearalgebralibraries(BLASandLAPACK)tobeinstalledoncomputer(commonnowadays,butcheck!)


Hector(Bos etal.,2013)• MuchthesameasCATSbutfasteralgorithm• Maximumlikelihoodestimatorforchosenmodel– Initialpositionandvelocity– Seasonalcycles(sumofperiodicterms)[optional]– Exponentofpowerlawnoisemodel– Also

• RequiresATLASlinearalgebralibrariestobeinstalledoncomputer

• LinuxpackageavailablebuttrickytoinstallfromsourceduetoATLASrequirement


sh_cats/sh_hector

• ScriptstoaidbatchprocessingoftimeserieswithCATSorHector

• RequiresCATSand/orHectortobepre-installed

• Outputs– Velocitiesin“.vel”-fileformat– Equivalentrandomwalkmagnitudesin“mar_neu”commandsforsourcinginglobk commandfile

• Cantakealong time!


Whitenoisevs flickernoisefromMaoetal. [1999]spectralanalysisof23globalstations

Short-cut(Maoetal,1998):Usewhitenoisestatistics(wrms)topredicttheflickernoise


“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)


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


Yellow:Daily(raw)Blue:7-dayaverages

UnderstandingtheFOGMEx algorithm:Effectofaveragingontime-seriesnoise

Notethedominanceofcorrelatederrorsandunrealisticrateuncertaintieswithawhitenoiseassumption:.01mm/yrN,E.04mm/yrU


Samesite,Eastcomponent(dailywrms 0.9mmnrms 0.5)

64-davgwrms 0.7mmnrms 2.0




Redlinesshowthe68%probabilityboundsofthevelocitybasedontheresultsofapplyingthealgorithm.

UsingTSVIEW tocomputeanddisplaythe“realistic-sigma”results

Noterateuncertaintieswiththe“realistic-sigma”algorithm:

0.09mm/yrN0.13mm/yrE0.13mm/yrU


Comparisonofestimatedvelocityuncertaintiesusingspectralanalysis(CATS)andGauss-Markovfittingofaverages(FOGMEx)

PlotcourtesyE.Calais


SummaryofPracticalApproaches

• Whitenoise+flickernoise(+randomwalk)tomodelthespectrum[Williamsetal.,2004]

• Whitenoiseasaproxyforflickernoise[Maoetal.,1999]• Randomwalktomodeltomodelanexponentialspectrum[Herring“FOGMEx”

algorithmforvelocities]• “Eyeball”whitenoise+randomwalkfornon-continuousdata______________________________________• OnlythelasttwocanbeappliedinGLOBKforvelocityestimation• Allapproachesrequirecommonsenseandverification


Externalvalidationofvelocityuncertaintiesbycomparingwithamodel- Simplecase:assumenostrainwithinageologicallyrigidblock

GMTplotat70%confidence

17sitesincentralMacedonia:4-5velocitiespierceerrorellipses


..samesolutionplottedwith95%confidenceellipses

1-2of17velocitiespierceerrorellipses


McCaffreyetal.2007

Externalvalidationofvelocityuncertaintiesbycomparingwithamodel- amorecomplexcaseofalargenetworkintheCascadiasubduction zone

Colorsshowslippingandlockedportionsofthesubductingslabwherethesurfacevelocitiesarehighlysensitivetothemodel;areatotheeastisslowlydeformingandinsensitivetothedetailsofthemodel


Velocitiesand70%errorellipsesfor300sitesobservedbycontinuousandsurvey-modeGPS1991-2004

Testarea(nextslide)iseastof238E


Residualstoelasticblockmodelfor73sitesinslowlydeformingregion

Errorellipsesarefor70%confidence:13-17velocitiespiercetheirellipse


CumulativehistogramofnormalizedvelocityresidualsforEasternOregon&Washington(70sites)

Noiseaddedtopositionforeachsurvey:0.5mmrandom1.0mm/sqrt(yr))randomwalk

Solidlineistheoreticalforachidistribution

PercentWithinRatio

Ratio(velocitymagnitude/uncertainty)

StatisticsofVelocityResiduals



PercentWithinRatio

Sameaslastslidebutwithasmallerrandom-walknoiseadded:

0.5mmrandom0.5mm/yrrandomwalk

(vs 1.0mm/sqrt(yr))RWfor‘best’noisemodel)

Notegreaternumberofresidualsinrangeof1.5-2.0sigma



PercentWithinRatio

Sameaslastslidebutwithlargerrandomandrandom-walknoiseadded:

2.0mmwhitenoise1.5mm/sqrt(yr))randomwalk

(vs 0.5mmWNand1.0mm/sqrt(yr))RWfor‘best’noisemodel)

Notesmallernumberofresidualsinallrangesabove0.1-sigma




Summary

• Allalgorithmsforcomputingestimatesofstandarddeviationshavevariousproblems:Fundamentally,ratestandarddeviationsaredependentonlowfrequencypartofnoisespectrumwhichispoorlydetermined.

• Assumptionsofstationarityareoftennotvalid

• FOGMEx (“realisticsigma”)algorithmisaconvenientandreliableapproachtogettingvelocityuncertaintiesinglobk

• Velocityresidualsfromaphysicalmodel,togetherwiththeiruncertainties,canbeusedtovalidatetheerrormodel


References

SpectralAnalysisLangbein andJohnson[J.Geophys.Res.,102,591,1997]Zhangetal.[J.Geophys.Res.,102,18035,1997]Maoetal.[J.Geophys.Res.,104,2797,1999]Dixonetal.[Tectonics,19,1,2000]Herring[GPSSolutions,7,194,2003]Williams[J.Geodesy,76,483,2003]Williamsetal.[J.Geophys.Res.109,B03412,2004]Langbein [J.Geophys.Res.,113,B05405,2008]Williams,S.[GPSSolutions,12,147,2008]Bos etal.[J.Geod.,87,351-360,2013]

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]
