time series and error analysis - geowebgeoweb.mit.edu › ~floyd › courses › gg ›...

Timeseriesanderroranalysis

T.A.HerringM.A.FloydR.W.KingMassachusettsInstituteofTechnology,Cambridge,MA,USA

UNAVCOHeadquarters,Boulder,Colorado,USA19–23June2017

http://web.mit.edu/mfloyd/www/courses/gg/201706_UNAVCO/MaterialfromR.W.King,T.A.Herring,M.A.Floyd(MIT)andS.C.McClusky (nowatANU)

IssuesinGNSSerroranalysis

• Whatarethesourcesoftheerrors?• Howmuchoftheerrorcanweremovebybettermodeling?• Dowehaveenoughinformationtoinfertheuncertaintiesfromthedata?• Whatmathematicaltoolscanweusetorepresenttheerrorsanduncertainties?

2017/06/21 Timeseriesanderroranalysis 1

DeterminingtheuncertaintiesofGNSSparameterestimates

• Rigorousestimateofuncertaintiesrequiresfullknowledgeoftheerrorspectrum,bothtemporalandspatialcorrelations(neverpossible)• Sufficientapproximationsareoftenavailablebyexaminingtimeseries(phaseand/orposition)andreweightingdata• Whatevertheassumederrormodelandtoolsusedtoimplementit,externalvalidationisimportant


ToolsforerroranalysisinGAMIT/GLOBK

GAMIT• “AUTCLNreweight=Y”(defaultinsestbl.)usesphaserms frompostfit edittoreweightdatawithconstant+

elevation-dependenttermsGLOBK• Rename(eq_file)to“_XPS”or“_XCL”toremoveoutliers• “sig_neu”addswhitenoisebystationandspan

• Bestwayto“rescale”therandomnoisecomponent• A largevaluecanalsosubstitutefor“_XPS”/“_XCL”renamesforremovingoutliers

• “mar_neu”addsrandom-walknoise• Principalmethodforcontrollingvelocityuncertainties

• Inthegdl-files,rescalevariancesofanentireh-file• Usefulwhencombiningsolutionsfromwithdifferentsamplingratesorfromdifferentprograms(Bernese,GIPSY)

Utilities• tsview andtsfit cangenerate“_XPS”commandsgraphicallyorautomatically• grw andvrw cangenerate“sig_neu”commandswithafewkeystrokes• FOGMEx (“realisticsigma”)algorithmimplementedintsview (MATLAB)andtsfit/ensum

• sh_gen_stats generates“mar_neu”commandsforglobk basedonthenoiseestimates

• sh_plotvel (GMT)allowssettingofconfidenceleveloferrorellipses• sh_tshist andsh_velhist (GMT)canbeusedtogeneratehistogramsoftimeseriesandvelocities


Sourcesoferror

• Signalpropagationeffects• Receivernoise• Ionosphericeffects• Signalscattering(antennaphasecenter/multipath)• Atmosphericdelay(mainlywatervapor)

• Unmodeledmotionsofthestation• Monumentinstability• Loadingofthecrustbyatmosphere,oceans,andsurfacewater

• Unmodeledmotionsofthesatellites


Epochs

12 3 45Hours

20

0mm

-20

Elevationangleandphaseresidualsforsinglesatellite

Characterizingphasenoise


Characterizingphasenoise


Monumenttypes


Walls

Poles

Reinforcedconcretepillars

Deep-bracing

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

Timeseriescharacteristics


Timeseriescomponents


observedposition

(linear)velocityterm

initialposition

observedposition


annualperiodsinusoid

initialposition



observedposition



semi-annualperiodsinusoid

initialposition

seasonalterm



observedposition



semi-annualperiodsinusoid

initialposition

seasonaltermε=3mmwhitenoise



Annualsignalsfromatmosphericandhydrologicalloading,monumenttranslationandtilt,andantennatemperaturesensitivityarecommoninGPStimeseries

Velocityerrorsduetoseasonalsignalsincontinuoustimeseries

TheoreticalanalysisofacontinuoustimeseriesbyBlewitt andLavallee (2002,2003)

Top: Biasinvelocityfroma1mmsinusoidalsignalin-phaseandwitha90-degreelagwithrespecttothestartofthedataspan

Bottom:Maximumandrms velocitybiasoverallphaseangles• TheminimumbiasisNOTobtainedwithcontinuousdataspanninganevennumberofyears• Thebiasbecomessmallafter3.5yearsofobservation


Characterizingthenoiseindailypositionestimates

Notetemporalcorrelationsof60-200daysandseasonalterms



Figure5fromWilliamsetal.(2004):Powerspectrumforcommon-modeerrorintheSOPACregionalSCIGNanalysis.Linesarebest-fitwhitenoiseplusflickernoise(solid=meanamplitude;dashed=maximumlikelihoodestimation)

Notelackoftaperandmisfitforperiods>1yr(frequencies<π× 10−8)

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


“White”noise

• Time-independent(uncorrelated)•Magnitudehascontinuousprobabilityfunction,e.g.Gaussiandistribution• Directionisuniformlyrandom


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

“Color”noise

• Time-dependent(correlated):power-law,first-orderGauss-Markov,etc.• Convergenceto“true”velocityisslowerthanwithwhitenoise,i.e.velocityuncertaintyislarger


“True”displacementpertimestepCorrelated(“colored”)noiseerror*Observeddisplacementaftertimestept(v=d/t)

*exampleis“randomwalk”(time-integratedwhitenoise)

Mustbetakenintoaccounttoproducemore“realistic”velocities

Thisisstatisticalandstilldoesnotaccountforallother(unmodeled)errorselsewhereintheGPSsystem

CATS(Williams,2008)

• CreateandAnalyzeTimeSeries• Maximumlikelihoodestimatorforchosenmodelsolvesfor• Initialpositionandvelocity• Seasonalcycles(sumofperiodicterms)[optional]• Exponentofpowerlawnoisemodel

• Requiressomelinearalgebralibraries(BLASandLAPACK)tobeinstalledoncomputer(commonnowadays,butcheck!)• InformationonM.Floyd’sexperienceofcompilingCATSathttp://web.mit.edu/mfloyd/www/computing/cats/

• Formerlyathttp://www.pol.ac.uk/home/staff/?user=WillSimoCats• However,abovewebpageandsourcecodenolongerseemtoavailable• PossiblyasignthatCATSissupersededbyHector?


Hector(Bos etal.,2013)

• MuchthesameasCATSbutfasteralgorithm• Maximumlikelihoodestimatorforchosenmodelsolvesfor• Initialpositionandvelocity• Seasonalcycles(sumofperiodicterms)[optional]• ExponentofpowerlawnoisemodelAlso,asofHectorversion1.6:• Changesinlinearvelocity• Non-linearmotions(logarithmicand/orexponentialdecays)

• RequiresATLASlinearalgebralibrariestobeinstalledoncomputer• LinuxpackageavailablebuttrickytoinstallfromsourceduetoATLASrequirement• http://segal.ubi.pt/hector/


sh_cats/sh_hector

• ScriptstoaidbatchprocessingoftimeserieswithCATSorHector• RequiresCATSand/orHectortobepre-installed• Outputs

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

• Cantakealong time!• ReadsGAMIT/GLOBKformats

• pos-file(s)asinput• eq-file(s)todefinediscontinuitiesforestimationofoffsets• tsfit commandfilecontaining“eq_file”,“max_sigma”,“n_sigma”and/or“periodic”optionsinsteadofspecifyingassh_cats/sh_hector options

• WritesfilesforGLOBK• apr-file(s),including“EXTENDED”termswhereperiodicand/ornon-linear(logrithmic and/orexponentialdecay)termshavebeenestimated

• “mar_neu”commandsforequivalentrandomwalkprocessnoise


WhitenoisevsflickernoisefromMaoetal.(1999)spectralanalysisof23globalstations

Approximations(Maoetal.,1999)


Usewhitenoisestatistics(wrms)topredicttheflickernoise

“Realisticsigma”algorithmforvelocityuncertainties

Motivation• Computationalefficiency• Handletimeserieswithvaryinglengthsanddatagaps• ObtainamodelthatcanbeusedinglobkConcept• Thedeparturefromawhite-noise(√n)reductioninnoisewithaveragingprovidesameasureofcorrelatednoise.

Implementation• Fitthevaluesofχ2 versusaveragingtimetotheexponentialfunctionexpectedforafirst-orderGauss-Markov(FOGM)process(amplitude,correlationtime)• Usetheχ2 valueforinfiniteaveragingtimepredictedfromthismodeltoscalethewhitenoisesigmaestimatesfromtheoriginal(least-squares)fitand/or

• FitthevaluestoaFOGMwithinfiniteaveragingtime(i.e.,randomwalk)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• Allapproachesrequirecommonsenseandverification


17sitesincentralMacedonia:4–5velocitiespierceerrorellipses

Externalvalidationofvelocityuncertaintiesbycomparingwithageophysicalmodel


Ifgeologicallyrigidmodelisvalid,70%ofsitesshouldshownostatisticallysignificantmotion,i.e.velocitylieswithinerrorellipse

GMTplotat70%confidence

Simplecase:assumenostrainwithinageologicallyrigidregion

Now1–2of17velocitiespierceerrorellipses



Samesolutionplottedwith95%confidenceellipses

McCaffreyetal.2007

AmorecomplexcaseofalargenetworkintheCascadiasubductionzone

Colorsshowslippingandlockedportionsofthesubducting slabwherethesurfacevelocitiesarehighlysensitivetothemodel;areatotheeastisslowlydeformingandinsensitivetothedetailsofthemodel



Velocitiesand70%errorellipsesfor300sitesobservedbycontinuousandsurvey-modeGPS1991-2004

Validationarea(nextslide)iseastof238°E


Residualstoelasticblockmodelfor73sitesinslowlydeformingregion

Errorellipsesarefor70%confidence:13-17velocitiespiercetheirellipse


Statisticsofvelocityresiduals

• CumulativehistogramofnormalizedvelocityresidualsforeasternOregonandWashington• 70sites

• Noiseaddedtopositionforeachsurvey:• 0.5mmrandom(“sig_neu”)• 1.0mm/sqrt(yr)randomwalk(“mar_neu”)

• Solidlineistheoreticalforaχ-distribution


Percen

twith

inra

tio

Ratio(velocitymagnitude/uncertainty)


• Sameaslastslidebutwithasmallerrandom-walknoiseadded:• 0.5mmrandom• 0.5mm/yr randomwalk• cf.1.0mm/sqrt(yr)RWfor“best”noisemodel

• Notegreaternumberofresidualsinrangeof1.5–2.0sigma


Percen

twith

inra

tio



• Sameaslastslidebutwithlargerrandomandrandom-walknoiseadded:• 2.0mmwhitenoise• 1.5mm/sqrt(yr))randomwalk• cf.0.5mmWNand1.0mm/sqrt(yr)RWfor“best”noisemodel

• Notesmallernumberofresidualsinallrangesabove0.1-sigma


Percen

twith

inra

tio


Summary

• Allalgorithmsforcomputingestimatesofstandarddeviationshavevariousproblems• Fundamentally,ratestandarddeviationsaredependentonlowfrequencypartofnoisespectrum,whichispoorlydeterminedwithoutverylongtimeseries(decades)

• Assumptionsofstationarity(constantnoisecharacteristicsovertime)areoften(usually?)notvalid• FOGMEx (“realisticsigma”)algorithmisaconvenientandreliableapproachtogettingvelocityuncertaintiesinglobk• Wearetestinghowreliable,incomparisontoothermethods,givengoodandbadtimeseries

• Velocityresidualsfromaphysicalmodel,togetherwiththeiruncertainties,canbeusedtovalidatetheerrormodel


References

SpectralAnalysis

• Langbein andJohnson(1997),J.Geophys.Res.,102,591–603,doi:10.1029/96JB02945.

• Zhangetal.(1997),J.Geophys.Res.,102,18035–18055,doi:10.1029/97JB01380.

• Maoetal.(1999),J.Geophys.Res.,104,2797–2816,doi:10.1029/1998JB900033.

• Dixonetal.(2000),Tectonics,19,1–24,doi:10.1029/1998TC001088.

• Williams(2003),J.Geod.,76,483–494,doi:10.1007/s00190-002-0283-4.

• Williamsetal.(2004),J.Geophys.Res.,109,B03412,doi:10.1029/2003JB002741.

• Langbein (2008),J.Geophys.Res.,113,B05405,doi:10.1029/2007JB005247.

• Williams(2008),GPSSolut.,12,147–153,doi:10.1007/s10291-007-0086-4.

• Bos etal.(2013),J.Geod.,87,351–360,doi:10.1007/s00190-012-0605-0.

Effectofseasonaltermsonvelocityestimates

• Blewitt andLavallée (2002),J.Geophys.Res.,107,2145,doi:10.1029/2001JB000570.Blewitt andLavallée (2003),J.Geophys.Res.,108,2010, doi:10.1029/2002JB002297.

RealisticSigmaAlgorithm

• Herring(2003),GPSSolut.,7,194–199,doi:10. 1007/s10291-003-0068-0.

• Reilinger etal.(2006),J.Geophys.Res.,111, B05411,doi:10.1029/2005JB004051.

Validationinvelocityfields

• McClusky etal.(2000),J.Geophys.Res.,105,5695–5719,doi:10.1029/1999JB900351.

• McClusky etal.(2001),Geophys.Res.Lett.,28, 3369–3372,doi:10.1029/2001GL013091.

• Davisetal.(2003),Geophys.Res.Lett.,30,1411, doi:10.1029/2003GL016961.

• McCaffreyetal.(2007),Geophys J.Int.,169,1315–1340,doi:10.1111/j.1365-246X.2007.03371.x.


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