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Addressing the problem of land motion at tide gauges

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Addressing the problem of land motion at tide gauges
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Doran, Kara
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Sea level
GPS
Error
Altimetry
TOPEX
Dissertations, Academic -- Marine Science -- Masters -- USF   ( lcsh )
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Abstract:
ABSTRACT: Estimation of global mean sea level change has become an area of interest for scientists in recent decades because of its importance as an indicator of climate change. Climate models predict varying degrees of change in global temperature and global sea level over the next 100 years. One way to check the validity of the models is to estimate sea level change over the last century and constrain the models to match these estimates. Traditionally, sea level change estimates have been calculated using long time series from tide gauges. There are some disadvantages to this approach however, since tide gauges have limited spatial coverage and make measurements relative to a land reference point that may be undergoing uplift or subsidence. Satellite altimetry has also been used in recent years to estimate sea level changes, but these measurements are subject to drift errors and must be calibrated. Mitchum (1998, 2000) has developed a method using the global network of tide gauges to calibrate altimeters that enables estimation of sea level change with a precision of 0.4 mm/yr. Errors in the estimates arise from a variety of sources, but the error of primary concern is that due to land motion at the tide gauge stations. In the present study we will investigate ways to improve the land motion estimate and thus reduce the error.
Thesis:
Thesis (M.S.)--University of South Florida, 2010.
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by Kara Doran.
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AddressingtheProblemofLandMotionatTideGauges by KaraJ.Doran Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofScience DepartmentofMarineScience CollegeofMarineScience UniversityofSouthFlorida MajorProfessor:GaryT.Mitchum,Ph.D. PeterA.Howd,Ph.D. RobertH.Weisberg,Ph.D. DateofApproval: November30,2009 Keywords:sealevel,GPS,TOPEX,error,altimetry c Copyright2010,KaraJ.Doran

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Acknowledgements Iwouldliketoexpressmysinceregratitudetomythesisadvisor,Dr.GaryMitchum, forhistimeanddedicationtothisprojectandmypersonaldevelopmentasascientist. Gary'smentoringandpatientinstructionprovidedmewithathoroughunderstandingof dataanalysisandphysicalprocessesthathasenabledmetobesuccessfulinmyprofessionalcareer.Iwouldalsoliketothankmycommitteemembers,Dr.PeterHowdandDr. RobertWeisbergforsharingtheirtimeandinsight.Theirfeedbackaboutthisresearch projecthasgreatlyimprovedtheworkpresentedhere.Fundingforthisresearchcame fromnumeroussources,includingtheVonRosenstielFellowshipprovidedgenerouslyby AnneandWernerVonRosenstiel,andgrantsfromtheNationalAeronauticsandSpace AdministrationandtheNationalOceanicandAtmosphericAdministration.Funding forconferencetravelwasprovidedbytheGraduateandProfessionalStudentCouncil ConferencePresentationGrantProgram.

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TableofContents ListofTablesiii ListofFiguresiv Abstractvi Chapter1Introduction1 1.1Motivation1 1.2OutlineoftheThesis9 Chapter2Overview10 2.1MethodsofMeasuringSeaLevel10 2.2TheMethodofAltimeterCalibrationUsingtheGlobalTideGauge Network13 2.3AddressingLandMotion17 2.4EarlierWork24 Chapter3MethodsofErrorEstimation28 3.1LimitationsofExistingErrorEstimationMethods28 3.2TheProposedMethodofErrorEstimation32 3.3TestingtheProposedMethodofErrorEstimation41 Chapter4MakinganOptimalGPSLandMotionEstimateataTideGauge54 4.1EvaluatingtheExternalEstimates55 4.2TheLandMotionGradientProblemandStationClassicationProposal58 4.3TheAlgorithmforMakinganExternalLandMotionEstimate67 Chapter5ApplicationofourMethods70 5.1ApplicationtoaNewSetofTideGaugeTimeSeries70 5.2Results74 5.3ConcludingRemarks75 i

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ListofReferences85 Appendices91 AppendixA:TablesofTideGaugeandGPSStationClassication92 ii

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ListofTablesTable 5.1 University of Hawaii Sea Level Center Fast Delivery Tide GaugesW ithInternalandExternalLandMotionEstimatesandErrors. 78Table A.1 T ide Gauge Categories 92 Table A.2 GPS Station Categories 99T ableA.3DetailsoftheCommentNumbersUsedinTablesA1and2. 107 iii

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ListofFigures Figure1.1GlobalMeanSeaLevelfromTideGauges.3 Figure1.2GlobalMeanSeaLevelfromTOPEXandJasonAltimeters.4 Figure1.3LandMotionatKodiakIsland,Alaska.6 Figure1.4LandMotionatGalvestonIsland,Texas.7 Figure1.5CurrentSetofTideGaugesandGPSStations.8 Figure2.1CurrentDriftSeriesforTOPEX/PoseidonandJason-1.14 Figure2.2Sealevel,modelt,andresidualsatPohnpei.20 Figure2.3EasterIslandGPStimeseries.21 Figure2.4TOPEXAlgorithmErrorIdentiedby Mitchum (1998).27 Figure3.1ExampleresidualspectrafromtheDarwin,Australiatidegauge(A) andtheAnnetteIsland,AlaskaGPSstation(B).29 Figure3.2RemovalofLow-FrequencyVariancebyFittingaTrendandSteps.30 Figure3.3ASpectrumCorrectedfortheRemovalofLow-FrequencyVariance byFittingaTrendandSteps.42 Figure3.4TestoftheProposedMethodforaTrend.44 Figure3.5DarwinTideGaugeSpectrumandModelFits.45 Figure3.6AnnetteIsland(AIS1)GPSSpectrumandModelFits.46 iv

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Figure3.7FittingaPowerLawSpectrumtoanAR(1)Spectrum.47 Figure3.8FittinganAR(1)SpectrumtoaPowerLawSpectrum.48 Figure3.9TestoftheProposedMethodforaTrendPlusaStepintheMiddleof theSeries.49 Figure3.10TestoftheProposedMethodforaTrendPlusaStepat25Percentof theRecordLength.50 Figure3.11TestoftheProposedMethodforaTrendPlusaStepat75Percentof theRecordLength.51 Figure3.12TestoftheProposedMethodforaTrendPlus3Steps.52 Figure3.13TestoftheProposedMethodforaTrendPlus5Steps.53 Figure4.1Darwin,AustraliaInternalandExternalLandMotionEstimates.56 Figure4.2Kushiro,JapanInternalandExternalEstimates.58 Figure4.3KeyWest,FloridaInternalandExternalLandMotionEstimates.59 Figure4.4TideGaugeandGPSNormalizedDifferencePairs.60 Figure4.5StationClassicationintoTectonicallyActiveorQuietStations.62 Figure4.6Category3NormalizedCorrelation.63 Figure4.7ICE5GModeledLandMotionDuetoGlacialIsostaticAdjustment.64 Figure4.8Category2NormalizedCorrelation.65 Figure4.9NormalizedCorrelationofCategory1Stations.67 Figure5.1Classicationofthe150FastDeliveryDatasetTideGauges.72 Figure5.2Classicationofthe164GPSStationsUsedinThisStudy.73 Figure5.3InternalandExternalLandMotionEstimatesfortheFastDeliverySet ofTideGauges.76 v

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AddressingtheProblemofLandMotionatTideGauges KaraJ.Doran ABSTRACT Estimationofglobalmeansealevelchangehasbecomeanareaofinterestforscientists inrecentdecadesbecauseofitsimportanceasanindicatorofclimatechange.Climate modelspredictvaryingdegreesofchangeinglobaltemperatureandglobalsealevelover thenext100years.Onewaytocheckthevalidityofthemodelsistoestimatesealevel changeoverthelastcenturyandconstrainthemodelstomatchtheseestimates.Traditionally,sealevelchangeestimateshavebeencalculatedusinglongtimeseriesfromtide gauges.Therearesomedisadvantagestothisapproachhowever,sincetidegaugeshave limitedspatialcoverageandmakemeasurementsrelativetoalandreferencepointthat maybeundergoingupliftorsubsidence.Satellitealtimetryhasalsobeenusedinrecent yearstoestimatesealevelchanges,butthesemeasurementsaresubjecttodrifterrors andmustbecalibrated. Mitchum (1998,2000)hasdevelopedamethodusingtheglobal networkoftidegaugestocalibratealtimetersthatenablesestimationofsealevelchange withaprecisionof0.4mm/yr.Errorsintheestimatesarisefromavarietyofsources, buttheerrorofprimaryconcernisthatduetolandmotionatthetidegaugestations.In thepresentstudywewillinvestigatewaystoimprovethelandmotionestimateandthus reducetheerror. vi

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Chapter1:Introduction 1.1Motivation Globalsealevelcanchangebytwofundamentalprocesses:changingthevolumeof thebasinandchangingthevolumeofwater.Changesinoceanbasinvolumebysedimentation,subsidence,tectonicactivity,andglacialisostaticadjustment( Harrison ,1990; LambeckandChappell ,2001)occurprimarilyonlonggeologictimescalesandarenot addressedhere.Watervolumecanchangebysequesteringandreleaseofwaterinundergroundandman-madereservoirs( HayandLeslie ,1990; Sahagianetal. ,1994; Gornitz 2001),meltingandaccumulationoficeincontinentalglaciersandicesheets( Meier 1984; Meieretal. ,2007),andthermalexpansionoftheoceans( Churchetal. ,1991; Cabanesetal. ,2001; Lombardetal. ,2005).Thermalexpansionandchangesincontinental icevolumearethetwolargestcontributorstooceanvolumechanges( Church ,2001)and aredirectlyrelatedtochangesinthetemperatureoftheearth. Estimationofglobalmeansealevelchangehasbecomeaproblemofinterestfor scientistsinrecentdecadesbecauseofitsimportanceasanindicatorofclimatechange anditspotentialeffectsoncoastalpopulationsandecosystems.Ofevengreaterconcern isthepossibilityofanaccelerationintherateofsealevelriseoverthetwentiethcentury. Climatemodelshavebeenusedtopredictanincreaseinglobalmeantemperatureof1to 6Celsiusoverthenextcentury,whichimpliesanincreaseinglobalsealevelof200to 1

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600mmormore( Bindoffetal. ,2007).Thesesealevelprojectionsincludetheeffectofa predictedaccelerationintheglobalmeansealevelriserate.Fromthelargerangeinthe predictedsealevelchange,however,itisobviousthatbetterestimatesoftheacceleration areessential. Bycomparinglong-termratesofglobalmeansealevelchangefromtidegaugeswith recentratesfromsatellitealtimetrywemaybeabletodetectasignicantacceleration intherateofglobalsealevelrise.However,thekeytodetectinganaccelerationinsea levelriseisgettingthebestlinearratesfromthetidegaugeandaltimetrytimeseries andproperlyassigninganerrorbartothoserates.Theprimaryfocusofthisthesisisto improvethepresentlyavailableerrorbars. Estimatesofglobalsealevelchangeoverthepastcenturyarebasedprimarilyonaveragingasubsetoftheglobaltidegaugenetwork(Figure1.1).Globalsealevelreconstructionsusingtidegaugeshavebeendonebymanyresearchers( e.g.,Douglas ,1991,1995; ChurchandWhite ,2006),butgettingatrueglobalaverageisproblematicbecauseofthe limitedspatialdistributionoflongrecords.Mostrecordswithgreaterthan50yearsof dataareinthenorthernhemispherewithfewrecordslongerthan50yearsinthesouthern hemisphere.Additionally,tidegaugesmeasuresealevelrelativetoland,soitisimpossible,basedonthetidegaugesalone,todeterminewhetheralong-termtrendinthedataisa changeinsealevelorlandmotionatthetidegauge. Theproblemofdatadistributioncanbeaddressedbyusingsatellitealtimetersto measureglobalsealevelchange(Figure1.2)( e.g.,Nerem ,1995; Minsteretal. ,1995; Cazenaveetal. ,1998; Cabanesetal. ,2001; LeulietteandMiller ,2009).Modernaltimeterssurveyalmosttheentireglobaloceaneverytendaysandcanresolvelowfrequency massredistributionsthatother insitu instruments,suchastidegauges,cannot.Unlike tidegauges,whichmeasureonlyonepointinspace,satellitealtimetersgetanalmost 2

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1860 1880 1900 1920 1940 1960 1980 2000 2020 150 100 50 0 50 100 150 Global Mean Sea Level (mm)Figure1.1:GlobalMeanSeaLevelfromTideGauges,modiedfrom ChurchandWhite (2006). instantaneous"snapshot"oftheglobaloceanandcandifferentiatebetweenglobaland localsealevelchangeduetomassredistribution.However,altimeterscanhavedrifterrorsthatmaketheestimationoflong-termglobalsealevelchangeirrelevantunlessthe driftsarecorrected.Onemethodofdetectingaltimeterdriftistousetheglobalnetwork oftidegaugesasacalibrationtool.Thismethodhasbeendevelopedby Mitchum (1994, 1998,2000),andhasbeenutilizedoverthepastdecadetoidentifyseveraldriftsinthe TOPEX/PoseidonandJason-1altimeters. Thebasicprincipleofthetidegauge-altimetercomparisonisthatboththetidegauge andnearbyaltimeterpassesaremeasuringthesameoceansignal,butthetidegaugemeasuressealevelrelativetoacrustalreferencepointthatmaybeinmotion( Douglas ,1995). 3

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-30 -20 -10 0 10 20 30 40 MSL(mm) 1994 1996 1998 2000 2002 2004 2006 2008 -30 -20 -10 0 10 20 30 40 MSL(mm) 1994 1996 1998 2000 2002 2004 2006 2008 -30 -20 -10 0 10 20 30 40 MSL(mm) 1994 1996 1998 2000 2002 2004 2006 2008 -30 -20 -10 0 10 20 30 40 MSL(mm) 1994 1996 1998 2000 2002 2004 2006 2008 -30 -20 -10 0 10 20 30 40 MSL(mm) 1994 1996 1998 2000 2002 2004 2006 2008 Univ of Colorado 2009_rel5 TOPEX Jason 60-day smoothing Inverse barometer applied Rate = 3.1 0.4 mm/yr Seasonal signals removed Figure1.2:GlobalMeanSeaLevelfromTOPEX(redpoints)andJason(bluepoints) Altimeters.Retrievedfrom http://sealevel.colorado.edu/ .Thedatahavebeencorrected fortheinvertedbarometereffectandseasonalsignalshavebeenremoved.Thetrendtto theglobalaveragesealevelis3 2 0 4mm/yr.Thelargestremaininguncertaintyinthis estimateislandmotionatthetidegauges. Thistidegaugedatamustbecorrectedforlandmotionbeforeitcanbeusedforaltimeter calibration,becausetheseasurfaceheightmeasuredbythealtimeterisnotaffectedby landmotionlikethetidegauges.Inmanycasesthelandmotionisverysmall,buteven thesmallestlandmotiontrendcancontributelargeerrorstothealtimetercalibration.In fact,landmotionatthetidegaugesisthelargestremainingerrorinthealtimeterdrift estimation(the0.4mm/yrgivenonFigure1.2). OneexampleoftheproblemoflandmotionattidegaugesisatKodiakIsland,Alaska. Tectonicupliftsincethe1964PrinceWilliamSoundearthquakeiscausingthelandto 4

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riseatarateofnearly10mm/yr.TheupperpanelofFigure1.3showsthesealeveltime seriesandtrendfromtheKodiakIslandtidegauge.Usingonlythetidegauge,itwould appearthatsealevelisfalling,insteadofrising.ThemiddlepanelofFigure1.3shows theGlobalPositioningSystem(GPS)verticaltimeseriesandtrend.Inthebottompanel, thelandmotiontrendhasbeenremovedfromthetidegaugesealevelseries.Nowthe sealeveltrendis-0.65mm/yr,avaluethatisofthesameorderastheacceptedglobalsea levelchangerateof1.8mm/yr( Douglas ,1995).Theoppositeproblemwithlandmotion occursinareasoflandsubsidence,suchasGalveston,Texas.InGalveston,groundwater pumpingandoilextractionhaveledtothelandsinkingatarateofabout-6mm/yr( Gallowayetal. ,1999).TheupperpanelofFigure1.4showstheGalvestontidegaugesea levelseriesandtheapparentsealevelriserateof6.78mm/yr.Whencorrectedforland motion,thetrendis-0.07mm/yr. Thetwoexamplesaboveillustratetheproblemoflandmotionatindividualtidegauges. Whilenoteverytidegaugehassuchdramaticlandmotion,itisnecessarytomakealand motionestimateateverytidegaugeinordertocorrectlyidentifybiasesinthealtimeter. OnewaytomakeanindependentlandmotionestimateatatidegaugeistousetheverticalcomponentofmotionfromacontinuouslyoperatingGPSorothergeodeticmeasurement( Bevisetal. ,2002).AcontinuouslyoperatingGPSreceivesathreedimensional positionxusing3ormoresatellitesevery30secondsorless.TheGPSsignalmustthen becalibratedandprocessedtoremoveatmosphericandantennaeffects( Woppelmann etal. ,2007).Theresultisatimeseriesofdailyorweeklyverticalpositionsthatcanbe usedtocomputealandmotionrate.Tidegaugeandgeodesyexpertsagreethattheideal locationfortheGPSreceiveriseithermountedwiththetidegauge,ifthetidegaugeis mountedonstableground,orwithin1kmofthetidegauge( Bevisetal. ,2002).Thisway, theGPSmeasuresthemotionofthelithospherearoundthetidegauge,notjustimmediate 5

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1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 400 200 0 200 400 Time (years)Tide Gauge Sea Level (mm) 2000 2000.5 2001 2001.5 2002 2002.5 2003 2003.5 2004 2004.5 2005 50 0 50 Time (years)GPS Vertical Position (mm) 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 400 200 0 200 400 Time (years)Tide Gauge Sea Level (mm) trend=9.43 mm/yr trend=8.78 mm/yr trend = 0.65 mm/yrFigure1.3:LandMotionatKodiakIsland,Alaska.Theupperpanelshowsthetimeseries oftidegaugesealevelfromtheKodiakIslandtidegauge.Thetrendinredshowsthatsea levelappearstobefallingatalmost10mm/yr.Thecenterpanelshowsthetimeseriesof verticalpositionsfromtheco-locatedGPSatKodiakIsland.Theverticallandmotion trendisupwardat8mm/yr.Inthebottompanel,thelandmotiontrendhasbeenremoved fromthetidegaugetimeseries.Theresultingsealeveltrendisnearlyzero. motionofthepierorothertidegaugemounting.Manyofthetidegaugesusedinthe altimetercalibrationhaveaGPSreceiverwithin10kmofthetidegauge(seeFigure2.4). HowtoestimatelandmotionatthetidegaugeswithoutanynearbyGPSisonefocusof thisresearch(Chapter4).Theothermajoraimistoplacerealisticerrorbarsontheland motionestimate(Chapter3). TheerrorbarsontrendsttobothGPSandtidegaugetimeseriesneedtobeinated forserialcorrelation.Lowfrequencysignalsinthetimeseriescontributetotheerror whenttingatrend.Previousmethodsforinatingtheerrorsforserialcorrelationwere 6

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1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 500 0 500 Time (years)Tide Gauge Sea Level (mm) 1999 1999.5 2000 2000.5 2001 2001.5 2002 2002.5 2003 2003.5 20 10 0 10 20 Time (years)GPS Vertical Position (mm) 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 500 0 500 Time (years)Tide Gauge Sea Level (mm) trend = 6.78 mm/yr trend = 6.85 mm/yr trend = 0.07 mm/yrFigure1.4:LandMotionatGalvestonIsland,Texas.Theupperpanelshowsthetime seriesoftidegaugesealevelfromtheGalvestonIslandtidegauge.Thetrendinred showsthatsealevelappearstoberisingatalmost7mm/yr.Thecenterpanelshowsthe timeseriesofverticalpositionsfromtheco-locatedGPSatGalvestonIsland.Thevertical landmotiontrendisdownwardat7mm/yr.Inthebottompanel,thelandmotiontrendhas beenremovedfromthetidegaugetimeseries.Theresultingsealeveltrendisnearlyzero. examinedby Mitchum (2000),butthesemethodsusetheresidualtimeseriesafteratrend hasbeenremovedand,aswewillshow,cansubstantiallyunderestimatetheerror.The GPStimeseriesalsohavemanydiscontinuities,oftenaccompaniedbyverticaloffsets. Theseoffsetsmustbetwithastepfunction.Fittingstepsalongwithatrendalsocontributesuncertainty.Wewilldescribeamethodforimprovingtheerrorinationforserial correlation,correctingforthevariancethatisremovedbyttingatrendandtakinginto accounttheeffectofttingstepsalongwithatrend. 7

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0 50 100 150 200 250 300 350 80 60 40 20 0 20 40 60 80 GPS Stations Tide Gauges Figure1.5:CurrentSetofTideGaugesandGPSStations.Shownonthemapisthesetof tidegaugesandGPSthatareincludedinthisstudy.Notethenumberoftidegauges(blue circles)thatareco-locatedwithaGPS(redtrianglesurroundingthebluecircle).Nearly halfofthe136tidegaugesusedinthisstudyhaveaGPSstationwithin100km. Byaddressingtheproblemoflandmotionatthetidegaugesandmakingthebestuse ofthecurrentsetofGPS,wehopetoreducetheerrorbaronthealtimeterdriftestimate andtheglobalsealevelratefromaltimetry.Assigningpropererrorbarstotidegaugesea leveltrendsandlandmotionestimateswillassistinformingthebestglobalaveragesea levelcurvefromtidegauges.Withthebestglobalsealevelchangeratesanderrorbars fromtidegaugesandaltimeters,wemaybeabletodetectwhetherornottherehasbeena signicantaccelerationintherateofsealevelriseoverthetwentiethcentury. 8

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1.2OutlineoftheThesis Inthefollowingsectionswewillintroducethebasicmethodofaltimeterdriftestimationusingtidegauges.Wewillthenpresentahistoryofsealevelmeasurements madebytidegaugesandaltimeters,alongwithahistoryofthedriftestimationmethod. InChapter3wethoroughlyexaminetheproblemofhowtoassignthebesterrorbarto atrendestimate.InChapter4wepresentthemethodformakingthebestlandmotion estimateusingknowledgeoftectonicsandlocallandmotion.Thenalchapterwillgive anexampleofmakingalandmotionestimateandassigninganerrortothatestimatefor anarbitrarysetoftidegaugeandGPSmeasurements. 9

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Chapter2:Overview 2.1MethodsofMeasuringSeaLevel Beforetheadventofsatellitealtimetry,estimatesofglobalmeansealevelchange werecalculatedusingcarefullychosensetsoftidegauges( e.g.,Douglas ,1991,1995; Douglasetal. ,2001).Tidegaugesprovidelongrecordsoflocalsealevelchangesand whenaveragedprovideaglobalsealevelchangerateovermanyyears.Tidegaugesare carefullymonitoredfordriftbytidestaffreadingsandgeodeticxingofbenchmarks (IOC,2006).Atidestaffisacalibratedpolethatisreadbyahumanobservertogive adirectmeasurementoftheheightoftheseasurface.Thesetidestaffreadingscanbe comparedtotidegaugemeasurementstodetectinstrumentdrift.Ifadriftisdetected inthedifferencesbetweenthetidestaffreadingsandthetidegaugemeasurementsthe instrumentwillbeinspectedandcalibrated.Tidestaffreadingscanbeusedtocalibrate thetidegaugeandensureaconsistentdatumforthegauge,evenwhentheinstrumentsare replaced. Althoughtidegaugesprovideverystablemeasurementsoflocalsealeveltrendsand uctuations,theseinstrumentsarenotidealformeasuringlongtermglobalsealevel changesbecauseofthelimitedspatialcoverageobtained.Becauseofthislimitation,some lowfrequencymassredistributions,duetoeventssuchastheElNinoSouthernOscillation(ENSO)orthePacicDecadalOscillation(PDO),maybemistakenfortrueglobal 10

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oceansignals( Barnett ,1984; Douglas ,1992; GrogerandPlag ,1993).Imagineasea levelrecordfromasingletidegauge.Inthisrecord,thereisaspanofseveralyearswhere thesealevelshowsasteadyrise.Usingonlythisonetidegaugeitwouldbeimpossibleto tellwhetherthisisasealevelriseduetotrueoceanvolumechangesorasealevelrisedue totheredistributionofwaterfromsomewhereelseintheocean.Evenwithalargenumber oftidegauges,theselowfrequencymassdistributionscannotberesolvedexceptbyvery longtimeseries.Asthelengthofthetimeseriesincreases,thelowfrequencyoscillations canbeidentiedandwillnotbewronglyinterpretedasglobaloceanvolumechanges. AsmentionedinChapter1,landmotionatthetidegaugesisanotherdisadvantageof usingtidegaugesforglobalsealevelestimates.AsillustratedbytheexamplesofGalveston,TexasandKodiakIsland,Alaska,landmotioncancauselocalsealeveltrendstobe quitedifferentfromtrueglobalsealevelchange.Eventheearliestestimatesofglobal meansealevelusingtidegaugesrecognizedtheimportanceoflandmotion.Therst globalaveragesealevelwascomputedby Gutenberg (1941)using71tidegaugesmostly intheNorthernhemisphere.Becauseofthedifcultyincomputingalinearregression, atrendwastbetweentherst10andlast10yearsofdatayieldingameansealevel changeof1 2 1 3mm/yr.ThisauthorwascarefultoeliminatetidegaugesinFennoscandia,eliminatingthepotentialproblemofincludingtidegaugeswithapost-glacialrebound signal.In1936theAssociationd'OceanographiePhysiquebeganpublishingtidegauge recordsforallavailableglobaltidegauges( EmeryandAubrey ,1991).ThetaskofpublishingtidegaugerecordsfelltothePermanentServiceforMeanSeaLevel(PSMSL)in the1950's. Inthelate1970'sinterestinsealevelriseasanindicatorofglobalwarmingprompted Emery (1980)tocompileaglobalsetoftidegaugestogetamediansealevelchange rateof3mm/yr.Thisratewastheresultofcarefulconsiderationoftidegaugesfornoise 11

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andlandmotion,butEmeryrealizedthelargescatterinthemeasurementsduetoland motionmadethisestimateunreliable. Barnett (1984)realizedthedifcultyofcomputing reliablemeansealevelestimatesfromtidegaugesalsomeantthatattributingchangesto man-inducedclimatechangewouldbenearlyimpossible.Morerecently, Douglas (1991, 1995)hascomputedmeansealevelchangeestimatesbyselectingtidegaugeswithlong timeseriesandlittlelandmotion,resultingintrendswithapurportedprecisionofafew tenthsofamillimeterperyear. Properlycalibratedaltimeterscanalsobeusedtocomputeglobalsealeveltrendstoa fewtenthsofamillimeterperyear( e.g.,Leulietteetal. ,2004; LeulietteandMiller ,2009; Merrieldetal. ,2009),eventhoughthehigh-precisionaltimetryrecordisonly15years long.AsmentionedinChapter1,altimeterssurveyalmosttheentireglobaloceanandcan resolvelowfrequencymassredistributionsthattidegaugescannot.Thebiggestproblem withusingthealtimetertoestimateglobalsealevelchangeisuncorrecteddrifterrors, thatcanbeofthesameorderastheglobalsealevelsignal.Amethodwasdeveloped ( WyrtkiandMitchum ,1990; Mitchum ,1994,1998,2000)tousetheglobaltidegaugeto identifydrifterrorsinthealtimeter.Thebasicmethodofaltimeterdriftestimationdevelopedby Mitchum (1994)istondtidegaugeswithlong,reliablesealevelrecordsand nearbyaltimetrypassesandtakethedifferencebetweenthetwotimeseries.Bytaking thedifference,oceansignalsthatarecommontobothrecordscancelout,leavingaseries dominatedbydrift,aswillbediscussedinthenextsection.Tidegaugetimeseriesmust becorrectedforlandmotionbeforeitcanbecomparedtothealtimetryseasurfaceheight timeseries.Thedifferenceseriesforindividualtidegaugesarethencombinedinaglobal averagetomakeadrifttimeseries.Linear,quadraticandotherfunctionsarethentto thedifferenceseriestoidentifythemagnitudeandcharacterofanyexistingerrors.The 12

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methodofusingtheglobaltidegaugenetworktocalibratealtimetershassuccessfully identieddrifterrorsinthepast,aswillbediscussedindetailinSection2.4. 2.2TheMethodofAltimeterCalibrationUsingtheGlobalTideGaugeNetwork Thesuccessofthecalibrationmethoddependsonselectingthelargestnumberoftide gaugesthatmeetasetofcriteriadenedby Mitchum (2000).TherstcriteriaforselectionisthetimeperiodoverlapwithboththeTOPEX/PoseidonandJason-1altimeters. Thetidegaugetimeseriesspecicallymustoverlaptheyear2002,whenJason-1and T/Pwereyingintandem,inordertodetermineanyoffsetbetweenthetwoaltimeters. AnexampledriftseriesshowinganoffsetbetweenT/PandJasonin2002isshownin Figure2.1.Thesecondcriteriaforstationselectionisgoodoceansignalcancelationin thetidegauge-altimeterdifferences.Poorsignalcancelationcouldbeduetolocaleffectsoftopography,localseasonalcycles,ornonlinearlandmotionatthetidegauge. Ingeneral,differenceserieswithastandarddeviationgreaterthan100mmorwhere thetidegauge-altimetercorrelationislessthan0.3areremovedfromtheset.Thenal criterionisthatthetidegaugeisundergoinglinearlandmotion.Becauseweonlyhave GPSmeasurementsspanningafewyears,wemustassumethatthelandmotionislinear andcanbeappliedtotheentiretimeseries. Beforethedifferenceseriesarecomputed,somepreprocessingisperformedonboth thetidegaugesealevelseriesandthealtimetryseasurfaceheightseries.Thetidalsignals atthetidegaugesareremovedbyrstttingandremovingatidemodel(IOC,2006), followedbylowpassltering.ThedominanttidesareremovedfromtheT/Pheightsby ttingandremovingharmonicswithperiodsof59and62days.Becausethe M2and S2tideshaveperiodsofcloseto12hours,thesealiasintothe9.9-dayrepeatperiodofthe 13

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Figure2.1:CurrentDriftSeriesforTOPEX/PoseidonandJason-1.TheTOPEXAandB timeseriesisnowconsideredtobefreefromdriftatthecurrentlevelofuncertainty.The prominentfeatureisthelarge(roughly150mm)offsetofJason-1thatneedstobe determinedinthecurrentcalibration. T/PandJason-1trackas62-dayand59-dayperiods( FuandCazenave ,2001).ThealtimetricheightdataarealsosmoothedalongtrackusingaGaussiansetofconvolutionlter weightsthatpasssignalsatgreaterthan50percentamplitudeforwavelengthsgreater than90km.Smoothingremovesshortwavelengthfeaturesduetonoiseintheionospheric correctionappliedtothealtimeterdata. Thetidegauge-altimetercalibrationmethodworksasfollows.Webeginbydening thealtimetricseasurfaceheight, hnt,asafunctionofstation, n ,andtime, t ,as hnt= true t+ snt+h nt(2.1) 14

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where true tisthetruealtimeterdriftthatispresentatallstationsandisdependenton timeonly.Thedriftisassumedtobeanerrorinasatellitemeasurementoralgorithmthat isgloballypersistentandthereforenotvariableinspace.The snttermistheoceansignal atonepointonthealtimetrytrack.Theerrortermh ntistherandomerrorassociatedwith thealtimeterinstrumentnoise. Thetidegaugemeasuresrelativesealevel,whichisthesealevelrelativetotheland onwhichthetidegaugeisxed.Thisisnotthesamesignalastheoceansignalfromthe altimeterbecauseoflandmotionatthetidegauge.Wecanwritetherelativesealevel nt, asafunctionoftheobservedoceansignalatthatstationandtimeminuslandmotion nt= sntŠtrue nt+ntwheretrue ntisthetruelandmotionatthetidegaugeandntistherandomerrordueto instrumentnoiseandthespatialvariationbetweenthetidegaugeandthealtimeter.The tidegaugesealevelisnotanabsolutemeasureofseasurfaceheight,aswiththealtimeter, becausethetidegaugeisreferencedtolandthatmaybeinmotion.Thisiscalledrelative sealevel.Inordertocorrectforlandmotionwemustmakeanestimate,est nt,oftheland motionatthetidegaugeandaddittotherelativesealeveltimeseries,whichgivesnt= nt+est nt= snt+nt+nt(2.2) wherent=est ntŠtrue ntistheerrorofthelandmotionestimate.Wewillinvestigatethiserrorterminthenext section. 15

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Thedifferenceseriesarecomputedbysubtracting(2.2)from(2.1)toobtainnt= true t+nt+nt(2.3) wherentisthecombinedrandomerrorofthetidegaugeandaltimetryseries. Thedifferenceseries,nt,isactuallycomputedfromanumber( M )ofaltimeterpasses (denotedbythesubscript m )nearthetidegauge.These M nearbydifferenceseriesare combinedinaweightedaveragetothesingletimeseriesforeachtidegaugestation,which wedenotebythe n subscript.Specically,nt=Mm = 1wmnmtwheretheweightsaredenedby wm=Š 2 m Š 2 mand2 misthevarianceofthedifferenceofthe mthaltimeterpassseriesfromthetidegauge series.Thischoiceoftheweightsensuresthattheweightedaverage( i.e. ,thentvalues) willhaveminimumvariance( Beers ,1962). Thevarianceofthenaldriftseries( Beers ,1962)foronetidegaugeis2 n=Mm = 1 Mk = 1wmwkmkmkwheremkisthecorrelationbetweenthe mthand kthpassdifferenceseries.Thesecorrelationsareknownfrompreviouswork( Mitchum ,1998). Wenowcombineallthestationsateachtimetoformthedrifttimeseriesasaweighted average,giving est t=nwnnt(2.4) 16

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where wnaretheweightsand est tisthedriftseries.Thedenitionoftheweightsdependsonhowwechoosetominimizetheerroronthedriftseries.Astudyby Bernierand Mitchum ( inprep )examinestheweightfunctionindetail,butthespecicchoicedoesnot matterfortheworkwewillpresenthere.Theonlyconditionontheweightsisthattheir sumisone.Substituting(2.3)forntgives est t= true t+nwn(nt+nt) (2.5) Thesecondtermin(2.5)istheerroronthedriftestimateasafunctionoftime,whichwe willrefertointhefollowingas Et=nwn(nt+nt) (2.6) Inthe Mitchum (1998)study,thelandmotionestimatewastakentobezero,butthe errorwasconsideredtobeasystematicerrorwithastandarddeviationof1mm/yr.The Mitchum (2000)studyimproveduponthelandmotionestimatebyusinggeodetic(GPS andDORIS)landmotionestimateswhenavailableandmakinganestimatefromthetide gaugetimeserieswhengeodeticmeasurementswerenotavailable.Thecontributionof theselandmotionimprovementsdecreasedthestandarddeviationofthelandmotion uncertaintyto0.4mm/yr.Thisstudyaimstomakeamoreskillfullandmotionestimate andtofurtherdecreasetheerroronthedriftestimate. 2.3AddressingLandMotion Inordertoevaluatemethodsofmakinglandmotionestimates,weneedtoknowthe varianceofthedriftestimateerror,2 Et,whichdependsontherandomerrortermsand 17

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theerrorinmakingthelandmotionestimate.Inordertoevaluatethevarianceofthedrift estimateerror,wemustconsiderthemethodsoflandmotionestimationmorecarefully. Inthe Mitchum (2000)study,theideaofmakingalandmotionestimateusingonly thetidegaugetimeserieswaspresented.Wewillrefertothisastheinternallandmotion estimate.Tocomputetheinternallandmotionestimateweusethelongesttimeseriesof sealevelfromeachtidegauge.Themajorityoftidegaugedataaremonthly-meansea leveltimeseriesfromthePermanentServiceforMeanSeaLevel(PSMSL)archives.If themonthlydataarenotavailableorthemonthlyseriesareshorterthan20years,then dailysealevelseriesareobtainedfromtheUniversityofHawaiiSeaLevelCenter'sFast DeliveryDataset.Wethentamodelwithatrendtotheseriesandassumethistrendis trueglobalsealevelchangeminusthelandmotion. Theinternallandmotionratenicanthenbecomputedasi n= oest tŠ orel nt(2.7) where orel ntisthettedrelativesealeveltrendand oest tisanestimatedglobalmeansea levelchangerateduetooceanvolumechangesoverthetimeperiodsspannedbythetide gaugerecord.Theglobalaveragesealevelchangerateisestimatedfromthe Church andWhite sealevelreconstruction( ChurchandWhite ,2006)availableonthePSMSL website.Theestimatedsealevelchangerateiscomputedbyttingatrendtothereconstructiondataspanningthedurationofthetidegaugetimeseries.Inthiswaywecan accountforanyaccelerationintheglobalsealevelchangerate,especiallyfortidegauges onlyspanningrecentdecades.The Mitchum (2000)studyuniversallyappliedaglobalsea levelchangerateof1.8mm/yr( Douglas ,1995)toalltimeseries,regardlessoflength. Thedifferencebetweenthe ChurchandWhite reconstructionandthepreviouslyusedrate of1.8mm/yrisgenerallysmallerthan0.25mm/yr,butimprovestheinternalestimate. 18

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Theerrorincomputingtheinternallandmotionestimateinthismannerisfromthree differentsources.Therstsourceoferrorisasystematicerrorequaltoanydifference between oest tandthetruegloballyaveragedsealevelriserate.Thiserroristhesameat everytidegaugeandwillnotaveragedown,whichiswhyweidentifyitasasystematic, orbias,error.Asecondsourceoferroristhedifferencebetweenthegloballyaveraged rateandthetruerateateachgauge;i.e.,theremaybespatialvariationsinthetruerateof sealevelchange.Theseerrorswillaveragedown.Thethirdsourceoferroristherandom errortermarisingfromthettingofatrendtothetidegaugesealevelseriesthatcontains variability. Someoftherandomerrorinttingatrendiscausedbylowfrequencysignalsinthe tidegaugerecordthatareduetomassredistributionsintheocean.Thesesignals,dueto oscillationssuchasENSO,canhavealargevariabilityandwillcontributealargerandom errortermtothelandmotionestimate.Theserandomerrorscouldbeminimizedbyusing longertimeseries,wherethelowfrequencyoscillationsaverageout,orbymodeling andremovinglowfrequencysignalsfromthesealevelseries.Tominimizetheerrors associatedwithENSOvariability,theSouthernOscillationIndex(SOI)andtheHilbert transformoftheSOIwereusedtomodeltheinterannualvariabilityinthetidegaugesea leveltrendregression.IncludingtheHilberttransformoftheSOIallowsforphasevariationofthesealevelresponsetoENSOatdifferenttidegauges.Themodelalsoincludes seasonalvariations.Resultsforsomestationsshowedincreasedwhiteningoftheresiduals ofthesealeveltrendtandareductionoftheerrorinthelandmotionestimate,while otherstationsshowedlittleimprovement.Forexample,applyingtheSOImodeltothe sealeveltrendtatPohnpei,inthetropicalPacic,capturedmuchoftheinterannual variabilityandreducedthevarianceby50percent,from9200to4400mm2(Figure2.2). 19

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1975 1980 1985 1990 1995 2000 400 200 0 200 400 Sea Level (mm)Sea level at Pohnpei and model incluing SOI 1975 1980 1985 1990 1995 2000 400 200 0 200 400 Residual sea level at PohnpeiSea Level (mm)Figure2.2:Sealevel,modelt,andresidualsatPohnpei.Intheupperpanel,thesolid bluelineisthemonthlymeansealevelfromthePohnpeitidegauge.Thereddashedline isthemodelincludingamean,trend,annualandsemiannualcycles,theSOIandthe HilberttransformoftheSOI.Thetresidualafterthemodelisremovedisshowninthe bottompanel.Theresidualshowsthatthevariabilityissignicantlyreducedbyincluding theSOIinthemodelt. Anotherwayofmakingalandmotionestimateisanexternal'estimate,byusing verticalpositionsfromgeodeticsatellitessuchasGPSorDORIS.Forthisstudy,GPS timeseriesandverticallandmotionrateswereprovidedbyGuyWoppelmannatUniversityLaRochelle.Thetimeserieshavebeenprocessedandanalyzedtoobtainthebest verticalpositionsandratespossible.Fordetailsontheprocessingsteps,see Woppelmann etal. (2007).Thenalstepintheprocessingisttingamodeltothedataincludinga trend.Themodelincludesamean,trend,annualandsemi-annualcycles,andinsome instancesstepfunctions.Verticalstepscausedbyequipmentchangesmustbeidentied andincludedinthemodelinordertotanaccuratetrendtotheGPStimeseries.The 20

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inclusionofthesestepfunctionsinthemodeltintroducesextrauncertaintytothetrend sincethetrendandstepfunctionsarenotindependent.AnexampleofaGPStimeseries withverticaloffsetsisEasterIsland(EISL)showninFigure2.3.EasterIslandhastwo verticaloffsets,onein1998andanotherin2003.SomeGPStimeserieshaveupto5 offsets,asequipmentchangesarecommon.Thelargerthenumberofoffsets,themore uncertaintyneedstobeaddedtothetrend.Amethodforestimatingtheerrorcontribution ofstepfunctionsisdiscussedinthefollowingchapter. 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 80 60 40 20 0 20 40 60 80 GPS Vertical Position (mm)Figure2.3:EasterIslandGPStimeseries.TheprocessedGPSverticalpositionsare showninblue,whilethemodeltincludingmean,trend,annualandsemi-annualcycles, andstepsisshowninred.Therearetwoverticaloffsets,onein1998andanotherin2003. ItisnotuncommonforGPStimeseriestohaveseveralverticaloffsets. Inmakingthebestlandmotionestimate,shouldtheinternalandexternalestimates beweightedequally?Inthe Mitchum (2000)study,theexternalestimatewascomputed bycombiningupto7GPSstationswithin1000kmofthetidegauge.Thenthenalland motionestimatewascomputedasaweightedaverageoftheinternalandexternalesti21

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mates.Thelandmotionestimateatanytidegauge(est n)isthuswrittenest n= wi ni n+( 1 Š wi n)e n(2.8) wherei nistheinternallandmotionestimate,ande nistheexternallandmotionestimate. Aswiththestationweights wnin(2.4),the wi nareconsideredknownforourpurposes. Wecannowderivethelandmotionerrorcontributionntasin(2.6)intermsofthe internalandexternalestimatesgivenin(2.8). Earlierwedescribedhowwecomputei nbyttingatrend, orel nt,totherelativesea levelseries(2.7).Weformanexpressionforthetruelandmotionrateateachstationtrue n,intermsofthettedtrend, orel nt,andthetruelocalsealevelchangeateachstation, otrue nt.true n= otrue ntŠ orel nt+ntwhichwecanrewriteastrue n=( oest tŠ orel nt) Š ( oest tŠ otrue t) Š ( otrue tŠ otrue nt)+nt( 1 )( 2 )( 3 )( 4 ) where otrue tistheunknowntrueglobalaveragesealevelchangerate.Thersttermisthe internallandmotionestimatefrom(2.7)andistheonlypartoftheaboveequationthat canbedirectlyobservedandcomputed.Term(2)representsthebiaserrorduetothefact thatthetrueglobalaveragesealevelchangemaydifferfromtheestimatewecompute fromthe ChurchandWhite (2006)reconstruction.Forsimplicity,wewillrefertothis biaserroras b0.Term(3)representstherandomerrorduetolocaluctuationsfromthe trueglobalaveragesealevelchangerate.Ifthetrueglobalaverageratewereknown,then 22

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theerrorswouldaveragetozero.Wewillrepresentthiserroras bn.Term(4)represents therandomerrorsduetoinstrumenterrorandttingerror.Substitutinginterms b0and bnforterms(2)and(3)andthedenitionfori nfrom(2.7)wehavetrue n=i nŠ b0Š bn+ntwhichwecanrearrangetoobtaintheinternallandmotionestimateasi n=true n+ b0+i nt(2.9) wherethe bntermhasbeenincludedintherandomerrortermnt.Inasimilarfashionwe canwritetheexternallandmotionestimatease n=true n+e nt(2.10) wheree ntisthearandomerrorarisingfromthettingofthetrendtotheGPSseriesthat containsvariability.Thisrandomerroralsoincludesanestimateofthereferenceframe z-translationerrorthatvarieswithlatitude,,as ( 1mm yr)2sin2( AltamimiandCollilieux 2009).Notethatthereisnobiaserrorincludedin(2.10).Thisassumptionmayneedtobe re-examinedinthefuture. Substituting(2.9)and(2.10)into(2.8)wegetanexpressionforthelandmotionestimateest n= wi n(true n+ b0+i nt)+( 1 Š wi n)(true n+e nt) Theerroronthelandmotionestimatedenedin(2.2)cannowbewrittenasnt= wi nb0+ wi ni nt+( 1 Š wi n)e nt23

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Thenalerrortermforthedriftseriesasin(2.6)becomes Et= b0nwnwi n+nwnwi ni nt+nwn( 1 Š wi n)e nt+nwnntWhiletherandomerrorsarereducedthroughaveragingovermanystations,thebiaserror isnot. Notehowtheerrorduetothelandmotiondependsontheerrorsduetottingtrends tothetidegaugeandGPSseries.Unlesstheseerrorsareestimatedrealistically,theerrors duetolandmotioncannotbe.Inturn,then,theerrorsonthesatellitedriftestimateswill alsonotbeproperlyassessed.Thisillustratestheimportanceofobtainingthebesterror estimatespossible,whichisthesubjectofthenextchapter.Beforedoingthat,however, wewillgiveabriefhistoryofhowthismethodwasdeveloped,andhowitcomparesto othercalibrationmethods. 2.4EarlierWork While Mitchum (1994)wasthersttoincorporateaglobalsetoftidegaugesforaltimetercalibration,othercalibrationswereperformedusingasingletidegaugeforhigh precisioncalibration.Singlecalibrationsitescanbeonetidegauge,suchastheTOPEX/Poseidon dedicatedsiteattheHarvestoilplatform( Christensenetal. ,1994),orasetoftidegauges inasmallareasuchasthetidegaugenetworkintheEnglishChannel( Murphyetal. 1996).Atsinglecalibrationsitesthesatellitepassesdirectlyoverthetidegauge.These sitesareoutttedwithpreciseinstrumentsandmeteorologicalsensorstodetectspecic errorsinthealtimetrymeasurement.Theadvantageofthesinglesiteapproachisthat itgivesanabsolutecalibrationandcanidentifyspecicinstrumenterrors,whereasthe globaltidegaugeapproachcanonlydetecttime-varyingerrors( i.e. ,drifts)inthenalsea 24

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levelmeasurement.Fordriftestimation,however,theglobalapproachhasbeenproven thebestmethod. Anotherapproachtosatellitecalibrationwasdevelopedby MorrisandGill (1994) usingtidegaugesintheGreatLakes.TheadvantageofusingtheGreatLakestidegauges isthatlakelevelsarewellmonitoredandthereareminimaltidessothatthevarianceof thedifferencebetweentidegaugeandaltimetertimeseriesissmallerthanistypicalatan openoceansite.Usingthissmallsetoftidegaugesissimilartothesinglesiteapproach, however,becausethetidegaugetimeseriesarehighlycorrelated,andarethusnottruly independent.Thismeansthatthevarianceofthealtimeterdriftestimateseriesdoesnot decreasethroughaveraging. Thereasonthattheglobaltidegaugeapproachispreferredisquitesimple.Ifeach tidegauge-altimeterdifferenceserieshasatypicalvarianceof2,thenbyaveraging N stations,thevariancebecomes2/ N ifthedifferentgaugeseriescanbeconsidered tobeindependent,whichhasbeenshowntobeagoodassumption.Thevarianceinthe differenceseriesatatypicalopenoceantidegaugesiteislargerthanatacalibrationsite, orintheGreatLakes,butthefactorof N provestobemuchmoreimportant. Therstcomparisonoftidegaugeseasurfaceheightswithsatellitealtimetrywas performedasawayofevaluatingtheGeodeticSatellite(GEOSAT)altimeter( Wyrtkiand Mitchum ,1990).Theseauthorsusedthetidegauge-altimetercomparisontoidentifya driftintheGEOSATseasurfaceheightmeasurementthatwasgeographicallycorrelated inthetropics.Thedrifthadastrongeast-westgradientandwasassociatedwiththeEl Ninoeventof1986-87.TheGEOSATaltimeterdidnothavearadiometertomakeatmosphericwatervaporcorrectionsandtheeast-westgradientofwatervaporduringthe strongElNinocausedacorrelateddriftinthesatelliteseasurfaceheight.Subsequent 25

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altimetrymissionsaddedaradiometeronboardthesatellitetomeasureatmosphericwater vapor. Thetidegaugecomparisonmethodwasexpandedgloballyandfurtherdeveloped afterthelaunchofthemorepreciseTOPEX/Poseidon(T/P)altimeterin1992. Mitchum (1994)madeapreliminarycomparisonoftidegaugeswiththerst30cyclesofT/Pdata thatillustratedthepotentialfortidegaugecalibrationontheT/Paltimeter.However,the methodofusingtidegaugestoactuallyestimatedriftofthealtimeterandtomonitorits stabilitywasnotwidelyaccepteduntilaT/Palgorithmerrorwascorrectlyidentiedby comparisonoftidegauge-altimetrydifferences( Mitchum ,1998).Thealgorithmerror wasanerrorintheprocessingofthedatathatproducedaslowtemporaldriftintheT/P seasurfaceheights.TheT/Pdriftseriesbeforeidenticationofthealgorithmerroris showninFigure2.4.Thedriftseriesclearlyshowstheshapeofthedriftcausedbythe algorithmerror.Afterthealgorithmerrorwascorrected,the Mitchum (1998)studyalso identiedaremainingdriftof Š 2 6 0 6mm/yrintheT/Pdata.Tocheckifthedrift mightbeduetosomeenvironmentalcorrection( i.e. ,watervapor)thetidegaugecalibrationwasrepeatedusingonlytidegaugeswithin15oftheequatorandthenagainwith tidegaugesmorethan15fromtheequator.Theresultswere Š 3 7 1 0mm/yrand Š 1 2 1 0mm/yr,respectively,indicatingameridionallyvaryingdriftrate.Thisdriftwas subsequentlyidentiedasanerrorinthewatervaporcorrection,givingthecommunity furthercondenceinthemethod. Mitchum (2000)furtherrenedandextendedtheglobaltidegaugeapproachformeasuringaltimeterdrift,thistimeaddinglandmotionestimatesfromgeodeticmeasurementssuchastheGlobalPositioningSystem(GPS)anditsFrenchcounterpart,DORIS (DopplerOrbitographyandRadiopositioningIntegratedbySatellite).Thiswasanimportantstepforwardbecause Mitchum (2000)alsoshowedthatlandmotionsatthetide 26

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Figure2.4:TOPEXAlgorithmErrorIdentiedby Mitchum (1998).Thesolidcirclesand errorbarsaretheresultsoftheTOPEXdriftanalysisusingtidegauges.Theidentied algorithmerrorisshownasthedashedline.The Mitchum (1998)studyprovedthe usefulnessofthecalibrationmethodbyshowingthataltimeter-tidegaugecomparison correctlyidentiedthealgorithmerror. gaugeswerethedominantsourceoferrorforsealevelriseestimatesmadefromthealtimetricdata.Atpresentthesealevelriseerrorbarisabout0.4mm/yr( Mitchum ,2000), andthisuncertaintyismostlyduetotheproblemofestimatinglandmotionatthetide gauges.Improvingthegeodeticlandmotionestimatesisthesecondmajorfocusofour work,andisdescribedinChapters4and5. 27

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Chapter3:MethodsofErrorEstimation AsdiscussedinChapter2,assigningrealisticerrorbarstotheinternalandexternalland motionestimatesiscriticalforformingthebestaltimeterdriftestimation.ThelandmotionerrorclearlyaffectsthetrendttothetidegaugeorGPStimeseries,butthets involveotherbasisfunctionssuchasseasonalcycles,theSOIindex,andstepfunctions. Itisimportanttoestimatetheerrorsontheseotherbasisfunctionsalongwiththetrend error,andtodeterminehowtheadditionalbasisfunctionsaffecttheerrorinthetted trend.Wewilladdresstheproblemofassigningpropererrorbarsasamultiplelinearregressionofatimeseriesonasetofarbitrarybasisfunctions,andaskaboutallparameter errorbars. InSection3.1wewillreviewseveralexistingmethodsforestimatingerrorsanddemonstratethelimitationsofthese.Insection3.2wewillderivethemethodthatweproposeto overcometheselimitations,andinSection3.3wewilltesttheexistingmethodsandour proposedmethodusingasetofnumericalsimulations. 3.1LimitationsofExistingErrorEstimationMethods Theclassicalapproachtodeterminingerrorsonthetparametersfromasetofbasis functionsillustratesthedifcultyinassigningarealisticerrorbartothelandmotion estimate.Assumingthatthetimeseriesisequaltothemodelpluswhitenoise,according 28

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totheclassicalapproach,giveswhatisoftenreferredtoastheformalerrorsofthet parameters.Therearetwoproblemswiththisassumption.First,thenoiseinthetide gaugeandGPStimeseriesistypicallyred,notwhite,becauseoflow-frequencysignals intheoceanandatmosphere(Figure3.1).Assumingthetimeseriesiswhitewhenit isredleadstooverestimationofthedegreesoffreedom,whichinturnleadstounderestimationoftheerrorbars.Thesecondproblemwiththeclassicalmethodisthatthe ttedmodelincludescontributionsfromnoise,whichwedeneasanyvariabilitynot includedinthemodel,sothattheresidualtimeseriesstatisticsarenotthesameasthe truenoiseseriesstatistics.Theunintendedremovalofnoisevariancebythemodelleads tounderestimationoftheerrorbars.Inthecaseofttingatrendthelowfrequenciesare particularlyaffected(Figure3.2).Wewillrefertothisproblemas"overtting",sincethe basisfunctionshaveremovednoiseinadditiontotheunderlyingmodel. 10 -410 310 210 110 010 310 210 110 010 110 2 FrequencyPower Spect ral Densit y 10 -410 310 210 110 010 -410 210 010 2 FrequencyPower Spect ral Densit y A BFigure3.1:ExamplespectrafromtheDarwin,Australiatidegauge(A)andtheAnnette Island,AlaskaGPSstation(B).Notethatthetidegaugeresidualspectrumhastheshape ofapowerlawspectrum,whiletheGPSresidualspectrumhasanAR(1)shape.These twospectraaretypicalexamplesofthelow-frequencyvariancethatispresentintide gaugeandGPSresidualtimeseries. 29

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100101102102103104105Fourier FrequencyPower Spectral Density Average Original Spectrum Average Residual Spectrum Figure3.2:RemovalofLow-FrequencyVariancebyFittingaTrendandSteps.Therst 100FourierFrequenciesoftheaveragespectrumof1000rednoise( fŠ 1)realizationsis plottedinblue.Theaveragespectrumoftheresidualsisplottedinred.Notethedecrease invarianceatthelowestfrequencieswheremostofthevarianceiscontained. Paststudieshaveattemptedtoaddresstheunderestimationoftheformalerrordue torednoiseintidegaugeandGPStimeseriesusinganumberofdifferentmethods.Via propagationoferror( Beers ,1962),wecanobtainestimatesoftheparametererrorsthat arecomputedbyintegratingthelaggedcovariancesequence.Thelaggedcovariancesequencecanbedirectlycomputedfromtheresiduals,buttheintegraloverthelaggedcovariancesequenceisnotareliablecalculation,aswillbediscussedbelow.Toavoidthis uncertainty,amodelcanbettothelaggedcovariancesequenceandthenintegratedto obtainanerrorestimate.Forexample, NeremandMitchum (2002)taGaussianfunction tothelaggedcovariancesequenceinordertoobtainmorereasonableerrors(ascompared 30

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totheformalerrors)fortrendsttotidegaugetimeseries.Amorecommonapproachis toassumethenoiseisarstorderautoregressive,AR(1),processinordertotthelagged covariancesequence( e.g. MaulandMartin ,1993).Theautoregressivemodelcanthen beintegratedanalyticallytoestimatetheeffectivedegreesoffreedomfordeterminingthe trenderrorbar. OtherstudieshaveaddressedtheproblemofrednoiseinatidegaugeorGPStime seriesusingthespectralequivalentofthelaggedcovariancemethoddescribedabove.The spectralapproachisbasedontheWeiner-Khinchinerelation( BendatandPiersol ,1986), whichprovesthattheFouriertransformofthelaggedcovariancesequenceisequaltothe spectrumofthetimeseries.Thismethodispreferredoverthelaggedcovariancemethod becausethelaggedcovarianceestimatesarenotindependentandhaveacomplicated, non-Gaussian,distribution( Bloomeld ,1976).Ontheotherhand,theestimatesofthe spectrumareapproximatelyindependentandareknowntobechi-squareddistributed. Bothofthesefeaturesareimportantadvantageswhenweneedtotamodeltothespectralestimates. Aswiththelaggedcovariancemethod,amodelcanbettothespectrumoftheresidualsandintegratedtoobtainanexpressionfortheeffectivenumberofdegreesoffreedom.TypicalmodelsfortidegaugeandGPStimeseriesincludeAR(1)( e.g. Mannand Lees ,1996)orpowerlaw( e.g. Maoetal. ; Williamsetal. ,2004).Thespectralapproach canalsobeusedinMonteCarlosimulationswhereamodelisttothespectrumofthe residualsandthenusedtogeneratemultiplerealizationsofthenoiseseries.Anexample ofthisapproachforapowerlawspectralmodelisgivenby Maoetal. .Anothermethod, originallysuggestedby Thompson (1973)andmorerecentlydescribedby Ebisuzaki (1997), hasrecentlybeenused( Leulietteetal. ,2004)tocorrecttrendestimatesforserialcorrelationinseasurfaceheighttimeseries.ThismethodcomputestheFouriertransform 31

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oftheresidualsandobtainsnoiserealizationsinfrequencyspacebyreplacingthephase withauniformrandomphase.TheinverseFouriertransformisthencomputedtoarrive atarealizationofanoiseseriesthatcanbeusedinMonteCarlosimulations.Allofthese methodshaveprovenusefulincorrectingforserialcorrelationinatimeseries,butnone addresstheproblemofoverttingaswewilldo. Inthefollowingsectionswewillderiveamethodofestimatingparametererrorbars foramultiplelinearregressionofatimeseriesonasetofarbitrarybasisfunctionsbased onthespectrumoftheresiduals.Thismethodwilladdresstheproblemoferrorunderestimationduetorednoiseandwillincludeacorrectionfactorforestimatingthenoise variancethatislostduetoremovalbythebasisfunctions.Wewillthencomparethe proposedmethodwiththepastmethodsdescribedaboveforavarietyofnoisetypesand basisfunctions. 3.2TheProposedMethodofErrorEstimation Ourproposedmethodoferrorestimationdoesnotrelyonsimulationorttingthe spectrum,butisdirectlycomputedfromthespectrumoftheresiduals.Theproposed methodalsoestimateshowmuchvarianceisremovedbyovertting,asdiscussedabove. Beforewebegin,letusdenesomeofthebasicmathematicalexpressionswewillbe usinginthissection.First,wedenetheFouriertransformofareal-valuedseries xnto be Xkwhere Xk=nxncnk+ inxnsnkand cnkand snktermsarethesineandcosinetermsthataredenedby cnk= cos 2nk N 32

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snk= sin 2nk N Wewillalsobeusingthespectrumoftheseries, Sk,denedas Sk= XkX kwheretheasteriskdenotesthecomplexconjugate. WedenetheobservabletimeseriesoftidegaugeorGPSheights, hr n,tobethetrue timeseriesplusonerealizationofthetruenoise.Wetanarbitrarysetofbasisfunctions asibr ifni(3.1) where br iarethetcoefcientsforthisonerealizationand fniarethebasisfunctions.We assumethatthebasisfunctionsareorthonormal,meaningthatnfni 2= 1nfnifnj= 0 Assumingthatthebasisfunctionsareorthonormaldoesnotrestrictthegeneralityof themethodbecauseweuseGram-Schmidtorthogonalization( Arfken ,1985)totransform thearbitrarysetofbasisfunctions.Wethenobtainthecovariancematrixforthetted orthonormalparameters.Sincetheparametersfortheoriginalsetofbasisfunctionsare linearcombinationsofthettedorthonormalparameters,wecanusepropagationoferrortoobtainthecovariancematrixfortheparametersfromthecovariancematrixofthe orthogonal,ttedparameters. 33

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Continuingwithourdenitions,if(3.1)isonerealizationofthettedmodel,thenwe candenethetruemodelparametersas btrue i= E [ br i] (3.2) wherethe E [] operatordenotestakinganexpectationvalue.Theparametererrorsforthe onerealizationthatweactuallyobservearethusr i= br iŠ btrue i. (3.3) Wealsoneedtoestimatethecovariancematrixoftheparametererrors,whichis2 ij= E [r ir j] (3.4) Becausethetrueparametersarenotknown,weneedamethodtoestimatethisexpression usingobservablequantities. Thetruenoiseforonerealizationofatimeseries hr ncanbedenedasr n hr nŠibtrue ifni(3.5) whereastheobservednoiseseriesfromoursinglerealizationis r n= hr nŠibr ifni(3.6) Takingthedifferenceof(3.5)and(3.6)gives r n=r nŠir ifni(3.7) 34

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whichquantiestheeffectofwhatwearecallingovertting.Wemustnowmakean estimateforthesecondtermin(3.7)inordertocorrecttheobservederrorstatistics(as estimatedfromtheresidualsfromthet)forthiseffect. Fromtheleast-squaressolutionthetparametersaregivenby br i=nhr nfni(3.8) becauseoftheorthonormalitycondition.Solving(3.5)for hr nandsubstitutinginto(3.8) yields br i=njbtrue jfnjŠr n fniwhichcanbeexpandedas br i=jbtrue jnfnjfniŠnr nfni(3.9) Becausethebasisfunctionsareorthonormal,thersttermof(3.9)isonlynon-zerofor i = j andwecansubstituteinto(3.3)toobtainanexpressionforr iintermsoftheerror ofeachrealizationandthebasisfunctions,whichgivesr i= br iŠ btrue i=nr nfni(3.10) Nowthatwehaveanexpressionforr iintermsofthe(unknown)trueerrorseriesandthe basisfunctions,wemustevaluatetheexpressionforthecovariancesgivenin(3.4). Substitutingtheexpressionforr ifrom(3.10)into(3.4)givesus E [r ir j]= E [nmr nr mfnifmj]=nmfnifmjE [r nr m] (3.11) 35

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and E [r nr m] canbewrittenusingthespectraldenitionofthelaggedautocorrelation accordingtotheWeiner-Khinchinetheorem( BendatandPiersol ,1986)as E [r nr m]= 1 N2 N / 2k = Š N / 2Skcos 2k N ( n Š m ) where Skisthetruenoisespectrum.Weexpandthecosinetermto E [r nr m]= 1 N2 N / 2k = Š N / 2Sk( cnkcmk+ snksmk) (3.12) where cnkand snkarethecosineandsinetermsasdenedatthebeginningofthissection. Substituting(3.12)into(3.11)yields E [r ir j]=nmfnifmj 1 N2kSk( cnkcmk+ snksmk) whichcanbewrittenas E [r ir j]= 1 N2kSknfnicnkmfmjcmk+nfnisnkmfmjsmk (3.13) ReferringtothedenitionoftheFouriertransformweseethattheterminbracketscanbe writtenmorecompactlyas Sij k= ( FkiF kj) (3.14) inordertowrite(3.13)as E [r ir j]= 1 N2kSkSij k. (3.15) Equation(3.15)isanexpressionforthevarianceofr iinthecasewhere i = j .For i = j weobtainthecovariancesbetweentheparameters. 36

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Ifweknewthespectrumofthetruenoise( Sk),wecouldcomputethecovariance matrixvia(3.15),butwedonotobservethisspectrum.Whatweobserveisthespectrum oftheonerealizationofnoise( r n)fromtheresidualsthatwehave.Wewillnowusethis toderiveanestimateofthetruenoisespectrum. WebeginbytakingtheFouriertransformof(3.7)toobtain Gr k= Gr kŠir iFkiwhere Gr k,and Gr karetheFouriertransformsof r nandr n,respectively,and Fkiisthe Fouriertransformof fni.Nowmultiplybythecomplexconjugatetoobtain Gr k G r k= Gr kG r k+ijr ir jFkiF kjŠ Gr kir iF kiŠ G kir iFki(3.16) Takingtheexpectationof(3.16)leadsto Sk= Sk+ijFkiF kjE [r ir j] ŠiF kiE [ Gr kr i] ŠiFkiE [ G r kr i] (3.17) where Skisthetrueunderlyingnoisespectrumand Skistheexpectationofthespectrum ofresidualsfromthemodeltthatisbiasedawayfrom Skbecauseofovertting. Wealreadyhaveanexpressionfor E [r ir j] from(3.15),soweonlyneedtoevaluate theothertwoexpectationsappearingin(3.17),namely E [ Gr kr i] and E [ G r kr i] .Because theparametersarereal-valued,thesecondexpectationvalueisjustthecomplexconjugate oftherst,whichmeansthatweonlyneedtoevaluatetherstexpectationandthentake thecomplexconjugateoftheresultinordertoobtainthethirdterm. 37

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Usingthedenitionofr ifrom(3.10)weexpandtheexpectation E [ Gr kr i] as E [ Gr kr i]= E nr nfnimr m( cmk+ ismk) Wecanrewritethisas E [ Gr kr i]=nmfni( cmk+ ismk) E [r nr m] andsubstitutefrom(3.12)for E [r nr m] toobtain E [ Gr kr i]=nmfni( cmk+ ismk) 1 N2pSp( cnpcmp+ snpsmp) Thiscanberearrangedto E [ Gr kr i]= 1 N2pSpnfnicnpmcmp( cmk+ ismk)+nfnisnpmsmp( cmk+ ismk) (3.18) WenowevaluatethetrigonometricsumsandsimplifythenotationusingtheKronecker delta,nm,whichisdenedbynm= 1 n = mnm= 0 n = m toobtainmcmpcmk= N 2 [pk+N Š p k]msmpsmk= N 2 [pkŠN Š p k] 38

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andmcmpsmk= 0msmpcmk= 0 Usingtheseresults(3.18)becomes E [ Gr kr i]= 1 N2pSpnfnicnp N 2 (pk+N Š p k) +nfnisnp i N 2 (pkŠN Š p k) whichwecanfurthersimplifybynotingthat cnppk= cnpN Š p k= cnkand snppk= Š snpN Š p k= snkbecauseofthesymmetryofthesineandcosinefunctions.Thisallowsustowrite E [ Gr kr i]= Sk Nnfni( cnk+ isnk) whichcanberecognizedas E [ Gr kr i]= 1 N Sr kFkiAsnotedabove,thenaltermthatweneedissimplythecomplexconjugateofthisone, meaningproportionalto F ki. 39

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Wenowhaveexpressionsforalloftheexpectationsappearingin(3.17)andwesubstitutethesetond Sk= Sk+ 1 N2pSpijFkiF kjSij pŠ 1 N SkiF kiFkiŠ 1 N SkiFkiF kiUsingtheearlierdenitionof Sij kfrom(3.14),thismaybewrittenas Sk= Sk+ 1 N2pSpijSij kSij pŠ 2 N SkiSii k. (3.19) Thisexpressionrelatesthespectrumofthetruenoise, Sk,totheexpectationvalueofthe (observable)spectrumobtainedfromthetresiduals, Sk. Wecannotsolve(3.19)for Sktoderiveaclosedformsolutionforthetruenoisespectruminterms Sk,butwecanobtainanapproximatesolutionbyassumingthatthetrue spectrumisproportionaltotheobservablespectrumwiththeproportionalityfactor,which wewillcall k,givenasafunctionoffrequency.Thatis,wedene Sk= k Sk. Specically,uponrearranging(3.19)wecanwrite Š 1 k= 1 + 1 N2ppijSij kSij pŠ 2 NiSii k. (3.20) Westillcannotsolvethisexplicitlyforthe kvalues,butwechosethisapproachbecause wefoundthatweobtainaccuratesolutionsto(3.20)withasimpleiterativeapproach.We rstassumethatthetruespectrum Spiswhitenoisewithaconstantspectrumdenedby Sp= N 2 40

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Wethencomputethe kvalues,whichareusedtoestimateanimproved Spspectrum. Wethendooneadditionaliterationtoobtainournal Spestimate.Overawiderange ofnumericalsimulationswefoundthatthisiterativeapproachworkedextremelywell despitethesimplicityoftheapproach.Wewilldiscussthisfurtherinthenextsection whenwetestourproposedmethod. Inthenextsectionwewillshowresultsfromtestingourproposedmethodagainstthe previouslyproposedmethodsthatwereviewedearlier.Beforedoingthis,however,we wanttogiveapreciseexplanationofwhatweareproposingtodo.Thederivationthat wehavedoneallowsustoestimatetheunderlyingnoisespectrum,andhencetomake accurateerrorcovarianceestimates, if wehaveanestimateofthe expectationvalue ofthe spectrumoftheresidualsfromthettedmodel.Inordertoapplyourmethod,wetakethe singlerealizationwehavefortheresidualsfromthet,computethespectrumandusethis asourestimateoftheexpectationvalue.Thisspectrumisthencorrectedforovertting inordertomakeourestimateofthetruenoisespectrumviathe kestimates.Whether thisapproachisanimprovementoverthepreviousmethodswillbedeterminedbyaset ofnumericalsimulationscomparingourmethodtotheothermethodswhenthecorrect answerisknown.Thisisthetopicofthefollowingsection. 3.3TestingtheProposedMethodofErrorEstimation Weshowedearlier(Figure3.2)theeffectofoverttingonthespectrumobtainedfrom theresidualtimeseries.Theobviousrstcheckiswhetherourcorrectionforovertting via(3.20)forthisiscorrect,atleastintheexpectationsense.InFigure3.3wehaveadded thecorrectedspectrumtoFigure3.2.Notethatatallfrequencies,thecorrectedspectrum isclosetothetruenoisespectrum.Thisisencouraging,butobviouslyfurthertestingis 41

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required.Wewillnowtestourmethodusingavarietyofnoisemodelsandbasisfunctions andcomparetheproposedmethodwiththeformalerrorandtheinatederrorsobtained byMonteCarlosimulationwithAR(1)andpowerlawspectralmodels. 100101102102103104105 Fourier FrequencyPower Spectral Density Average Original Spectrum Average Residual Spectrum Average Corrected Spectrum Figure3.3:ASpectrumCorrectedfortheRemovalofLow-FrequencyVariancebyFitting aTrendandSteps.Therst100FourierFrequenciesoftheaveragespectrumof1000red noise fŠ 1realizationsisplottedinblue.Theaveragespectrumoftheresidualsisplotted inred.Theaveragecorrectedspectrumisplottedingreen.Notethatthedecreasein varianceatthelowestfrequencieshasbeencorrectedbyusingthenewmethodoferror ination. Thetestingprocedureworksasfollows.Wechooseasetofbasisfunctionsanda known,whatwewillcalltrue,errormodel.Theerrormodelcanbewhitenoise,AR(1) spectrum,orpowerlawspectrum.FortheAR(1)andpowerlawcases,arangeofcoefcientsareusedtosimulatenoisethatvariesfromnearlywhitetoveryred.Wethen generateatimeseriescomprisedofthemodelplusonerealizationfromthegiventrue 42

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noisemodelandtthemodelparametersusingleastsquares.Wekeepthettedtrend estimateandalsomakeestimatesofthevarianceofthetrendbasedontheresidualsfrom thetusingourmethod,theEbisuzakimethodandmethodsbasedonttingeitheran AR(1)orpowerlawspectrum.Werepeatthis10000timesandestimatethe true variance asthevarianceofthe10000trendestimatesaccordingto(3.2).Finally,weestimatethe variancesonewouldmostlikelyobtainfromeachofthedifferingmethodsasthemean valueoverthe10000casesofthevariancesobtainedforeachmethod.Theresultswillbe presentedastheratioofthecalculatedvariancestothetruevariances. Firstwetestthemethodonthesimplestmodel,ameanandatrend.Theresultsfor theproposedmethod,AR(1)model,powerlawmodelandformalerrorareshownin Figure3.4.Thevarianceestimatedbyourproposedmethodiswithin5percentofthe truevarianceofthetrendforallnoisemodels.Theformalerrorgrosslyunderestimates forallnoisetypesexceptwhitenoise,asexpected.TheEbisuzakimethod,sinceitdoes notdependonthenoisemodel,underestimatesnearlyequallyforallnoisetypes.The underestimationisbecausethismethoddoesnotmakeanycorrectionforovertting.For theAR(1)andpowerlawmethods,ifthettedmodelmatchesthenoisetype,theresults aresimilartotheEbisuzakimethod.Ifthewrongnoisemodelisselected,thevariance canbeoverorunderestimatedbyafactorof5ormore.BecausetheAR(1)andrednoise simulationsdependonchoosingamodel,itisnotsurprisingthatAR(1)andpowerlaw simulationsunderandoverestimatethetrenderror.Anexampleoftidegaugeresidual spectrumatDarwin,Australiaillustratestheproblemoftherednoiseoverestimate(Figure3.5).TheARmodelisareasonableestimateofthespectrumwhilethespectralslope estimateoverestimatesforthelowestfrequencies.Theresultingsimulatedrednoiseseries arenotrepresentativeoftheoriginaldataandresultinoverestimatederrors.Thespectrum oftheAnnetteIslandGPSresidualsillustratestheproblemoftheAR(1)modelunderesti43

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2.5 2 1.5 1 0.5 0 0.5 1 log( 2 calculated/ 2 true) Ebisuzaki AR(1) Power Law Formal Error Proposed Method =0 =0.7 =0.5 =0.9 =0.5 =1 =1.5Figure3.4:TestoftheProposedMethodforaTrend.Thelogoftheratioofthecalculated variancetothetruevarianceisplottedonthey-axisforeachmethodofcomputingthe variance(proposedmethod,Ebisuzaki,AR(1)andrednoise)andtheformalerror estimate.Eachmethodispresentedforavarietyofnoisetypes:AR(1)withcoefcients of= 0 0 5 0 7 0 9andpowerlawwithspectralslopes= Š 0 5 Š 1 Š 1 5. mation(Figure3.6).TheAR(1)tunderestimatesthelowfrequencyendofthespectrum, resultinginsimulatedseriesthatarelessredthanthetruenoisemodel. Thereisalsoapeculiarstructure(Figure3.4)fortheresultsoftheAR(1)andpower lawsimulationswhenthewrongmodelischosen:amaximumvarianceoverestimateat= 0 7whenanAR(1)modelist,andamaximumvarianceunderestimateat= Š 1 whenapowerlawmodelist.Atlowvaluesofandthespectrumisclosetowhite andcanbetwellbyeithermodel.Asthenoisebecomesveryred,theAR(1)spectrum approachesapowerlawshapeandagaineithermodelchoicewilltfairlywell.The 44

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104 103 102 101 100 106 104 102 100 102 104 FrequencyPower Spectral Density Spectrum Spectral Slope AR model Figure3.5:DarwinSpectrumandModelFits.Thespectrumoftheresidualsofthe Darwintidegaugeafterremovalofamean,trendandharmonicsisplottedinblack.The rednoisemodelisplottedinredandtheAR(1)modelinblue.Therednoisemodel greatlyoverestimatesthelowfrequencyvariancecontribution. maximumdisagreementbetweentheAR(1)andpowerlawspectra,therefore,occursat intermediatevaluesofand(Figures3.7-3.8). Forthissimple,butimportant,model,theproposedmethodoutperformsallother methodsoferrorestimation.Wenexttestedusingmorecomplicatedmodels,beginning withameanandtrendplusonestepfunction.Byexperimentingwithavarietyoflocationsforthesteps,wefoundthatthelargesterrorsoccurredwhenweplacestepsat25 and75percentofthetimeserieslength,andthesmallesterrorsoccurwhenthestepisin themiddleofthetimeseries.Forastepinthemiddleoftheseries(Figure3.9)theresults 45

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104 103 102 101 100 103 102 101 100 101 102 FrequencyPower Spectral Density PSD Spectral Slope AR model Figure3.6:AnnetteIsland(AIS1)GPSSpectrumandModelFits.Thespectrumofthe residualsoftheAnnetteIslandGPSafterremovalofamean,trendandharmonicsis plottedinblack.TherednoisemodelisplottedinredandtheAR(1)modelinblue.The AR(1)noisemodelgreatlyunderestimatesthelowfrequencyvariancecontribution. aresimilartotheresultsfromthetrendonlysimulation,exceptthatthepowerlawmethod nowoverestimatesforallnoisetypes.Theproposedmethodagainoutperformsallofthe othermethods.Forstepsat25(Figure3.10)and75percent(Figure3.11)oftherecord length,theproposedmethodalsooutperformsallothermethods. Thefullrangeofsimulationswasalsorunformodelsconsistingofatrendplus3 steps,randomlyplacedintherst,middleandlastthirdsofthetimeseries,and5steps randomlyplacedthroughoutthetimeseries.Tendifferentrandomlyplacedstepswere usedandtheratioofcalculatedvariancetotruevariancepresentedinFigures3.12and 3.13istheaverageratiofromthetendifferentmodels.Resultsvariedbetweentheten 46

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 Variance of Residuals from Power Law FitAR(1) CoefficientFigure3.7:FittingaPowerLawSpectrumtoanAR(1)Spectrum.Thevarianceofthe residualsfromttingapowerlawspectrumtoanAR(1)spectrumwithvarious coefcients.Themaximummistoccursat= 0 7. casesbyonlyabout5percentfortheproposedmethod.Theimportantresultsisthatthe proposedmethodagainoutperformsalloftheothermethods. Tosummarizetheresultsfromthesesimulations,wenotethatoverabroadrangeof truenoisemodels,ourproposedmethodtypicallyreproducesthetruetrendvarianceto withinabout10%evenwhensimultaneouslyttingasetofstepfunctions,asisoftenthe casewiththeGPStimeseriesthatweneedtousefortheexternallandmotionestimates. Theformalerrors,ontheotherhand,badlyunderestimatethetrendvariance.Forfairly rednoisespectra,whichoftenoccurwhenttingtrendstothetidegaugeseriesinorder toobtaintheinternallandmotionestimate,theunderestimatecaneasilybeafactorof 10to100.FittingAR(1)orpowerlawmodelstothespectrumoftheresidualtimeseries 47

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Variance of Residuals from AR(1) FitPower Law CoefficientFigure3.8:FittinganAR(1)SpectrumtoaPowerLawSpectrum.Thevarianceofthe residualsfromttinganAR(1)spectrumtoapowerlawspectrumwithvariousslopes. Themaximummistoccursat= Š 1 1.Astheslopevaluesapproach-2,thenoise becomesnonstationaryandcannotbemodeledbyanautoregressivespectrum. without apriori knowledgeofthetruespectralshapecaneasilyresultintrendvariance estimatesthataretoolargeortoosmallbyafactorof3-10.Notethatoverestimatingthe varianceisasdamagingasunderestimatingitsincetheproperweightingbetweenthe internalandexternallandmotionestimateswillbewrongineithercase.Finally,while theEbisuzakimethodhastheadvantageofbeingrelativelyinsensitivetothetruenoise model'sspectralshape,thelackofacorrectionforoverttingleadstounderestimatingthe trendvariancesbyafactorof2-4whenstepfunctionsmustbeincludedinthemodelthat wet. 48

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2.5 2 1.5 1 0.5 0 0.5 1 log( 2 calculated/ 2 true) Proposed Method Ebisuzaki AR(1) Power Law Formal Error =0 =0.5 =0.7 =0.9 =0.5 =1 =1.5Figure3.9:TestoftheProposedMethodforaTrendPlusaStepintheMiddleofthe Series.Thelogoftheratioofthecalculatedvariancetothetruevarianceisplottedonthe y-axisforeachmethodofcomputingthevariance(proposedmethod,Ebisuzaki,AR(1) andrednoise)andtheformalerrorestimate.Eachmethodispresentedforavarietyof noisetypes:AR(1)withcoefcientsof= 0 0 5 0 7 0 9andpowerlawwithspectral slopes= Š 0 5 Š 1 Š 1 5. Aswehaveshownthroughthesetestsoveravarietyofnoisetypesandmodelfunctions,theproposedmethodoferrorestimationcancorrectforthelowfrequencyvariance removedbyttingandremovingamodelfromatimeseries.Ourmethodhasanadded advantageinthatitdoesnotdependonchoosingamodelforthespectrum.Wecanapply thenewmethodoferrorestimationwithoutlengthysimulationstoanytimeserieswith justafewlinesofcode.Thederivationofthisnewmethodoferrorestimationhasgreatly increasedourcondenceinthelandmotionerrorestimatesforboththeinternalandexternallandmotionestimates.Notealsothatalthoughwehavefocusedonestimatingtrend 49

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2.5 2 1.5 1 0.5 0 0.5 1 log( 2 calculated/ 2 true) Proposed Method Ebisuzaki AR(1) Power Law Formal Error =0 =0.5 =0.7 =0.9 =0.5 =1 =1.5Figure3.10:TestoftheProposedMethodforaTrendPlusaStepat25Percentofthe RecordLength.Thelogoftheratioofthecalculatedvariancetothetruevarianceis plottedonthey-axisforeachmethodofcomputingthevariance(proposedmethod, Ebisuzaki,AR(1)andrednoise)andtheformalerrorestimate.Eachmethodispresented foravarietyofnoisetypes:AR(1)withcoefcientsof= 0 0 5 0 7 0 9andpowerlaw withspectralslopes= Š 0 5 Š 1 Š 1 5. variancesbecauseoftheparticularapplicationwearemaking,thederivationisinfact completelygeneralandcanbeappliedtoanysetofbasisfunctions. Wewillnowmoveontotheproblemofhowtomakethebestlandexternal(i.e.,GPSbased)landmotionestimateforaarbitrarytidegauge.Theseestimatescanthenbecombinedwiththeinternallandmotionestimatesinanoptimalwaynowthatwehaverealistic estimatesofthevariancesofthesequantities. 50

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2.5 2 1.5 1 0.5 0 0.5 1 log( 2 calculated/ 2 true) Proposed Method Ebisuzaki AR(1) Power Law Formal Error =0 =0.5 =0.7 =0.9 =0.5 =1 =1.5Figure3.11:TestoftheProposedMethodforaTrendPlusaStepat75Percentofthe RecordLength.Thelogoftheratioofthecalculatedvariancetothetruevarianceis plottedonthey-axisforeachmethodofcomputingthevariance(proposedmethod, Ebisuzaki,AR(1)andrednoise)andtheformalerrorestimate.Eachmethodispresented foravarietyofnoisetypes:AR(1)withcoefcientsof= 0 0 5 0 7 0 9andpowerlaw withspectralslopes= Š 0 5 Š 1 Š 1 5. 51

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2.5 2 1.5 1 0.5 0 0.5 1 log( 2 calculated/ 2 true) Proposed Method Ebisuzaki AR(1) Power Law Formal Error =0 =0.5 =0.7 =0.9 =0.5 =1 =1.5Figure3.12:TestoftheProposedMethodforaTrendPlus3Steps.Thelogoftheratioof thecalculatedvariancetothetruevarianceisplottedonthey-axisforeachmethodof computingthevariance(proposedmethod,Ebisuzaki,AR(1)andrednoise)andthe formalerrorestimate.Eachmethodispresentedforavarietyofnoisetypes:AR(1)with coefcientsof= 0 0 5 0 7 0 9andpowerlawwithspectralslopes= Š 0 5 Š 1 Š 1 5. 52

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2.5 2 1.5 1 0.5 0 0.5 1 log( 2 calculated/ 2 true) Proposed Method Ebisuzaki AR(1) Power Law Formal Error =0 =0.5 =0.7 =0.9 =0.5 =1 =1.5Figure3.13:TestoftheProposedMethodforaTrendPlusa5Steps.Thelogoftheratio ofthecalculatedvariancetothetruevarianceisplottedonthey-axisforeachmethodof computingthevariance(proposedmethod,Ebisuzaki,AR(1)andrednoise)andthe formalerrorestimate.Eachmethodispresentedforavarietyofnoisetypes:AR(1)with coefcientsof= 0 0 5 0 7 0 9andpowerlawwithspectralslopes= Š 0 5 Š 1 Š 1 5. 53

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Chapter4:MakinganOptimalGPSLandMotionEstimateataTideGauge Earlierwedenedtheinternallandmotionestimatebasedonthetidegaugeseriesitself andtheexternalestimatebasedonindependentGPSmeasurements.Combiningthesein anoptimalwayrequiresaccuratevariancesestimatesfortheserates,andwehavederived amethodfordoingthatinthepreviouschapter.Thenextstepistodeterminethebest externalratetouseatanarbitrarytidegauge.Theinternalrate,whichisbasedonthe tidegaugetimeseries,isavailableateverygauge.Theexternalrate,ontheotherhand, isnotnecessarilyavailablebecausethereisnotacontinuousGPSco-locatedwithevery tidegauge.LimitingthetidegaugestothesubsetwhereGPSdataareavailableisnota solution.Asweexplainedearlier,thetrendestimatesfromaltimetryarebestconstrained usingthelargestnumberoftidegaugespossible.Also,detectingspatially-varyingerrors inthealtimetricdatarequiresthebestglobaltidegaugecoveragepossible. Theproblemweaddressinthischapterishowtoassignanexternalratetoatide gaugethatdoesnothaveaco-locatedGPSreceiver.Basically,thequestionis,howfar awaycanaGPSreceiverbebeforeweconsideritunacceptable? Intherstsectionofthischapterweshowhowintercomparisonsbetweentheinternal andexternallandmotionratescanbeusedtoevaluatetheappropriatenessofanalgorithm forassigningGPSratestothetidegaugelocations.Inthenextsectionwearguethatthe mainproblemisidentifyingtidegaugesinareaswherethelandmotiongradientislarge andwedevelopaclassicationsystemthatwilltakethelandmotiongradientintoaccount 54

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inordertodeterminehowtheGPS-derivedexternalratesareappliedtothetidegauges.In thenalsectionwesummarizetheoverallmethodwehaveadopted.Inthenalchapter ofthisthesis,wewilldemonstrateourmethodviaanapplicationtoaspecicsetoftide gaugelocationsandthepresentlyavailablesetofGPSlandmotiontimeseries. 4.1EvaluatingtheExternalEstimates Inordertodeterminewhetheragivenexternalrateistrustworthy,werequiresome sortofindependentcheck.Wewillusecomparisonsofpossibleexternalrateestimates withtheinternalrateateachtidegaugeforthispurpose.Theimportantpointisthatthe internalandexternallandmotionrateestimatesarecompletelyindependentofoneanother.Therefore,wecanevaluateanyproposedalgorithmforassigningexternalrates tothetidegaugesbyaskingwhetherthereisasignicantcorrelationwiththeinternal rateestimates.Sincethetwoareindependent,asignicantcorrelationcanonlyoccur ifbothestimateshavestatisticallysignicantskill.Basically,weassertthatifboththe internalandexternallandmotionestimateshavereasonablysmallerrorbarsandagree towithinthoseerrorbars,wecanhavecondenceinbothestimates.Ifthetwoestimates donotagreewewillexaminebothestimatestodeterminethecauseofthediscrepancy,if possible. Therststepinidentifyingthecauseofthedisagreementbetweentheinternaland externallandmotionestimatesistoexaminetheinternalestimate.Ashorttimeseries mayhaveaninterannualsignalthatcausestheinternallandmotionratetobeoverorunderestimated.AttheDarwin,Australiatidegauge,forexample,thedisagreementbetween internalandexternallandmotionestimatesisattributedtotheinternalestimate,because oflow-frequencyvariabilityinthetimeseries.ThetopleftpanelofFigure4.1showsa 55

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mapoftheDarwintidegauge(bluecircle)andthenearbyGPS(redtriangle).TheGPS isonly53kmawayfromthetidegauge,buttheexternalrate(3 23 1 22mm/yr)does notagreewellwiththeinternalrate( Š 1 77 1 94mm/yr).Atthislocationtheproblem appearstobethatthetidegaugetimeseriescontainsnonlinear,decadal-scalevariabilitythatisnotaccountedforinthetrendttingmodel.Themainpointofthisexample, however,isthatwedonotuniformlyindicttheGPSestimatewheneveradiscrepancyis found.Generally,though,theproblemsthatwehavefoundareusuallyduetomakingthe external,orGPS,landmotionestimate. 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 400 200 0 200 Internal rate = 1.77 mm/yr (1.94)mm 129.5 130 130.5 131 131.5 132 132.5 13.5 13 12.5 12 11.5 125 130 135 140 20 15 10 5 53 km, 3.23 (1.22) Figure4.1:Darwin,AustraliaInternalandExternalLandMotionEstimates.Thetopleft panelshowsamapoftheDarwintidegaugeandthenearbyGPS.Therightpanelisa3 by3degreezoomofthesamemap.Thetidegaugeseriesusedtottheinternalland motionrateisplottedinthelowerpanelwiththeinternallandmotiontrendinred. 56

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TheexternalratecanbeproblematiciftheGPSstationisfarfromthetidegaugeand ismeasuringalocalizedphenomenon,suchastectonicactivity.AttheKushiro,Japan tidegauge,theinternalestimateisderivedfroma50-yearlongtidegaugesealevelserieswithaclearlineartrend,andnoobviouslow-frequencyvariability(Figure4.2).We considerthisinternallandmotionestimatetobereliable.Weseeinthetoprightpanel ofthisgurethattherearenogeodeticstationswithin100kmofthetidegauge.The nearestGPShasaratethatisofthesamemagnitudeastheinternalratebutoppositein sign.ExternalratesfromGPSstationsaroundJapanshowlargespatialgradientsinthe estimatesoflandmotioninthisarea.BecauseofthetectonicactivityintheJapanese Peninsula,thedistancescaleforcoherentverticallandmotionisverysmall,ontheorder of10km( FujiiandXia ,1993). Atthemajorityofstationsexamined,however,wedonotseemajordisagreement betweentheinternalandexternalestimates.Oneexampleofastationwithexcellentinternal,externallandmotionagreementisKeyWest,Florida(Figure4.3).Onceagain, thetidegaugesealevelseriesisover50yearslong,withaclearlineartrend.TheGPS stationatKeyWestis16kmfromthetidegaugeandagreeswiththeinternalestimate towithintheerrorbars.TheGPSstation200kmnorthofKeyWestalsoagreeswiththe internalrate.EventheratherdistantGPSstationatCharlestonagreeswellwiththeKey Westinternalrate.TheGPSstationatMobile,AlabamadoesnotagreewiththeKeyWest internalrate,butthismaybeduetolandsubsidencearoundthewesternGulfofMexico ( Gornitz ,2001).KeyWestisnottheonlyplacewendsuchagreement;therearemany tidegauge/GPSpairsthathaveexcellentinternal,externallandmotionrateagreement, evenwhenthetidegaugesandtheGPSreceiversareseparatedbyhundredsofkilometers. 57

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1950 1960 1970 1980 1990 2000 400 200 0 200 400 mmInternal Rate 8.03 mm/yr (0.23) 130 135 140 145 150 30 35 40 45 50 143 143.5 144 144.5 145 145.5 146 42 42.5 43 43.5 44 44.5 45 117 km, 10.39 (3.90) 470 km, 1.64 (0.72) 491 km, 0.63 (1.53) 583 km, 2.36 (0.32) 505 km, 11.69 (2.86) 847 km, 0.39 (0.76) 909 km, 0.99 (0.60) 917 km, 0.23 (0.33) 977 km, 0.25 (0.51) Figure4.2:Kushiro,JapanInternalandExternalLandMotionEstimates.Thetopleft panelshowsamapoftheKushirotidegaugeandallGPSstationswithin1000km.The rightpanelisa3by3degreezoomofthesamemap.Thetidegaugeseriesusedtotthe internallandmotionrateisplottedinthelowerpanelwiththeinternallandmotiontrend inred. 4.2TheLandMotionGradientProblemandStationClassicationProposal Weexaminedplotslikethoseshownjustaboveatalloftheavailabletidegaugelocations.Wecannotshowallofthese,butweconcludethatlargespatialgradientsinlocal landmotionarethemajorcontributingfactortodisagreementbetweentheinternaland externallandmotionestimates.Figure4.4showseachtidegaugeandtheabsolutevalue ofthedifferencesbetweeninternalandexternallandmotionestimateswithin3000km, normalizedbythecombinederror.TidegaugesthatarelocatedonPacicislands,the 58

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1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 200 100 0 100 200 Internal rate = 0.22 mm/yr (0.30)mm 270 275 280 285 290 15 20 25 30 35 16 km, 0.26 (0.44) 210 km, 0.92 (0.65) 867 km, 2.30 (0.43) 932 km, 0.76 (0.82) 277 277.5 278 278.5 279 279.5 280 23 24 25 26 27 Figure4.3:KeyWest,FloridaInternalandExternalLandMotionEstimates.Thetopleft panelshowsamapoftheKushirotidegaugeandallGPSstationswithin1000km.The rightpanelisa3by3degreezoomofthesamemap.Thetidegaugeseriesusedtotthe internallandmotionrateisplottedinthelowerpanelwiththeinternallandmotiontrend inred.NotetheexcellentagreementwithGPSbothnearandfarfromthetidegauge. UnitedStateseastcoast,Europe,andAustralia(bluecircles)generallyhavenormalized differencessmallerthan2withfewoutliers.Alloftheseregionsoftheworldaretectonicallyquietandfarawayfromactivecontinentalmargins.ThetidegaugeandGPSpairs plottedinredarewherethenormalizeddifferencesaregenerallylargerthan2.These stationsareinJapan(82-95),Alaska(23-25),andthePaciccoastsofNorthandCentral America(99-110).SomeofthelargestoutliersareinJapan,wherethenormalizeddifferencescanbeaslargeas 25.AlloftheseregionsarenearthetectonicallyactivePacic "RingofFire"wherelargespatialgradientsinlandmotionareexpected.Intectonically 59

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activeareas,thespatialgradientoflandmotionislarge,sothedistancescaleofcoherent landmotionissmall.Inotherareas,thespatialgradientoflandmotionissmall,sothe distancescaleofcoherentlandmotionislarge. 0 20 40 60 80 100 120 0 5 10 15 20 25 Tide Gauge Station Number| / | Non Pacific Rim Pacific Rim Figure4.4:TideGaugeandGPSNormalizedDifferencePairs.Foreachtidegauge,the absolutevalueofthenormalizeddifferenceiscomputedforallGPSstationswithin3000 km.StationsonthePacicRimareplottedinredwhilenonPacicRimstationsarein blue. This"RingofFire"exampleisperhapsobvious,butthereareotherreasonswhythere canbeshortwavelengthlandmotionchanges.Wemustthereforequantifythedistance scaleofcoherentlandmotionsothateachtidegaugeandGPSstationcanbeclassied accordingtotheexpectedmagnitudeofthespatialgradientoflandmotionnearthestation.TherstcriterionthatweimposeisthatinnocasedowepairaGPSratewithatide 60

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gaugeifthisrequirescrossingatectonicplateboundary,ormovingtoadifferentocean basin.Developingadditionalcriteriaisthefocusoftheremainderofthissection. Alongtectonicallyactiveplatemargins,earthquakescauselocalupliftandsubsidence onsmalldistancescales.TidegaugesandGPSstationsneartheseboundariesneedtobe constrainedbyasmalldistancescaleformakingexternallandmotionestimates.How doweidentifywhichstationsshouldhavethisrestriction,sincenotallplateboundaries aretectonicallyactive?Wecomputeatectonicactivityindexbaseduponthenumber andintensityofearthquakes.Insteadofcomputingtheindexforeachstation,wecomputeanindexforeachpointontheboundary.WeobtainedthepositionofplateboundariesfromtheUniversityofTexasInstituteforGeophysicsandearthquakemagnitudes andlocationsfromtheUnitedStatesGeologicalSurveyandclassiedeachpointalong theplateboundaryasactiveorquiet.Anactivelocationonaplateboundaryisdened ashavinganearthquaketotalenergyofgreaterthan3 x 1013Jandanaverageenergyof greaterthan5 x 1011J.Theseenergiesarecomputedateachpointbytotalingandaveragingtheenergiesfromallearthquakesofmagnitude4orgreaterwithina200kmradius. Allotherlocationsalongaplateboundaryarecalledquiet.Wethenusethisactive/quiet denitionandthedistanceofeachstationtotheplateboundarytomakeaclassication thatseparatesthetidegauge/GPSpairsintoatectonicandnon-tectonicset.Inorderto beclassiedasnon-tectonicastationmustbefurtherthan250kmfromanactiveplate boundarylocationandfurtherthan100kmfromaquietplateboundary.Thestations classiedastectonicornon-tectonicareshowninFigure4.5alongwithmagnitude4or higherearthquakesandplateboundariesclassiedasactiveorquiet. Todeterminetheappropriatedistancescaleformakingtheexternallandmotionestimate,wetakethesetoftidegaugesclassiedashavinglargespatialgradientsinland motionduetotectonicactivity(denotedasCategory3)andcomputeexternallandmotion 61

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0 50 100 150 200 250 300 350 80 60 40 20 0 20 40 60 80 Mag > 4 Earthquakes Quiet Boundary Active Boundary NonTectonic Station Tectonic Station Figure4.5:StationClassicationintoTectonicallyActiveorQuietStations.The tectonicallyactivestationsareplottedinredtriangles,andthequietstationsinbluedots. Thebasisfortheclassication,earthquakeenergyatthenearestboundary,isillustrated bythenumberofmagnitude4orhigherearthquakes(green)andtheclassicationofthe boundaryasactive(magenta)orquiet(black). rateswithGPSstationsindistancebinsfrom0to200km.Thewidthofthebinincreases withdistance.Thecorrelationofinternalandexternallandmotionestimates,normalized bytheformalerrorforeachdistancebinisshowninFigure4.6.Thecorrelationisnot signicantbeyondadistanceof10km,soCategory3tidegaugescanonlyuseaGPS stationformakinganexternallandmotionifitiswithin10km.Thissameclassication appliestoGPSstationsaswell,sothataGPSstationinatectonicallyactiveareacannot bepairedwithanon-tectonictidegaugemorethan10kmaway.Thesetoftidegauges andGPSthatareclassiedasCategory3arelistedinAppendixA,Tables2and3. 62

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0 20 40 60 80 100 120 140 160 180 200 1 0.5 0 0.5 1 1.5 2 2.5 3 Distance (km)Normalized CorrelationFigure4.6:Category3NormalizedCorrelation.Thecorrelationhasbeennormalizedby theformalerror, 1 N,whereNisthenumberofstationpairs.Thedashedlineindicates the95percentcondencelevel. Afteridentifyingthetectonicstations,therearestillsomelargetidegauge/GPSnormalizeddifferencesthatcannotbeexplainedbytectonics.Intheseinstances,welookfor othercausesoflargespatialgradientsinlocallandmotion.Onesuchcauseissubsidence duetouidwithdrawalfromthesurroundingsediments,orsedimentcompaction.We canidentifypotentialareasofsubsidencebylookingatthevariabilityintheinternaland externallandmotionratessurroundingaparticularstation.Ifthereissignicantspatial variabilityinthelandmotionestimates,wesearchtheliteratureforanydocumentation ofsubsidencenearthetidegaugeorGPS.Ifthereisdocumentedsubsidence,thestation isclassiedasCategory2,astationwithalargespatialgradientinlocallandmotion. Incaseswherethereisnodocumentedsubsidenceatthestation,weleavethestation classiedasCategory1,astationwithnolargespatialgradientsoflandmotion. Largespatialgradientsinlocallandmotioncanalsobecausedbyglacial-isostatic adjustment(GIA),whichisverticalmotionoflandmassesinresponsetotheremoval 63

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LongitudeLatitude 0 50 100 150 200 250 300 350 80 60 40 20 0 20 40 60 80 15 10 5 0 5 10 15 Stations with large gradients in GIA All other stations Figure4.7:ICE5GModeledLandMotionDuetoGlacialIsostaticAdjustment.Thecolor contoursindicate2mm/yrintervalsoflandmotion.NotetheareasofhighlyvariableGIA inNorthAmerica,NorthernEuropeandAntarctica.StationswithlargegradientsofGIA landmotionaredenotedbymagentacircles. ofthecontinentaliceloadattheendofthelastglacialperiod( PeltierandTushingham 1989).TodeterminestationswhereGIAmightbethecauseoflocallandmotionweutilizethe1-degreegriddedICE5GGIAmodelsealevelchange( Peltier ,2004).Inverting thesealevelchangegivesusanestimateoflandmotionduetoGIA.TheICE5Gmodel landmotionduetoGIAandthelocationsoftidegaugesandGPSstationsareshownin Figure4.7.Ifthegradientoflandmotionfromthegridcellnearesttothestationtoanyof theadjacentgridcellsisgreaterthan0.5mm/yrthestationisidentiedasCategory2,a stationwithalargespatialgradientinlocallandmotion. 64

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ThecorrelationofinternalandexternallandmotionestimatesforCategory2tide gaugesversusdistanceisshowninFigure4.8.Again,thenormalizedcorrelationiscomputedbetweeninternalandexternallandmotionestimatesindistancebinsfrom0to500 kmwherethewidthofthebinincreaseswithdistance.Thecorrelationisnotsignicant beyondadistanceof300km,soCategory2tidegaugescanonlyuseaGPSstationfor landmotionifitiswithin300km.ThissameclassicationappliestoGPSstationsas well,sothataGPSstationinanareaofsubsidencecannotbepairedwithatidegauge morethan300kmaway.ThesetoftidegaugesandGPSthatareclassiedasCategory2 arelistedinAppendixA,Tables2and3. 0 50 100 150 200 250 300 350 400 450 500 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Distance (km)Normalized CorrelationFigure4.8:Category2NormalizedCorrelationThecorrelationhasbeennormalizedby theformalerror, 1 N,whereNisthenumberofstationpairs.Thedashedlineindicates the95percentcondencelevel. AlltidegaugeandGPSstationsthatdonotfallintoCategories2or3areclassied asCategory1,meaningnon-tectonicstationswithoutknownlargelandmotionspatial gradients.BecauseCategory1stationsarenotneartectonicallyactiveregions,orregions withlocalizedlandmotion,thedistancescaleofcoherentlandmotionshouldbelarge 65

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andwecanlookforGPSstationsrelativelyfarfromthetidegauge.Todeterminethebest distancescalewecomputedthecorrelationofallreliablelandmotionestimates(Category 1tidegaugesandGPSwitherrorslessthan2mm/yr)withinvaryingdistancebins.We incorporatednotonlytidegauge/GPSpairs,butalsotidegauge/tidegaugeandGPS/GPS pairstoobtainalargernumberofstations.TheresultsofthisanalysisareshowninFigure 4.9.Thecorrelationplottedinredhasbeennormalizedbytheformalerror, (1 N) ,where Nisthenumberofstationpairs.Becausestationscanpairmultipletimes,however,the formalerrormaysomewhatunderestimatethenumberofindependentpairs.Inanattempt tocorrectforthiseffect,wealsoshowthecorrelation(plottedinblue)afternormalizing byanestimateoferrorthatincorporatesthedegreeofindependencederivedfromMonte Carlosimulations.Bothcorrelationsyieldsimilarresults.Thenormalizedcorrelations areabovethe95percentcondencelevel(normalizedcorrelationof2)outto500km. Althoughthe1000kmdistanceappearstobesignicant,itdoesnotmakesensetochoose thisdistancegiventhebehaviorofthenormalizedcorrelationfrom600kmupto1000 km. Thisisperhapsthemostsignicantresultofthisstudy.IthasbeenarguedthatGPSderivedestimatesoflandmotionareuselessunlessthereceiverisco-locatedwiththe gauge.Theseresults,however,suggestthatifoneiscarefultoavoidareaswithknown largespatialgradientsinlandmotion,thenthereissufcientspatialcoherenceinthe landmotioneldtoallowusefulestimatesatmanytidegaugesthatdonotyethaveacolocatedGPSseries.Wedonotarguethatthisapproachisideal.ClearlyhavingGPSat everytidegaugeusedforaltimeterdriftestimationispreferred,butuntilthathappensthis resultwillallowustosubstantiallyreducetheerrorbaronaltimeterdriftestimates,and consequentlyonsealevelriseestimatesfrombothaltimetersandfromtidegaugesalone. 66

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0 500 1000 1500 2 1 0 1 2 3 4 5 Distance (km)Normalized Correlation / / sim Figure4.9:NormalizedCorrelationofCategory1Stations.Thecorrelationplottedinred hasbeennormalizedbytheformalerror, 1 N,whereNisthenumberofstationpairs. Thecorrelationplottedinbluehasbeennormalizedbyanestimateoferrorthat incorporatesthedegreeofindependencederivedfromMonteCarlosimulations.The dashedlineindicatesthe95percentcondencelevel. 4.3TheAlgorithmforMakinganExternalLandMotionEstimate Weconcludethischapterwithabriefsummaryofouralgorithmforplacinganexternal,orGPS-derived,landmotionestimateonatidegaugetimeseriesatanarbitrary location.Note,though,thatisitpossiblewewillconcludethatitisimpossibletomake suchanassignmentgiventhepresentarrayofGPStimeseries. First,thetidegaugemustbeclassiedintowhatwehavedenedasCategory1,2, or3.Therststepintheclassicationalgorithmistodeterminethetectonicplateand oceanbasinforeachstation.Then,anycausesforlargespatialgradientsinlandmotion mustbeidentied.Welookforanynearbytectonicactivityastheprimarycauseofgradientsinlandmotion.UsingtheplateboundariesfromUniversityofTexasInstitutefor 67

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GeophysicsandthetectonicboundaryindexdescribedinSection4.2,wecomputethe distancetothenearestboundaryandnotetheboundaryclassicationofthatpoint( i.e. activeorquiet).Ifthestationislessthan250kmfromaquietboundaryor100kmfrom anactiveboundary,itisclassiedasCategory3andalandmotionestimateismadeusing aGPSwithin10km.IfnoGPSisfoundwithin10km,thennoexternallandmotion estimateismade. IfthegaugeisnotclassiedasCategory3,thenthenextstepistoidentifyanyother causesoflocallandmotion(subsidenceorGIA).ToidentifyGIA-inducedlandmotion wecomputethegradientoflandmotionfromtheICE5Gmodelfortheareasurroundingthegaugeanddetermineifthegradientislargerthan0.5mm/yr.Ifso,thegaugeis classiedasCategory2.IfthereisnolargegradientinGIA,thenwecomparetheinternal estimatetoexternalestimatesaroundthetidegauge.Iftherearelargedifferences(indicatingspatialgradientsinlandmotion),wesearchtheliteratureforanydocumentedland motions.Ifsuchdocumentationisfound,thenthetidegaugeisclassiedasCategory2. AnexternallandmotionestimatecanbemadeforaCategory2tidegaugebylookingfor Category1or2GPSstationsoutto300km.IftherearenoCategory1or2GPSwithin 300km,thennoexternallandmotionestimateismade. AllstationsthatarenotclassiedasCategory2or3areplacedintoCategory1.An externallandmotionestimatecanbemadeforaCategory1tidegaugebylookingfor Category1GPSstationsouttoadistanceof500km,orCategory2GPSstationstoa distanceof300km.IftherearenoCategory1GPSwithin500km,thennoexternalland motionestimatecanbemade. Notethatalloftheprecedingmatchingcriteriaareallsubjecttotheoverallconstraint thatnoGPSstationcanbematchedtoatidegaugelyingonadifferenttectonicplateorin adifferentoceanbasin. 68

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Nowthatwehavedescribedthealgorithmindetail,wewilldemonstrateitsuseby applyingittotheUniversityofHawaiiSeaLevelCenterJointArchiveforSeaLevelFastDeliverysetoftidegauges.WewilluseGPSverticallandmotionratescomputedbyGuy Woppelmann( Woppelmannetal. ,2007)butwitherrorbarscomputedusingthemethod describedinChapter3. 69

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Chapter5:ApplicationofourMethods InChapter4,wedevelopedanalgorithmforchoosinganexternallandmotionestimate atanarbitrarytidegaugelocation.Thisalgorithmwasdevelopedusingaspecicsetof tidegauges,butthemethodmustbeappliedinthefuturetoavaryingsetoftidegauge locationsandanevolvingsetofGPSlocations.Inordertodemonstratetheapplicationof thealgorithm,wewillnowapplythemethodsdescribedinChapters3and4tothemost recent(August2009)setoftidegaugesthatareavailablefromtheUniversityofHawaii SeaLevelCenterastheFastDeliverydataset.ThesetofGPSstationsisthesameasthat usedinChapter4. 5.1ApplicationtoaNewSetofTideGaugeTimeSeries Therststepinmakingthelandmotionestimatetocomputetheinternallandmotion rateateachtidegaugelocationsasdescribedinChapter2,Section4.Briey,werst checktoseeifmonthlymeansealeveltimeseriesfromthePermanentServiceforMean SeaLevel(PSMSL)areavailableforeachstationintheFastDeliveryset.Ifmonthlydata areavailablewedeterminetheinternalratebyttingamean,trend,seasonalcycleand ENSOindextothesealevelseriesspanningfrom1946toJuly,2009.Ifmonthlydataare notavailableweusethedailytimeseries.Asdiscussedearlier,thetrendttothetime seriesisinterpretedastherateofmeansealevelchangeminuslandmotion.Toestimate 70

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therateofglobalsealevelchangewetatrendtothesealevelreconstructionof Church andWhite (2006)overthespanofeachofthetidegaugetimeseries.Anerrorbaron therateofglobalsealevelchangeiscomputedfromtheresidualsofthedenedspanof ChurchandWhite (2006)datausingthemethoddescribedinChapter3.Theglobalmean sealevelchangerateisremovedfromthettedtrendandthenthetrendisinvertedsothat positivelandmotionisupwardandnegativelandmotionisdownward.Theresidualsfrom themodeltarethenusedtoestimatetheerrorontheinternallandmotionrateusingthe methodoferrorestimationdescribedinChapter3.Thiserrorisaddedtotheerrorofthe globalmeansealevelchangeestimatetoobtainthetotalerroroftheinternallandmotion estimate.TheinternallandmotionrateanderrorforeachtidegaugearegiveninTable1. WenextusethealgorithmdevelopedinChapter4tocategorizeeachtidegaugebased onthemagnitudeofspatialgradientsinlandmotion.Onceeachtidegaugeisclassied, wecanassignanexternalestimatebysearchingforGPSstationswithinthespecied distanceforeachcategory.Werstusethetectonicboundaryindextoplacetidegauges intoCategory3.Tidegaugeswithin250kmofanactiveboundaryor100kmofaquiet boundaryareclassiedasCategory3.Additionally,tidegaugesthatarenearactivevolcanos,butnotnearatectonicboundaryareclassiedasCategory3.ThesestationsincludeHiloandKawaihaeinHawaiiandthevolcanicislandsofSanFelix,JuanFernandez andLasPalmas.ThetidegaugeclassicationsareshowninFigure5.1. AfterCategory3stationshavebeenidentied,wenowlookforanyothercausesof largespatialgradientsinlocallandmotion.ToidentifyGIA-inducedlandmotionwe computethegradientoflandmotionfromtheICE5Gmodelfortheareasurroundingthe gaugeanddetermineifthegradientislargerthan0.5mm/yr.Ifso,thegaugeisclassied asCategory2.SeventidegaugesfromtheFastDeliverysetareidentiedasCategory2 basedonlargegradientsinGIA:Goteborg,Sweden,Tono,BritishColumbia,Churchill, 71

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0 50 100 150 200 250 300 350 80 60 40 20 0 20 40 60 80 Category 1 Category2 Category 3 Figure5.1:Classicationofthe150FastDeliveryDatasetTideGauges. Manitoba,Ketchikan,Alaska,Lerwick,Scotland,Stornoway,Scotland,andPrinceRupert,BritishColumbia(Figure5.1). IfthereisnolargegradientinGIA,wethencomparetheinternalestimatetotheexternalestimatesaroundeachtidegauge.Iftherearelargespatialgradients,wesearch theliteratureforanydocumentedsubsidenceoruplift.Ifanydocumentationisfound, thenthetidegaugeisclassiedasCategory2.ThetidegaugeatDutchHarbor,Alaskais undergoingupliftduetoearthquakerebound( BridgesandGao ,2006)andisclassiedas Category2.TidegaugesinPatagonia(PuertoMadrynandPortStanley)areundergoing upliftthatcannotbeexplainedbyGIAalone( Rostamietal. ,2000).Duck,NorthCarolinaandAtlanticCity,NewJerseyareincludedintheChesapeakeBaysubsidencezone ( Kearney ,2008)wheresedimentcompactioniscausingthelandtosink.Thetwotide 72

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gaugesinGalveston,TexasarealsoplacedinCategory2becauseoflocalizedsubsidence duetouidwithdrawal( Gabrysch ,1990). AlltidegaugesthatarenotclassiedasCategory2or3aredenotedasCategory1 (Figure5.1).Thesestationsarefarfromplateboundariesandinlocationswherespatial gradientsinlandmotionareexpectedtobesmall. WealsoneedtomakeasimilarclassicationoftheGPSrates.InthisparticularapplicationweareusingthesamesetofGPSstationsthatwereusedinChapter4todevelop thealgorithm,butinthefuturethiswillchange.TheclassicationoftheGPSstations basedontheworkdescribedinChapter4aresummarizedinAppendixA,Table2andare nowalsoshowninFigure5.2. 0 50 100 150 200 250 300 350 80 60 40 20 0 20 40 60 80 Category 1 Category 2 Category 3 Figure5.2:Classicationofthe164GPSStationsUsedinThisStudy. 73

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5.2Results UsingthedistancecriteriaspeciedforeachcategoryinChapter4(Category3= 10km,Category2=300km,Category1=500km),wemakeanexternalestimateby pairingthetidegaugewiththenearestGPSthatmeetsthedistance,categoryandsame plate,sameoceanbasinrequirements.Theresultsofapplyingthelandmotionalgorithm anderrorestimationtothesetof150tidegaugesfromHawaii'sFastDeliverydatasetare givenbelowinTable1.Ofthe150tidegauges,90havesuccessfullybeenassignedan externallandmotionrate.Thisisagreatimprovementovertheearlierworkof Mitchum (2000),whereonly51gaugeswereassignedexternalratesevenwithamuchlessstringentcriterionfortheallowabledistancebetweenthetidegaugesandtheGPSlocations. InthatstudyGPSrateswereusedouttoadistanceof1000kmfromthetidegauges, asopposedtoourmaximumdistanceof500km.Weclearlyhavemoreexternalrates availableatthetidegauges,butwealsoneedtoaskwhethertheseratesarereliable,at leastinastatisticalsense. Inordertoaddressthisquestion,wefocusonstationswhereboththeinternaland externalestimateshaveerrorslessthan2mm/yr,whichresultsinasetof60pairs.We willusethesetoevaluatetheskillofouralgorithminassignedexternalratestothetide gaugetimeseries.Recallourearlierargumentthatagreementbetweentheseratesindicatesskillinbothsincetheseestimatesarecompletelyindependent.This(reduced)setof internalandexternalratepairshasacorrelationof0.68(Figure5.3).Ifthe60pairscan betreatedasindependent,thiscorrelationis5standarddeviationsfromzero.Ifatleast10 pairscanbetreatedasindependent,thecorrelationisstillsignicantlydifferentfromzero with95%condence.Thereareoutliersintheset,buttheoverallagreementdemonstrates thatthereisskillinbothestimates. 74

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Asafurthercheckonthesignicanceofthiscorrelation,wedevisedanothertestthat isnotsensitivetooutliers.Inthistestwesimplycountthenumberofpairswherethe internalandexternalestimateshavethesamesignandcomparethattothenumberof pairswheretheinternalandexternalestimateshaveoppositesigns.Ifthepairshaveno truerelationship,theweexpectthatthenumberofsamesignandoppositesignpairswill (statistically)beequal.Ontheotherhand,ifthepairsareperfectlyrelated,allwillhave thesamesign. Forthesetof60pairsshowninFigure5.3thereare40pairswiththesamesignand 20pairswiththeoppositesign.Weevaluatethesignicanceofthisexcessofsamesignto oppositesignpairswithaMonteCarlosimulation.Wegenerate60randomindependent pairswitherrorbarsequaltothoseoftherealdataandcountthenumberofsamesign pairs.Thisprocessisrepeated10000timesinordertoestimatetheprobabilitydistributionfunctionforthenumberofsamesignpairsthatcanoccurwhenthereisnotrue relationshipbetweenthepairs.Wendthatthelikelihoodofobtaining40ormoresame signpairsfromasetof60independentpairsislessthan1%,whichconrmsourconclusionfromtheclassicalcorrelationcoefcientthatthereisskillinbothofourlandmotion estimates. 5.3ConcludingRemarks Inconclusion,wecontendthatwehavedemonstratedtheeffectivenessofthemethodswehavedeveloped,andthatthesemethodswillsubstantiallyimproveourabilityto quantifysealevelriseestimatesfrombothtidegaugesandfromsatellitealtimeters. First,wehavedevelopedamethodtoestimatetheparametererrorsforageneralmultiplelinearregressionmodelthatdealswithbothcorrelatednoiseseriesandtheproblem 75

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15 10 5 0 5 10 15 15 10 5 0 5 10 15 Internal Land Motion Estimate (mm/yr)External Land Motion Estimate (mm/yr) Cat 1 TG and GPS Cat2 TG and GPS Cat 3 TG and GPS Cat 1 TG with Cat 2 GPS Cat 2 TG with Cat 1 GPS + + ++ = 0.68 N=60 ++, + + = 40 20Figure5.3:InternalandExternalLandMotionEstimatesfortheFastDeliverySetofTide Gauges.Theinternallandmotionratesanderrorsareplottedonthex-axisandthe externalratesanderrorsareplottedonthey-axis.Thepairshavebeencoloredtomatch thecategoryofeachtidegaugeandGPSseeninFigures5.1and5.2.Thegreencircles areCategory1tidegaugeswithCategory2GPSandCategory2tidegaugeswith Category1GPS.Eachquadrantofthegureisalsomarkedwitharedsignpair, indicatingthequadrantofthesameandoppositesignedpairsthatwerecountedaspartof thestatisticaltestdescribedinSection5.2. ofestimatingthetrueunderlyingnoisemodelsfromthetresiduals,whicharecontaminatedbywhatwecallovertting.Thisallowsustobetterquantifysealevelriseestimates computedfromtidegaugesalonebecausewewillnowhaveimprovedinformationon thereliabilityoftheratesdeterminedateachgaugearoundtheglobe.Wewillalsohave 76

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bettersealevelriseestimatesfromsatellitealtimetersbecausethismethodwillallowan optimalweightingoftheinternalandexternallandmotionestimates,thusreducingwhat isatpresentthedominanterrorinthesatellitesealeveltrendestimates. Second,wehavedevelopedanalgorithmformakingwhatwewillargueisthebest useoftheexternal,orGPS-derived,landmotionestimates.IdeallyweshouldhaveGPS measurementsateverytidegaugeandweshouldhavemanyyearsofdata.Inrealitywe haveco-locatedGPSseriesatasmallnumberofgaugesandtimeseriesthatareoften onlyafewyearsinlength.Theoverallstatisticsofthetidegaugeestimateofaltimeter driftdependsonalargenumberofgaugesbeingused,sowecannotsimplyeliminate gaugesthatdonothaveaco-locatedGPS.ThismeansthatwemustuseGPSratesthat arenotalwaysclose'tothegauges,andwehavedevelopedaschemethatallowsusto quantitativelysaywhatclosemeans. Thenetresultisthatthankstothisworkourcommunitywillbeableto: € Putaccurateerrorbarsonsealevelchangesestimatesfromtidegaugestations.This willallowoptimalweightingofthetidegaugeestimatesinordertoformamore precisecalculationoftwentiethcenturysealevelchange. € Determinethebestdriftestimationforsatellitealtimetersviathebestuseofthe growingnetworkofGPSlandmotionrates.Thisinturnallowsthebestestimateof sealevelchangeoverthepast15years. € Evaluatepossiblesealevelriseratechangesbydifferencingthetidegaugerates withthealtimeterriserate.Quantifyingthisratechange,andmoreimportantlyits errorbar,iskeytoassessingclimatechangemodels. 77

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Table5.1:UniversityofHawaiiSeaLevelCenterFastDeliveryTideGaugesWith InternalandExternalLandMotionEstimatesandErrors.Thecolumnsare(i)Station name,(ii)internallandmotionrate,(iii)internalrateerror,(iv)externallandmotionrate, (v)externalrateerror.Ifnoexternalrateiscomputedforagiventidegauge,theexternal ratebecomes0mm/yrwithanerrorof10mm/yr. StationName Int(mm/yr) Err(mm/yr) Ext(mm/yr) Err(mm/yr) POHNPEI -0.49 0.92 0.69 2.93 BETIO -1.56 1.07 -1.97 2.78 BALTRA -0.89 0.60 0.12 1.62 NAURU 3.13 1.13 0 10 MAJURO -1.42 0.89 0.04 2.36 MALAKAL 0.41 1.00 0 10 YAP 1.38 0.22 0 10 HONIARA 3.66 0.50 0 10 CHRISTMASISLAND 1.25 0.53 0 10 KANTON 1.96 0.96 0 10 PAPEETE -1.12 0.63 -0.29 0.58 RIKITEA 0.06 0.72 0 10 NOUMEA 4.93 2.42 2.06 0.73 EASTERISLAND 1.04 1.14 2.14 0.95 RAROTONGA -1.85 0.38 0.60 3.60 PENRHYN 0.45 1.06 0 10 FUNAFUTI 0.02 1.03 2.87 1.83 SAIPAN 1.33 1.01 -3.15 0.94 KAPINGAMARANGI -0.22 0.19 0 10 SANTACRUZ 0.18 0.77 0.12 1.62 78

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-continuedfrompreviouspage CABOSANLUCAS 1.24 1.17 0 10 NUKU'ALOFA -4.49 0.83 3.12 0.66 KODIAKISLAND 12.19 0.77 8.61 1.15 ADAKISLAND 3.26 0.55 0 10 DUTCHHARBOR 7.75 0.72 0 10 PORTVILA -1.49 0.48 -2.84 1.57 CHICHIJIMA -3.37 0.51 0.05 3.15 MIDWAYISLAND 1.23 0.63 0 10 WAKEISLAND -0.25 0.33 0 10 GUAM 0.21 0.27 2.14 0.47 KWAJALEIN 0.04 0.16 0.04 2.36 PAGOPAGO -0.91 0.63 3.65 1.53 HONOLULU 0.43 0.16 -0.05 0.98 NAWILIWI 0.49 0.25 2.79 1.05 KAHULUI 0.20 0.57 -0.05 0.98 HILO -1.39 0.34 -0.41 0.85 LALIBERTAD 3.39 0.30 0 10 CALLAO -0.12 1.17 0 10 MOMBASA 2.23 0.58 0.27 0.44 PORTLOUIS 1.84 1.60 2.60 2.46 RODRIGUES 1.88 1.44 0 10 HULHULE -0.32 1.22 3.10 3.99 GAN -3.28 1.38 0 10 SALALAH 0.90 0.38 0 10 79

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-continuedfrompreviouspage POINTLARUE -5.55 0.43 -1.23 2.47 KOTAPHAONOI 0.49 0.86 0 10 LAMU 2.36 1.86 0.27 0.44 ZANZIBAR 5.54 0.94 0.27 0.44 BROOME -2.52 2.94 0 10 DARWIN -1.00 2.52 3.23 1.24 COCOSISLAND -3.87 2.11 3.61 2.29 FREMANTLE 0.69 0.22 0.83 1.41 ESPERANCE -1.43 1.65 0 10 KERGUELEN -2.72 2.39 1.68 1.13 DURBAN 1.22 0.43 2.76 0.97 PORTELIZABETH -0.89 0.55 0 10 PONTADELGADO -0.82 0.23 -2.36 0.71 KEYWEST -0.20 0.25 0.26 0.61 SANJUAN 0.39 0.25 0 10 NEWPORT -0.49 0.20 -0.18 0.81 BERMUDA 0.27 0.15 -0.74 0.72 DUCKPIER -2.35 0.58 -3.12 0.99 CHARLESTON -0.62 0.34 0.76 0.99 ATLANTICCITY -2.37 0.18 -3.34 0.92 CHURCHILL 12.47 0.79 8.31 1.92 HALIFAX -0.76 0.21 -1.57 0.86 ST-JOHN'S 0.07 0.27 -1.19 0.76 ILHAFISCAL -1.53 1.29 0 10 80

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-continuedfrompreviouspage LERWICK 2.17 0.46 0 10 NEWLYN 0.31 0.17 -1.04 0.82 STORNOWAY -0.03 0.74 0.15 0.94 BALBOA 0.09 0.53 0 10 KOLAK 2.41 0.46 0 10 QUARRYBAY 0.82 0.63 0 10 BRISBANE 0.28 0.66 0 10 BUNDABERG 1.55 0.24 0 10 FORTDENISON 0.65 0.21 2.06 0.76 TOWNSVILLE 0.61 0.21 2.68 0.62 SPRINGBAY 0.17 0.58 1.57 1.05 BOOBYISLAND 9.98 0.55 0 10 ABASHIRI 0.50 0.61 10.32 3.96 HAMADA -3.09 0.57 0.92 0.93 TOYAMA -1.72 0.76 2.16 1.58 KUSHIRO -7.85 0.18 0 10 OFUNATO -2.47 0.22 -0.63 1.66 MERA -1.89 0.28 -0.25 0.77 KUSHIMOTO -1.26 0.76 -1.18 2.43 ABURATSU 0.18 0.71 -1.07 1.10 NAHA -0.02 0.44 1.54 0.99 MAISAKA 3.96 0.70 0 10 WAKKANAI -1.96 0.39 5.19 2.36 NAGASAKI 0.09 0.32 -0.45 2.06 81

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-continuedfrompreviouspage HAKODATE 1.99 0.60 3.72 1.43 ISHIGAKI -0.54 0.98 1.54 0.99 MANZANILLO 8.60 5.05 7.67 0.61 LOMBRUM -3.02 0.66 2.44 2.08 LAUTOKA -1.59 2.12 1.51 1.64 PRINCERUPERT 0.46 0.28 0.41 1.43 SANFRANCISCO 0.04 0.29 -0.54 1.46 CRESCENTCITY 2.91 0.21 0 10 NEAHBAY 4.05 0.27 4.84 0.82 SITKA 3.73 0.23 0 10 SEWARD 4.44 0.72 0 10 SANDIEGO -0.03 0.23 -1.36 2.22 YAKUTAT 8.39 1.16 0 10 KETCHIKAN 2.05 0.28 0.41 1.43 SANDPOINT 1.47 1.30 0 10 SOUTHBEACH 0.30 0.23 1.47 0.87 NOME -6.36 1.15 0 10 TANJONG -1.07 1.04 -13.78 2.34 FORTPULASKI -0.84 0.23 0.76 0.99 VIRGINIAKEY 1.26 1.25 0.92 0.79 PENSACOLA 0.27 0.22 -3.38 0.81 GALVESTONI -4.98 0.43 -6.85 1.08 GALVESTONII -4.43 0.44 -6.85 1.08 GOTEBORG 1.82 0.43 0 10 82

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-continuedfrompreviouspage BREST 0.13 0.35 -1.18 0.85 MARSEILLE 1.39 0.38 -0.32 0.98 CUXHAVEN -1.65 0.98 -0.16 0.85 PUERTOMADRYN 6.81 2.30 0 10 TOFINO 3.67 0.57 2.02 0.96 CALDERA -1.22 0.49 0 10 ANTOFAGASTA 2.95 0.15 0 10 WELLINGTON -0.37 0.18 -2.06 0.78 CEUTA 1.34 0.10 0 10 LANGKAWI 0.38 0.30 0 10 PORTLAND 1.46 0.57 0 10 MACQUARIE -3.07 1.77 0 10 FRENCHFRIGATE 0.58 2.76 0 10 JUANFERNANDEZ 8.20 0.33 0 10 SANFELIX 9.58 3.04 0 10 BLUFF 0.10 0.18 0 10 VALPARAISO 3.41 1.31 0 10 MOSSELBAY -0.57 5.80 2.42 2.39 KNYSNA 0.19 0.48 2.42 2.39 EASTLONDON 2.24 1.22 0 10 RICHARD'SBAY 0.31 1.78 2.76 0.97 LASPALMAS -15.01 1.20 0 10 SIMON'STOWN 0.19 0.32 2.42 2.39 DAKAR 3.87 0.59 0 10 83

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-continuedfrompreviouspage PORTSTANLEY 1.27 0.50 0 10 NAKANO -2.06 0.33 -0.45 2.06 NASE -1.27 0.53 1.54 0.99 NISHNOO -0.08 0.14 0 10 KAWAIHAE -3.60 0.82 0 10 LAJOLLA -0.17 0.24 -1.36 2.22 PORTNOLLOTH 0.52 0.26 0 10 SALDAHNA -1.28 1.20 2.42 2.39 CAPETOWN 0.57 2.31 2.42 2.39 LACORUNA 1.03 0.99 -3.91 1.25 84

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Hay,W.,andM.Leslie, SeaLevelChange ,chap.Couldpossiblechangeinglobalgroundwaterreservoircauseeustaticsealeveluctuations?,pp.161170,NationalAcademies Press,1990. Kearney,M., SuddenandDisruptiveClimateChange ,chap.TheNorthernGulfof MexicoCoast:HumanDevelopmentPatterns,DecliningEcosystems,andEscalating VulnerabilitytoStormsandSeaLevelRise,pp.85100,Earthscan,2008. Lambeck,K.,andD.Chappell,Sealevelchangethroughthelastglacialcycle, Science 292 ,679686,2001. Leuliette,E.,andL.Miller,Closingthesealevelrisebudgetwithaltimetry,Argoand GRACE, Geophys.Res.Lett. 36 ,L04,608,2009. Leuliette,E.,R.Nerem,andG.Mitchum,CalibrationofTOPEX/PoseidonandJason altimeterdatatoconstructacontinuousrecordofmeansealevelchange, Marine Geodesy 27 ,7994,2004. Lombard,A.,A.Cazenave,P.LeTraon,andM.Ishii,Contributionofthermalexpansion topresentdaysealevelchangerevisted, GlobalandPlan.Change 47 ,116,2005. Mann,M.E.,andJ.M.Lees,Robustestimationofbackgroundnoiseandsignaldetection inclimatictimeseries, ClimaticChange 33 ,409445,1996. Mao,A.,C.Harrison,andT.Dixon,Noisein(gps. Maul,G.A.,andD.M.Martin,SealevelriseatKeyWest,Florida,1846-1992:America'slongestinstrumentrecord?, Geophys.Res.Lett. 20 ,19551958,1993. Meier,M.,Thecontributionofsmallglacierstosealevel, Science 226 ,14181421,1984. Meier,M.,M.Dyurgerov,U.Rick,S.O'Neel,W.Pfeffer,R.Anderson,S.Anderson,and A.Glazovsky,Glaciersdominateeustaticsea-levelriseinthe21stcentury, Science 317 ,10641067,2007. 88

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Appendices 91

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AppendixA:TablesofTideGaugeandGPSStationClassication TableA.1:TideGaugeCategories.Thecolumnsare(i)Stationname,(ii)latitude,(iii) longitude,(iv)category,(v)commentnumber.Thecategorynumberscorrespondtothe acceptabledistanceformatchingtidegaugeswithGPSstations.Category1stationscan bematchedwithGPSupto3000kmaway,aslongastheyareonthesametectonicplate andsameoceanbasin.Category2stationscanbematchedwithGPSupto100kmaway, duetolocalized,non-tectoniclandmotion.Category3stationscanbematchedwithGPS only10kmawayduetotectoniclandmotion.Category4stationshavebeenrejectedand arenotusedinthisstudy.SeeTableA.3fordetailedcomments. StationName Latitude Longitude Category Comment POHNPEI 6.99 158.24 1 1 BETIO 1.36 172.93 1 1 BALTRA -0.44 269.72 1 1 NAURU -0.53 166.91 1 1 MAJURO 7.11 171.37 1 1 MALAKAL 7.33 134.46 3 3 YAP 9.51 138.13 3 3 HONIARA -9.43 159.96 3 3 CHRISTMASISLAND 1.98 202.53 1 1 KANTON -2.81 188.28 1 1 92

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Appendix:(continued) -TableA.1-continuedfrompreviouspage PAPEETE -17.53 210.43 1 1 RIKITEA -23.13 225.05 1 1 NOUMEA -22.29 166.44 1 1 EASTERISLAND -27.15 250.55 1 1 RAROTONGA -21.21 200.23 1 1 PENRHYN -8.98 201.95 1 1 FUNAFUTI -8.53 179.20 1 1 SAIPAN 15.23 145.74 1 1 KAPINGAMARANGI 1.10 154.78 1 1 SANTACRUZ -0.75 269.69 1 1 CABOSANLUCAS 22.88 250.09 3 3 NUKU'ALOFA -21.13 184.83 1 1 KODIAKISLAND 57.73 207.49 3 3 ADAKISLAND 51.86 183.37 3 3 DUTCHHARBOR 53.88 193.46 2 4 PORTVILA -17.77 168.30 3 3 CHICHIJIMA 27.10 142.18 1 1 MIDWAYISLAND 28.22 182.63 1 1 WAKEISLAND 19.28 166.62 1 1 GUAM 13.43 144.65 3 3 KWAJALEIN 8.73 167.73 1 1 PAGOPAGO -14.28 189.32 3 3 93

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Appendix:(continued) -TableA.1-continuedfrompreviouspage HONOLULU 21.31 202.13 1 1 NAWILIWI 21.96 200.65 1 1 KAHULUI 20.90 203.53 1 1 HILO 19.73 204.93 3 3 LALIBERTAD -2.20 279.08 3 3 CALLAO -12.05 282.85 3 3 MOMBASA -4.07 39.66 1 1 PORTLOUIS -20.16 57.50 1 1 RODRIGUES -19.67 63.42 1 1 HULHULE 4.18 73.53 1 1 GAN -0.69 73.15 1 1 SALALAH 16.94 54.01 1 1 POINTLARUE -4.67 55.53 1 1 KOTAPHAONOI 7.83 98.43 1 1 LAMU -2.27 40.90 1 1 ZANZIBAR -6.16 39.19 1 1 BROOME -18.00 122.22 1 1 DARWIN -12.47 130.85 1 1 COCOSISLAND -12.12 96.90 1 1 FREMANTLE -32.05 115.73 1 1 ESPERANCE -33.87 121.90 1 1 ST.PAUL -38.71 77.54 4 7 94

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Appendix:(continued) -TableA.1-continuedfrompreviouspage KERGUELEN -49.35 70.22 1 1 DURBAN -29.88 31.00 1 1 PORTELIZABETH -33.96 25.63 1 1 PONTADELGADO 37.74 334.33 3 3 KEYWEST 24.55 278.19 1 1 SANJUAN 18.46 293.88 3 3 NEWPORT 41.51 288.67 1 1 BERMUDA 32.37 295.30 1 1 DUCKPIER 36.18 284.26 2 5 CHARLESTON 32.78 280.07 1 1 ATLANTICCITY 39.36 285.58 2 5 CHURCHILL 58.78 265.80 2 2 HALIFAX 44.67 296.42 1 1 ST-JOHN'S 47.57 307.28 1 1 ILHAFISCAL -22.90 316.84 1 1 LERWICK 60.15 358.86 2 2 NEWLYN 50.11 354.46 1 1 STORNOWAY 58.21 353.61 1 1 BALBOA 8.96 280.43 1 1 KOLAK 11.80 99.82 1 1 QUARRYBAY 22.3 114.22 1 1 BRISBANE -27.37 153.17 1 1 95

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Appendix:(continued) -TableA.1-continuedfrompreviouspage BUNDABERG -24.83 152.35 1 1 FORTDENISON -33.85 151.23 1 1 TOWNSVILLE -19.25 146.83 1 1 SPRINGBAY -42.55 147.93 1 1 BOOBYISLAND -10.60 141.92 1 1 ABASHIRI 44.02 144.28 1 1 HAMADA 34.90 132.06 1 1 TOYAMA 36.77 137.22 1 1 KUSHIRO 42.97 144.38 3 3 OFUNATO 39.07 141.72 3 3 MERA 34.92 139.83 3 3 KUSHIMOTO 33.47 135.78 3 3 ABURATSU 31.57 131.42 3 3 NAHA 26.22 127.67 1 1 MAISAKA 34.68 137.62 3 3 WAKKANAI 45.40 141.68 1 1 NAGASAKI 32.73 129.87 3 3 HAKODATE 41.78 140.73 1 1 ISHIGAKI 24.33 124.15 1 1 MANZANILLO 19.05 255.67 3 3 LOMBRUM -2.03 147.37 1 1 LAUTOKA -17.60 177.43 1 1 96

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Appendix:(continued) -TableA.1-continuedfrompreviouspage PRINCERUPERT 54.32 229.67 2 2 SANFRANCISCO 37.81 237.54 3 3 CRESCENTCITY 41.75 235.82 3 3 NEAHBAY 48.37 235.38 3 3 SITKA 57.05 224.67 3 3 SEWARD 60.12 210.57 3 3 SANDIEGO 32.72 242.83 3 3 YAKUTAT 59.55 220.27 3 3 KETCHIKAN 55.33 228.34 2 2 SANDPOINT 55.34 199.50 3 3 SOUTHBEACH 44.63 235.96 3 3 NOME 64.50 194.57 1 1 ESPERANZA -63.40 303.01 1 1 TANJONG 1.26 103.85 1 1 FORTPULASKI 32.03 279.10 1 1 VIRGINIAKEY 25.73 279.84 1 1 PENSACOLA 30.40 272.79 1 1 GALVESTONI 29.29 265.21 2 5 GALVESTONII 29.31 265.21 2 5 GOTEBORG 57.68 11.80 2 2 BREST 48.38 355.50 1 1 MARSEILLE 43.30 5.35 1 1 97

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Appendix:(continued) -TableA.1-continuedfrompreviouspage CUXHAVEN 53.87 8.72 1 1 PUERTOMADRYN -42.77 294.97 2 6 TOFINO 49.15 234.08 2 2 CANANEIA -25.02 312.07 1 1 FUNCHAL 32.63 343.10 1 1 PORTHEDLAND -20.30 118.58 1 1 CARNARVON -24.88 113.62 1 1 CALDERA -27.07 289.17 3 3 ANTOFAGASTA -23.65 289.60 3 3 WELLINGTON -41.28 174.78 3 3 CEUTA 35.90 354.68 3 3 LANGKAWI 6.43 99.77 1 1 PORTLAND -38.33 141.60 1 1 MACQUARIE -54.48 158.97 3 3 LORDHOWE -31.53 159.07 1 1 BLUFF -46.60 168.33 3 3 98

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Appendix:(continued) TableA.2:GPSStationCategories.Thecolumnsare(i)Stationname,(ii)latitude,(iii) longitude,(iv)category,(v)commentnumber.Thecategorynumberscorrespondtothe acceptabledistanceformatchingtidegaugeswithGPSstations.Category1stationscan bematchedwithGPSupto3000kmaway,aslongastheyareonthesametectonicplate andsameoceanbasin.Category2stationscanbematchedwithGPSupto100kmaway, duetolocalized,non-tectoniclandmotion.Category3stationscanbematchedwithGPS only10kmawayduetotectoniclandmotion.Category4stationshavebeenrejectedand arenotusedinthisstudy.SeeTableA.3fordetailedcomments. StationName Latitude Longitude Category Comment AIS1 55.07 228.40 2 2 ASC1 -7.95 345.59 1 1 ASPA -14.33 189.28 4 7 BAY1 55.19 197.29 3 3 BAY2 55.19 197.29 3 3 BRMU 32.37 295.30 1 1 CHA1 32.76 280.16 1 1 CNMI 15.21 145.75 4 7 TAEJ 36.40 127.37 4 7 DARW -12.84 131.13 1 1 DUCK 36.18 284.25 2 5 EISL -27.15 250.62 1 1 GALA -0.70 269.70 1 1 GUAM 13.59 144.87 3 3 GUS2 58.42 224.30 2 2 HILO 19.72 204.95 3 3 99

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Appendix:(continued) -TableA.2-continuedfrompreviouspage HNLC 21.30 202.14 1 1 HOB2 -42.80 147.44 1 1 KERG -49.35 70.26 1 1 KGN0 35.71 139.49 3 3 KGN1 35.71 139.49 3 3 KODK 57.73 207.50 3 3 KOKB 22.22 199.89 4 7 KWJ1 8.72 167.73 1 1 KYW1 24.58 278.35 1 1 LAE1 -6.67 146.99 3 3 MAC1 -54.50 158.94 3 3 MALD 4.19 73.53 4 7 MIZU 39.14 141.13 3 3 NEAH 48.2979 235.38 3 3 NOUM -22.27 166.41 1 1 PBL1 37.86 237.58 3 3 PDEL 37.75 334.34 3 3 PERT -31.80 115.89 2 5 PLO3 32.67 242.76 3 3 PUR3 18.46 292.93 3 3 SEY1 -4.67 55.48 4 7 STR1 -35.32 149.01 1 1 100

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Appendix:(continued) -TableA.2-continuedfrompreviouspage SUVA -18.15 178.43 1 1 TOW2 -19.27 147.06 1 1 TRND 41.05 235.85 3 3 USUD 36.13 138.36 1 1 YSSK 47.03 142.72 1 1 P194 39.19 139.55 3 3 ABER 57.14 357.92 1 1 MANZ 19.06 255.70 3 3 TAIW 25.02 121.54 3 3 TUVA -8.53 179.20 4 7 ACOR 43.36 351.60 1 1 AJAC 41.93 8.76 1 1 ALAC 38.34 359.52 1 1 ALBH 48.39 236.52 2 2 ALEX 31.20 29.91 1 1 ALME 36.85 357.54 3 3 AUCK -36.60 174.83 3 3 BARH 44.40 291.78 1 1 BORK 53.56 6.75 1 1 BRST 48.38 355.50 1 1 BRUS 50.80 4.36 1 1 CAGL 39.14 8.97 1 1 101

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Appendix:(continued) -TableA.2-continuedfrompreviouspage CANT 43.47 356.20 1 1 CASC 38.69 350.58 1 1 CEDU -31.87 133.81 1 1 CEUT 35.90 354.69 3 3 CHAT -43.96 183.43 1 1 CKIS -21.20 200.20 1 1 COCO -12.19 96.83 1 1 CRO1 17.76 295.42 3 3 DGAR -7.27 72.37 1 1 DUBR 42.65 18.11 4 7 DUNT -45.81 170.63 3 3 EPRT 44.91 293.01 1 1 FORT -3.88 321.57 1 1 FTS1 46.21 236.04 3 3 GAL1 29.33 265.26 2 5 GENO 44.42 8.92 1 1 GODE 39.02 283.17 2 5 GOUG -40.35 350.12 3 3 GRAS 43.76 6.92 1 1 HELG 54.17 7.89 1 1 HLFX 44.68 296.39 1 1 HNPT 38.59 283.87 2 5 102

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Appendix:(continued) -TableA.2-continuedfrompreviouspage HOFN 64.27 344.80 3 3 HOLB 50.64 231.87 3 3 JAB1 -12.66 132.89 1 1 KARR -20.98 117.10 1 1 KEN1 60.68 208.65 2 4 KIRI 1.36 172.92 1 1 KOUR 5.25 307.19 1 1 LAGO 37.10 351.33 1 1 LAMP 35.50 12.61 1 1 LAUT -17.61 177.45 1 1 LPGS -34.91 302.07 2 6 LROC 46.16 358.78 1 1 LYTT -43.61 172.72 3 3 MALI -2.30 40.19 1 1 MALL 39.55 2.63 1 1 MARS 43.28 5.35 1 1 MAS1 27.76 344.37 1 1 MATE 40.65 16.70 4 7 MKEA 19.80 204.54 3 3 MOB1 30.23 271.98 2 5 NAIN 56.54 298.31 2 2 NANO 49.30 235.91 2 2 103

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Appendix:(continued) -TableA.2-continuedfrompreviouspage NEWL 50.10 354.46 1 1 NKLG 0.35 9.67 1 1 NPRI 41.51 288.67 1 1 NSTG 55.01 358.56 1 1 NTUS 1.35 103.68 4 7 P201 45.41 141.69 4 7 P202 44.02 144.29 1 1 P204 41.78 140.73 1 1 P205 39.02 141.75 3 3 P206 34.92 139.83 3 3 P207 36.76 137.23 4 7 P208 33.48 135.77 3 3 P209 34.90 132.07 4 7 P210 32.74 129.87 3 3 P211 31.57 131.41 3 3 P212 26.21 127.67 1 1 P213 27.09 142.20 1 1 PETP 53.07 158.61 1 1 PIMO 14.64 121.08 3 3 PNGM -2.04 147.37 1 1 POHN 6.96 158.21 1 1 QAQ1 60.72 313.95 2 2 104

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Appendix:(continued) -TableA.2-continuedfrompreviouspage RBAY -28.80 32.08 1 1 REUN -21.21 55.57 1 1 REYK 64.14 338.05 3 3 RIOG -53.79 292.25 3 3 RWSN -43.30 294.89 2 6 SANT -33.15 289.33 3 3 SEAT 47.65 237.69 1 1 SFER 36.46 353.79 3 3 SHEE 51.45 0.74 1 1 SIMO -34.19 18.44 1 1 SIO3 32.87 242.75 3 3 STAS 59.02 5.60 2 2 STJO 47.60 307.32 1 1 THTI -17.58 210.39 1 1 TIDB -35.40 148.98 1 1 TONG -21.15 184.82 1 1 TORS 62.02 353.24 1 1 TSKB 36.11 140.09 1 1 VALE 39.48 359.66 1 1 VTIS 33.71 241.71 3 3 WGTT -41.29 174.78 3 3 WSRT 52.92 6.61 1 1 105

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Appendix:(continued) -TableA.2-continuedfrompreviouspage VANU -17.74 168.32 3 3 OHI2 -63.32 302.10 2 2 CHUR 58.76 265.91 2 2 AOML 25.74 279.84 1 1 GLPT 37.25 283.50 2 5 GOLD 35.43 243.11 3 3 KELS 46.12 237.10 1 1 METS 60.22 24.40 2 2 NEWP 44.59 235.94 3 3 SAMO -13.85 188.26 3 3 SELD 59.45 208.29 2 4 SOL1 38.32 283.55 2 5 VBCA -38.70 297.73 2 6 YAR1 -29.05 115.35 1 1 106

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Appendix:(continued) TableA.3:DetailsoftheCommentNumbersUsedinTablesA1and2 Number Comment 1 NoComment. 2 Landmotionduetolocalglacialisostaticadjustment(GIA).Stationsareput intocategory2whenthegradientofthelandmotionfromtheICE-5GGIA model( Peltier ,2004)fromthegridcellnearesttothestationtoanyofthe adjacentgridcellsisgreaterthan1mm/yr. 3 Locallandmotionduetotectonicactivity.Stationsareputintocategory3 iftheyarecloserthan250kmfromatectonicallyactiveboundaryor100 kmfromaquietboundary.Thedetailsoftheboundaryclassicationare dicussedinChapter4. 4 Upliftduetofar-aeldpost-earthquakerebound( BridgesandGao ,2006) 5 Subsidenceduetouidwithdrawaland/orsedimentcompaction( Gornitz 2001) 6 UpliftduetorecentdeglaciationnotincludedintheICE-5Gmodel( Rostami etal. ,2000). 7 Stationsthatarenotusedinthisstudyduetoknownissueswiththestation (location,instrumenterrors,levelingproblems,non-linearities) 107


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Doran, Kara.
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Addressing the problem of land motion at tide gauges
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ABSTRACT: Estimation of global mean sea level change has become an area of interest for scientists in recent decades because of its importance as an indicator of climate change. Climate models predict varying degrees of change in global temperature and global sea level over the next 100 years. One way to check the validity of the models is to estimate sea level change over the last century and constrain the models to match these estimates. Traditionally, sea level change estimates have been calculated using long time series from tide gauges. There are some disadvantages to this approach however, since tide gauges have limited spatial coverage and make measurements relative to a land reference point that may be undergoing uplift or subsidence. Satellite altimetry has also been used in recent years to estimate sea level changes, but these measurements are subject to drift errors and must be calibrated. Mitchum (1998, 2000) has developed a method using the global network of tide gauges to calibrate altimeters that enables estimation of sea level change with a precision of 0.4 mm/yr. Errors in the estimates arise from a variety of sources, but the error of primary concern is that due to land motion at the tide gauge stations. In the present study we will investigate ways to improve the land motion estimate and thus reduce the error.
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