Viii hotine marussi symposium on mathematical geodesy proceedings of the symposium in rome 17 21 jun

VIII

Hotine Marussi Symposium on Mathematical Geodesy Proceedings of the Symposium in Rome 17 21 June 2013 1st Edition Nico Sneeuw

Visit to download the full and correct content document: https://textbookfull.com/product/viii-hotine-marussi-symposium-on-mathematical-geod esy-proceedings-of-the-symposium-in-rome-17-21-june-2013-1st-edition-nico-sneeuw /

More products digital (pdf, epub, mobi) instant download maybe you interests ...

IX Hotine-Marussi Symposium on Mathematical Geodesy: Proceedings of the Symposium in Rome, June 18 – 22, 2018 Pavel Novák

https://textbookfull.com/product/ix-hotine-marussi-symposium-onmathematical-geodesy-proceedings-of-the-symposium-in-romejune-18-22-2018-pavel-novak/

Cone Penetration Testing IV : Proceedings of the 4th International Symposium on Cone Penetration Testing (CPT 2018), June 21-22, 2018, Delft, The Netherlands First Edition Hicks

https://textbookfull.com/product/cone-penetration-testing-ivproceedings-of-the-4th-international-symposium-on-conepenetration-testing-cpt-2018-june-21-22-2018-delft-thenetherlands-first-edition-hicks/

International Symposium on Advancing Geodesy in a Changing World Proceedings of the IAG Scientific Assembly Kobe Japan July 30 August 4 2017 Jeffrey T. Freymueller

https://textbookfull.com/product/international-symposium-onadvancing-geodesy-in-a-changing-world-proceedings-of-the-iagscientific-assembly-kobe-japan-july-30-august-4-2017-jeffrey-tfreymueller/

Foundations and Practice of Security 6th International Symposium FPS 2013 La Rochelle France October 21 22 2013 Revised Selected Papers 1st Edition Jean Luc Danger

https://textbookfull.com/product/foundations-and-practice-ofsecurity-6th-international-symposium-fps-2013-la-rochelle-franceoctober-21-22-2013-revised-selected-papers-1st-edition-jean-lucdanger/

Proceedings of the 5th International Symposium on Biological Control of Arthropods 1st Edition Peter Mason

https://textbookfull.com/product/proceedings-of-the-5thinternational-symposium-on-biological-control-of-arthropods-1stedition-peter-mason/

Proceedings of the 12th International Symposium on Computer Science in Sport IACSS 2019 Martin Lames

https://textbookfull.com/product/proceedings-of-the-12thinternational-symposium-on-computer-science-in-sportiacss-2019-martin-lames/

Recent Developments in Acoustics Select Proceedings of the 46th National Symposium on Acoustics Mahavir Singh

https://textbookfull.com/product/recent-developments-inacoustics-select-proceedings-of-the-46th-national-symposium-onacoustics-mahavir-singh/

Progress in Optomechatronic Technologies Proceedings of International Symposium on Optomechatronic 2018 Amalia Martínez-García

https://textbookfull.com/product/progress-in-optomechatronictechnologies-proceedings-of-international-symposium-onoptomechatronic-2018-amalia-martinez-garcia/

Advances in Neural Networks ISNN 2017 14th

International Symposium ISNN 2017 Sapporo Hakodate and Muroran Hokkaido Japan June 21 26 2017 Proceedings Part I 1st Edition Fengyu Cong

https://textbookfull.com/product/advances-in-neural-networksisnn-2017-14th-international-symposium-isnn-2017-sapporohakodate-and-muroran-hokkaido-japan-june-21-26-2017-proceedingspart-i-1st-edition-fengyu-cong/

Nico Sneeuw

Pavel Novák

Mattia Crespi

Fernando Sansò Editors

VIII Hotine-Marussi Symposium on Mathematical Geodesy

Proceedings of the Symposium in Rome, 17-21 June, 2013

InternationalAssociation ofGeodesySymposia

ChrisRizos,SeriesEditor PascalWillis,AssistantSeriesEditor

InternationalAssociation ofGeodesySymposia

ChrisRizos,SeriesEditor

PascalWillis,AssistantSeriesEditor

Symposium101:GlobalandRegionalGeodynamics Symposium102:GlobalPositioningSystem:AnOverview Symposium103:Gravity,Gradiometry,andGravimetry Symposium104:SeaSurfaceTopographyandtheGeoid Symposium105:EarthRotationandCoordinateReferenceFrames Symposium106:DeterminationoftheGeoid:PresentandFuture Symposium107:KinematicSystemsinGeodesy,Surveying,andRemoteSensing Symposium108:ApplicationofGeodesytoEngineering Symposium109:PermanentSatelliteTrackingNetworksforGeodesyandGeodynamics Symposium110:FromMarstoGreenland:ChartingGravitywithSpaceandAirborneInstruments Symposium111:RecentGeodeticandGravimetricResearchinLatinAmerica Symposium112:GeodesyandPhysicsoftheEarth:GeodeticContributionstoGeodynamics Symposium113:GravityandGeoid Symposium114:GeodeticTheoryToday Symposium115:GPSTrendsinPreciseTerrestrial,Airborne,andSpaceborneApplications Symposium116:GlobalGravityFieldandItsTemporalVariations Symposium117:Gravity,GeoidandMarineGeodesy Symposium118:AdvancesinPositioningandReferenceFrames Symposium119:GeodesyontheMove Symposium120:TowardsanIntegratedGlobalGeodeticObservationSystem(IGGOS) Symposium121:GeodesyBeyond2000:TheChallengesoftheFirstDecade Symposium122:IVHotine-MarussiSymposiumonMathematicalGeodesy Symposium123:Gravity,GeoidandGeodynamics2000 Symposium124:VerticalReferenceSystems Symposium125:VistasforGeodesyintheNewMillennium Symposium126:SatelliteAltimetryforGeodesy,GeophysicsandOceanography Symposium127:VHotineMarussiSymposiumonMathematicalGeodesy Symposium128:AWindowontheFutureofGeodesy Symposium129:Gravity,GeoidandSpaceMissions Symposium130:DynamicPlanet-MonitoringandUnderstanding... Symposium131:GeodeticDeformationMonitoring:FromGeophysicaltoEngineeringRoles Symposium132:VIHotine-MarussiSymposiumonTheoreticalandComputationalGeodesy Symposium133:ObservingourChangingEarth Symposium134:GeodeticReferenceFrames Symposium135:Gravity,GeoidandEarthObservation Symposium136:GeodesyforPlanetEarth Symposium137:VIIHotine-MarussiSymposiumonMathematicalGeodesy Symposium138:ReferenceFramesforApplicationsinGeosciences Symposium139:EarthontheEdge:ScienceforaSustainablePlanet Symposium140:The1stInternationalWorkshopontheQualityofGeodetic ObservationandMonitoringSystems(QuGOMS’11) Symposium141:Gravity,GeoidandHeightsystems(GGHS2012)

VIIIHotine-MarussiSymposiumon

MathematicalGeodesy

ProceedingsoftheSymposiuminRome, 17–21June,2013

Editedby

PavelNovák

MattiaCrespi

FernandoSansò

NicoSneeuw

VolumeEditorsSeriesEditor

NicoSneeuw InstituteofGeodesy UniversityofStuttgart

Stuttgart Germany

PavelNovák

DepartmentofMathematics

UniversityofWesternBohemia

Pilsen

CzechRepublic

MattiaCrespi

GeodesyandGeomaticsDivision UniversityofRome"LaSapienza"

Rome Italy

FernandoSansò

DipartimentodiIngegneriaCivileeAmbientale PolitecnicodiMilano Milano Italy

ChrisRizos UniversityofNewSouthWales

Sydney NewSouthWales

Australia

AssociateEditor

PascalWillis

Institutnationaldel’Information géographiqueetforestière ServicedelaRecherche etdel’Enseignement

Saint-Mandé France

ISSN0939-9585ISSN2197-9359(electronic)

InternationalAssociationofGeodesySymposia

ISBN978-3-319-24548-5ISBN978-3-319-30530-1(eBook) DOI10.1007/978-3-319-30530-1

LibraryofCongressControlNumber:2016935295

©SpringerInternationalPublishingSwitzerland2016

Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation, reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval, electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse.

Thepublisher,theauthorsandtheeditorsaresafetoassume thattheadviceandinformationinthisbookarebelieved tobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgiveawarranty, expressorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthatmayhavebeen made.

Printedonacid-freepaper

ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland

Preface

ThisvolumecontainstheproceedingsoftheVIIIHotine-MarussiSymposiumonMathematicalGeodesy,whichwasheldJune17to21,2013.Forasecondtimeinrow,theSymposium tookplaceattheFacultyofEngineeringoftheUniversityofRome“LaSapienza”,Italy.Again, thesymposiumwashostedinthebeautifulancient chiostro oftheBasilicaofS.Pietroin Vincoli,famouslyknownforitsstatueofMosesbyMichelangelo.

Thetraditionalname mathematicalgeodesy fortheseriesofHotine-MarussiSymposia maynotfullydojusticetotheSymposium’sbroadscopeoftheoreticalgeodesyingeneral. However,thenamefortheserieshasbeenusedsince1965,i.e.thedaysofAntonioMarussi, whichisagoodreasontoadheretoit.ThevenueoftheHotine-MarussiSymposiahas traditionallybeeninItaly,asexemplifiedinthehistoricaloverviewandmaponthenextpages.

Since2006theseriesisundertheresponsibilityoftheIntercommissionCommitteeon Theory(ICCT),across-commissionentitywithintheInternationalAssociationofGeodesy (IAG).TheoverallgoaloftheHotine-MarussiSymposiahasalwaysbeentheadvancement oftheoreticalgeodesy.ThisgoalisalignedwiththeobjectivesoftheICCT,whichhasthe developmentsingeodeticmodellinganddataprocessinginthelightofrecentadvancesof geodeticobservingsystemsaswellastheexchangebetweengeodesyandneighbouringEarth sciencesasitscentralthemes.Indeed,thecurrentproceedingsaretestimonytothewidthand vibrancyoftheoreticalgeodesy.

TheSymposiumattracted90participantswhocontributed88papers(71oraland17poster), organizedineightregularsessionsplusthesessionattheAccademiaNazionaledeiLincei.To alargeextent,thesessions’topicsweremodelledonthestudygroupstructureoftheICCT. ThechairsoftheICCTstudygroups,whoconstitutedtheSymposium’sScientificCommittee, wereatthesametimeresponsiblefororganizingthesessions:

1. Geodeticdataanalysis

W.Kosek,R.Gross,C.Kreemer

2. Geopotentialmodeling,boundaryvalueproblemsandheightsystems

P.Novák,M.Schmidt,C.Gerlach

3. Atmosphericmodelingingeodesy

T.Hobiger,M.Schindelegger

4. GravityfieldmappingmethodologyfromGRACEandfuturegravitymissions

M.Weigelt,A.Jäggi

5. Computationalgeodesy

R. ˇ Cunderlík,K.Mikula

6. Theoreticalaspectsofreferenceframes

A.Dermanis,T.VanDam

7. DigitalTerrainModeling,SyntheticApertureRadarandnewsensors:theoryandmethods

M.Crespi,E.Pottier

8. Inversemodeling,estimationtheory

P.Xu

AspecialsessionwasorganizedattheAccademiaNazionaledeiLinceibyFernandoSansò, himselfamemberofthisvenerableacademy,withthreekeynoteaddresses.WhatFernandodid

notknowatthispointwasthattheorganizingcommitteehaddecidedtodedicatethe VIIIHotine-MarussiSymposiuminhishonour.ToputFernandointothespotlightfour briefspeechesfollowedbyfourrenownedgeodesistsandold-timecolleaguesofFernando: MicheleCaputo,SakisDermanis,ErikGrafarendandChristianTscherning.Pleasenotethat thelatterthreenamesrepresenttheuniversitiesofThessaloniki,StuttgartandCopenhagen, respectively,fromwhichFernandohasbeenawardedhonorarydoctorates.Eachofthese gentlemenreminiscedabouttheirlong-termcollaborationandfriendshipwithFernando, buttheyalsocharacterizedhimbyentertaininganecdotes.Thusthelong-termcommitment anddedicationofProf.Sansòwasacknowledged,whohasbeenthedrivingforcebehind theseriesofHotine-MarussiSymposiaoverthepastdecades.Whetherthishonourmight beaburdenatthesametime,asthecartoonseemstosuggest,well:futurewilltell.

Credits:RiccardoBarzaghi

Wewanttoexpressourgratitudetoallofthosewhohavecontributedtothesuccess oftheVIIIHotine-MarussiSymposium.Theaforementionedstudygroupchairs(Scientific Committee)putmucheffortinorganizingattractivesessionsandconveningthem.Theyalso tookaleadingroleinthepeer-reviewprocess,whichwasmanagedbytheIAGproceedings editorDr.PascalWillis.Weequallyowethankstoallreviewers.Althoughmuchofthereview processitselfremainsanonymous,thecompletelistofthereviewersisprintedinthisvolume asatokenofourappreciationoftheirdedication.

FinancialandpromotionalsupportwasgivenbytheFacultyofEngineeringoftheSapienza UniversityofRome.

ButmostofourthanksareduetoMattiaCrespiandhisteamoftheAreaofGeodesy andGeomatics(AGG),whichispartoftheDepartmentofCivil,BuildingandEnvironmental Engineering(DICEA),whohostedtheSymposium.Itiswellknownthatthequalityofa

LocalOrganizingCommittee(LOC)isdecisivetoasuccessfulscientificmeeting.Beyond responsibilityforwebsite,registration,technicalsupportandallkinds ofotherarrangements, theLOCorganizedavisittotheVillaFarnesinaandagreatsocialeventtotheCapitoline Hill,includingaguidedmuseumtourandaroof-topdinnerwithanastonishingviewover theeternalcity.ThroughtheirableorganizationandimprovisationskillsMattiaCrespiand histeam(ElisaBenedetti,MaraBranzanti,PaolaCapaldo,GabrieleColosimo,NicoleDore, FrancescaFratarcangeli,AugustoMazzoni,AndreaNascetti,JolandaPatruno,Francesca PieraliceandMartinaPorfiri)havedonemorethantheirshareinbringingtheVIIIHotineMarussiSymposiumtosuccess.

StuttgartNicoSneeuw October2014PavelNovák

MattiaCrespi FernandoSansò

FiftyYearsofHotine-MarussiSymposia

In1959,AntonioMarussi,incooperationwiththeItalianGeodeticCommission,starteda seriesofsymposiainVenice.Thefirstthreeofthesecoveredtheentiretheoreticaldefinition of3Dgeodesy,asdelineatedindiscussionswithrenownedcontemporaryscientists:

•16–18July1959,Venice,1stSymposiumonThreeDimensionalGeodesy,publishedin BollettinodiGeodesiaeScienzeAffini,XVIII,Nı 3,1959

•29May–1June1962,Cortinad’Ampezzo,2ndSymposiumonThreeDimensionalGeodesy, publishedinBollettinodiGeodesiaeScienzeAffini,XXI,Nı 3,1962

•21–22April1965,Turin,3rdSymposiumonMathematicalGeodesy,publishedbyCommissioneGeodeticaItaliana,1966

Fromtheverybeginning,MartinHotineprovidedessentialinspirationtothesesymposia. Afterhisdeathin1968,thefollowingsymposiabearhisname:

•28–30May1969,Trieste,1stHotineSymposium(4thSymposiumonMathematical Geodesy),publishedbyCommissioneGeodeticaItaliana,1970

•25–26October1972,Florence,2ndHotineSymposium(5thSymposiumonMathematical Geodesy),publishedbyCommissioneGeodeticaItaliana,1973

•2–5April1975,Siena,3rdHotineSymposium(6thSymposiumonMathematicalGeodesy), publishedbyCommissioneGeodeticaItaliana,1975

•8–10June1978,Assisi,4thHotineSymposium(7thSymposiumonMathematical Geodesy),publishedbyCommissioneGeodeticaItaliana,1978

•7–9September1981,Como,5thHotineSymposium(8thSymposiumonMathematical Geodesy),publishedbyCommissioneGeodeticaItaliana,1981

AfterMarussi’sdeath,in1984,thesymposiawerefinallynamedtheHotine-Marussi Symposia:

•3–6June1985,Rome,IHotine-MarussiSymposium(MathematicalGeodesy)

•5–8June1989,Pisa,IIHotine-MarussiSymposium(MathematicalGeodesy)

•29May–3June1994,L’Aquila,IIIHotine-MarussiSymposium(MathematicalGeodesy, GeodeticTheoryToday),publishedbySpringer,IAG114

•14–17September1998,Trento,IVHotine-MarussiSymposium(MathematicalGeodesy), publishedbySpringer,IAG122

•17–21June2003,Matera,VHotine-MarussiSymposium(MathematicalGeodesy),publishedbySpringer,IAG127

•29May–2June2006,Wuhan,VIHotine-MarussiSymposium(TheoreticalandComputationalGeodesy,1sttimeunderICCT),publishedbySpringer,IAG132

•6–10July2009,Rome,VIIHotine-MarussiSymposium(MathematicalGeodesy),publishedbySpringer,IAG137

•17–21June2013,Rome,VIIIHotine-MarussiSymposium(MathematicalGeodesy), publishedbySpringer,IAG142

PartILinceiSession

OpeningRemarksfortheVIIIHotine-MarussiSymposium

MicheleCaputo

FernandoSansòLaudation ..................................................7

MicheleCaputo

GlobalReferenceSystems:TheoryandOpenQuestions ........................9

AthanasiosDermanis

PartIIGeodeticDataAnalysis

NoiseAnalysisofContinuousGPSTimeSeriesofSelectedEPNStations toInvestigateVariationsinStabilityofMonumentTypes .......................19 AnnaKlos,JanuszBogusz,MariuszFigurski,andWieslawKosek

ImprovementofLeast-SquaresCollocationErrorEstimatesUsingLocalGOCE Tzz SignalStandardDeviations ..................................................27 C.C.Tscherning

MultivariateIntegerCycle-SlipResolution:ASingle-ChannelAnalysis ..........33 P.J.G.TeunissenandP.F.deBakker

TheoryofEarthRotationVariations .........................................41 RichardS.Gross

VariableSeasonalandSubseasonalOscillationsinSeaLevelAnomalyData andTheirImpactonPredictionAccuracy ....................................47 W.Kosek,T.Niedzielski,W.Popi ´ nski,M.Zbylut-Górska,andA.Wn˛ek PermanentGPSNetworksinItaly:AnalysisofTimeSeriesNoise ................51

R.Devoti,G.Pietrantonio,A.R.Pisani,andF.Riguzzi

VADASE:StateoftheArtandNewDevelopmentsofaThirdWaytoGNSS Seismology ................................................................59

E.Benedetti,M.Branzanti,G.Colosimo,A.Mazzoni,andM.Crespi

OntheSpatialResolutionofHomogeneousIsotropicFiltersontheSphere .......67 BalajiDevarajuandNicoSneeuw

OnTime-VariableSeasonalSignals:ComparisonofSSAandKalmanFiltering BasedApproach ...........................................................75

Q.Chen,M.Weigelt,N.Sneeuw,andT.vanDam

ExtensiveAnalysisofIGSREPRO1CoordinateTimeSeries ....................81 M.Roggero

PartIIIGeopotentialModeling,BoundaryValueProblemsandHeightSystems

DeterminationofW0 fromtheGOCEMeasurementsUsingtheMethod ofFundamentalSolutions ...................................................91

Róbert ˇ Cunderlík

CombinationofGOCEGravityGradientsinRegionalGravityFieldModelling UsingRadialBasisFunctions ................................................101

VerenaLieb,JohannesBouman,DeniseDettmering,MartinFuchs, andMichaelSchmidt

RosboroughRepresentationinSatelliteGravimetry ............................109 NicoSneeuwandMohammadA.Sharifi

CombiningDifferentTypesofGravityObservationsinRegionalGravityModeling inSphericalRadialBasisFunctions ..........................................115 KatrinBentelandMichaelSchmidt

HeightDatumUnificationbyMeansoftheGBVPApproachUsingTideGauges ...121 E.Rangelova,M.G.Sideris,B.Amjadiparvar,andT.Hayden

PartIVAtmosphericModelinginGeodesy

ComputationofZenithTotalDelayCorrectionFieldsUsing Ground-BasedGNSS .......................................................131

B.Pace,R.Pacione,C.Sciarretta,andG.Bianco

RigorousInterpolationofAtmosphericStateParametersforRay-Traced TroposphericDelays ........................................................139 CamilleDesjardins,PascalGegout,LaurentSoudarin,andRichardBiancale ComparisonofDifferentTechniquesfor TroposphericWetDelayRetrievalOver SouthAmericaandSurroundingOceans ......................................147

A.Calori,G.Colosimo,M.Crespi,andM.V.Mackern

PartVGravityFieldMappingMethodologyfromGRACEandFutureGravity Missions

TheRoleofPositionInformationfortheAnalysisofK-BandData:Experiences fromGRACEandGOCEforGRAILGravityFieldRecovery ...................157

A.Jäggi,G.Beutler,U.Meyer,H.Bock,andL.Mervart

GravityFieldMappingfromGRACE:DifferentApproaches—SameResults? .....165 ChristophDahle,ChristianGruber,ElisaFagiolini,andFrankFlechtner

TheEffectofPseudo-StochasticOrbit ParametersonGRACEMonthlyGravity Fields:InsightsfromLumpedCoefficients ....................................177

U.Meyer,C.Dahle,N.Sneeuw,A.Jäggi,G.Beutler,andH.Bock

OnanIterativeApproachtoSolvingtheNonlinearSatellite-FixedGeodetic Boundary-ValueProblem ...................................................185 MarekMacák,KarolMikula,ZuzanaMinarechová,andRóbert ˇ Cunderlík

PartVIComputationalGeodesy

AnOpenCLImplementationofEllipsoidalHarmonics .........................195 OtakarNesvadbaandPetrHolota

ARemarkontheComputationoftheGravitationalPotentialofMasses withLinearlyVaryingDensity ...............................................205

MariaGraziaD’Urso

TheObservationEquationofSpiritLevelinginMolodensky’sContext ...........213

B.Betti,D.Carrion,F.Sacerdote,andG.Venuti

PartVIITheoreticalAspectsofReferenceFrames

ReferenceStationWeightingandFrameOptimalityinMinimallyConstrained Networks ..................................................................221

C.Kotsakis

AtmosphericLoadingandMassVariationEffectsontheSLR-Defined Geocenter .................................................................227

RolfKönig,FrankFlechtner,Jean-ClaudeRaimondo,andMargaritaVei

RadargrammetricDigitalSurfaceModelsGenerationfromHighResolution SatelliteSARImagery:MethodologyandCaseStudies .........................233

AndreaNascetti,PaolaCapaldo,FrancescaPieralice,MartinaPorfiri, FrancescaFratarcangeli,andMattiaCrespi

PartVIIIDigitalTerrainModeling,SyntheticApertureRadarandNewSensors: TheoryandMethods

PrinciplesandApplicationsofPolarimetricSARTomographyforthe CharacterizationofComplexEnvironments ...................................243 LaurentFerro-Famil,YueHuang,andEricPottier

Re-griddingandMergingOverlappingDTMS:ProblemsandSolutions inHELI-DEM .............................................................257 LudovicoBiagiandLauraCarcano

Single-EpochGNSSArrayIntegrity:AnAnalyticalStudy ......................263 A.KhodabandehandP.J.G.Teunissen

PartIXInverseModeling,EstimationTheoryVIIIHotine-Marussi:Geodetic DataAnalysis

GlobaltoLocalMohoEstimateBasedonGOCEGeopotentialModelandLocal GravityData ..............................................................275 R.Barzaghi,M.Reguzzoni,A.Borghi,C.DeGaetani,D.Sampietro,andA.M.Marotta

AnOverviewofAdjustmentMethodsforMixedAdditiveandMultiplicative RandomErrorModels ......................................................283 YunShi,PeiliangXu,andJunhuanPeng

CycleSlipDetectionand CorrectionforHeadingDeterminationwith Low-CostGPS/INSReceivers ...............................................291 PatrickHenkelandNaoyaOku

AdjustingtheErrors-In-VariablesModel:LinearizedLeast-Squaresvs.Nonlinear TotalLeast-Squares ........................................................301 BurkhardSchaffrin

MultivariateGNSSAttitudeIntegrity:TheRoleofAffineConstraints ...........309 GabrieleGiorgiandPeterJ.G.Teunissen

IntegratingGeologicalPriorInformationintotheInverseGravimetricProblem: TheBayesianApproach .....................................................317 L.Rossi,M.Reguzzoni,D.Sampietro,andF.Sansò

EffectsofDifferentObjectiveFunctionsinInequalityConstrainedand Rank-DeficientLeast-SquaresProblems ......................................325 LutzRoese-KoernerandWolf-DieterSchuh

ListofReviewers ............................................................

Author Index

OpeningRemarksforthe VIIIHotine-Marussi Symposium

MicheleCaputo

Abstract

Openingremarksforthe2013Hotine-MarussiSymposiumSpecialSessionatthe AccademiaNazionaledeiLincei.

Keywords

Anelasticity•Geoidalondulations•Gravity•Rheology

AtthepreviousHotine-MarussisymposiuminRome,2009, ItoldoftheMarussipendulums,ofthegravityabsorption experiments,ofthegravitonsandoftheimportantdevelopmentsofthemodelsofthegravityfieldoftheEarthproduced anddiscussedacenturyagobyPizzettiandSomigliana, eventuallypresentedintheAccademia,andfoundersofthe wellknownelegantmodellingofthegravityfieldofthe Earth.

ThenIaskedmyselfwhatcomesnext?TomysurpriseI hadnotmentionedMarussi’s IntrinsicGeodesy.Thereason probablyistheCayley-Darbouxtheorem,whichwasprecededbytheimportantworksofMainardiandofCodazzi onthemappingofrevolutionsurfaces.

Marussiknewallthissowellthatheassignedthetheorem assubjectforathesisaskingmetobe correlatore since hewasverybusyandoftenaway(Milani 1961;Sansòand

Caputo 2008).Probablybecauseofthistheorem,Marussi neverencouragedmetoworkinthisfieldnorwentanyfurtherwiththeconceptofintrinsicgeodesywhich,practically, extendstothegravityfieldtheclassicprincipleofembedding anyprobleminitsmostappropriatecoordinatesystem.

Inthiscasethepossiblecoordinatesystemwouldhave beenthefamilyofequipotentialsurfacesofthegravity fieldwhich,unfortunately,didnotturnouttosatisfythe conditionsofthetheorem,exceptlocally.

InthissenseitwasGrafarend(1988)toseethatthetheory couldbeappliedlocally.InanothersenseIfeelthatIshould quoteBocchio(1974)whowroteontheextensionofthe conceptof intrinsic togeophysics.

PhilosophicallyIseetheword intrinsic astheoperationofimbeddingtheprobleminitsreality.Digintothe problem,deepenoughuntilyoufindthe pepitadeoro,or thetruffledependingonyourtaste,whichshowsyouthe path.The pepita maybetheappropriategeometricspace, ortheappropriatefunctionalspace,ortheappropriatesensor’stypeand/orthebestintrinsicspacedistributionofthe data.

FernandoSansòandMattiaCrespikindlyaskedmetoopentheworks ofthe2013Hotine-MarussisymposiumandspeakofGeodesyandof theAccademiadeiLincei,andgavemethegreatpleasuretobehere amongmyfriendssinceGeodesyandtheAccademiahavefilledagood partofmylife.

M.Caputo( ) CollegeofGeosciences,TexasA&MUniversity,CollegeStation Texas,Rome,Italy

e-mail: mic.caput27@gmail.com

2GeodesyandRheology

LetmenowgobackhalfacenturywhenIwassharingthe office,withthepersonwhowasthenmymentor,hisdesk betweenthetwowindowsinthecorner,mydeskinfrontof hisbutintheoppositecorner.Theinstitutehadbeenfounded byMarussi,director,andmembers:meandatechnician;in

anoldbuildingofTriestethenstilloccupiedbytheallied troops.ThelibraryoftheinstitutewasMarussi’soffice.On theshelvesonthesideofmydeskwasabookwiththetitle Rheology,thenmysteriousforme,whichatthattimeseemed tohaveverylittletodowithgeodesyorgeophysics.

AllIknewafter5yearsofstudiesoftheGreeklanguage inhighschoolwasthat rheo meansmovement;diggingout ofmymemory,Iarrivedto pantarei;butthatwascertainly notenoughtosatisfyanylevelofcuriosity.

Itwasa sign onthewall.LaterIrealisedthatrheologywouldbeimpliedinthefutureofgeophysicsand geodesytothepointtomakeitimportantforsomerealistic results.IfImaymakeacomparisonitwouldbeastudy ofmacroeconomicproblemswithouttakingintoaccount memory(Demaria 1976,Caputo 2012a,Caputo 2012b). Rheologywasthelastdisciplinetojointhesetformingthe interdisciplinaryroyalcrownofGeodesyandGeophysics.

Forinstancetodaywearenotsoconcernedwithelevationsbutwiththeirtimevariationseitherinthecaseofthe coastlinesandofsealevelasseparatematters.

Wemeasureearthtidestostudy,withverypoorresolution asamatteroffact,someaverageelasticpropertiesoftheinterioroftheEarth,butthemajorexpectedresultwastheestimate(Slichteretal. 1964)ofthephaselagoftheMoonrelativetothebulgesitcreatesintheEarth,involvingtheinelasticpropertiesoftheEarth;inthiscasemoreinterestingthan theelasticproperties,and,inturn,wefoundthatitisrheologymodellingandrulingthehistoryoftheEarth–Moonsystem:anditsfutureaswellasthoseofallplanetarysystems.

Alwaysconcernedwithrheology,onemysteryisthe distributionofthecontinentalmassesonthesurfaceofthe Earth,apparentlydisordered,althoughnotcasually,butwith littlelongitudinalandlatitudinalsymmetry,whichdoesnot seemcompatiblewiththestabilityoftheEarthrelativetothe axisofrotation(CaputoandCaputo 2013).Itistherheology whichadjuststhedistributionofmassesintheinteriorin ordertokeeptheEarthinthesamepositionrelativetothe axisofrotation,atleastintherecentdecades.Theabsence ofthepossibleandinevitableassociatedprecessionsrelative totheaxisofrotationprovesit.

Naturallyalloccursinobedienceofthesecondprinciple ofthermodynamics,whichisembeddedintheconstitutive equationsofrheologyandmakesthegeneralequationsof elasticityphysicallyconsistentwhenrheologyisincluded andfinallyacceptable.Idonotmeantheelasticityasiscommonlyusedinseismologyorthatexistingonlyinelementary booksbutthatwhichconsidersalsothedissipationofelastic energyorifyoulikethesecondprincipleofthermodynamics.

Atthattime,ImeanwhenIwasinMarussi’soffice,the GeoidwastheTanni’s(1940)Geoidorlittlemore.Limitedto severallineshundredsofkilometreslonginnorthernEurope. PracticallyinthefirstpartofthetwentiethcenturytheGeoid

wasanunreachablefluidspectrumandnoattentionwas giventotheeffectsofrheology.YoucouldseeGeoid,since itwasthereonallcoastlinesbut,sinceitdidnothaveany mathematicalrepresentation,itwasasforMartinEdenin JackLondon’sbook: theinstantheknew, heceasedto know

TodaywehavetheGeoidandobservethetides,alsofor thesolidEarth.

Thequestionisthenposed:ifweacceptthatallis changingduringandbetweenrepeatedmeasurementswhat shouldwechangeinourstrategyandtechniquesinobserving andinterpretingtheresults?WillGeomaticsaloneanswer thisquestion?Orshoulditbeinterdisciplinary?

Howtogofurtherintheconceptofembeddingthe problem,ormoregenerallythematter,initsreality?How aboutsymmetries?

The pepita concerningthefunctionalspaceofrheology seemstobeinthevarioustypesofmathematicalformalisms, practicallyrepresentingthesecondprincipleofthermodynamics,whoseuseinrecentdecadeshasspreadinmany fieldsofscience.

Appliedinfieldssuchastheoreticalphysics(e.g.Naber 2004),biology(e.g.Cesaroneetal. 2004),geodesy(e.g. Baleanuetal. 2009),medicine(e.g.ElShahed 2003),diffusion(e.g.Mainardi 1996),chemistry(e.g.Martinelletal. 2006),plasmasinboundeddomains(e.g.Agrawal 2002), geophysics(e.g.Iaffaldanoetal. 2006),ineconomyand financeandinplasmaturbulence(e.g.DelCastilloNegrete etal. 2004).

3SymmetryandtheDesign ofNetworks

Aparticularlyimportantstepconcerningintrinsicgeodesyor geophysicswastofindanappropriatenetworkfortheobservationandtheneededobservableswhichobviouslyshould concernGeomatics.Anetworkintrinsicoftheproblemor, better,dictatedbytheproblemandbythecharacteristics oftheneededobservableassuggestedinthe,perhapsonly aseminal,workofCaputo(1979a, b)followedbythe interestingandimportantmeetingOptimizationandDesign ofGeodeticNetworks(GrafarendandSansò 1985).

Inaccordanceandwiththehelpwhichmaycomefromthe principlesofsymmetry(Weyl 1983),Iconsiderednetworks resultingfromthe pflastersatz

Forinstancetheselectionsofseismographicnetworks distributedamongthosesuggestedbythe pflastersatz may improvetheknowledgeofthedepthofearthquakewhichis sopoorlyknownandalsoestimatetheamountoflostinformationinthesitesunreachedbythenetwork.Atleastsuch networkwouldensurehomogeneityinthedatacollection.

Perhapsnotsurprisingly,fromthedistributionofthestation dependsthefrequencyofcertaintypesofeventsandtypesof signalsrecorded.

Oneclearexampleisthesurveyofthestresstensorin thePhlaegreanFieldsnearNapleswhichshowedsurface deformationsoftheorderof10 4 fromwhichwasinferred thepossibilityofanearthquakeofmagnitude5(Caputo 1979c).

Iapologiseforusingmyworkforthisexample,notonly becauseitwaseasiertodigoutthansomeoneelse’swork butthepointisthatknowingitbetterIcandigoutitsmost importantaspecthere:myregretaboutit,isthatIdidnot finishthework,theemblemofwhatIamtryingtosay.The undergroundisoftenrich,Imaynotsayfull,ofcavities andthecavityeffect(Neuber 1937;Harrison 1976),orthe anomaliesoftheelasticmaterialsmayalsoaffectthestress fieldwhichweareobservingonthesurfaceoftheEarthand, therefore,wemaynotbeextrapolatedtotheunderground withoutadditionalinformation.

Thisinturncastssomedoubtsonthereliabilityofthe estimateofthemagnitudeofanearthquakepossiblydueto thereleaseoftheestimatedelasticenergystoredinthelocal portionofthecrust.Infactwasalsomissinganaccurate knowledgeofthelocalvaluesoftheelasticparameters.

Astressfieldestimatebasedondatataken40yearslater wouldmissalsotheinformationonthebalancebetween therelaxationoftheelasticenergyalreadystoredandthat accumulatedintheadditional40years.Obviouslythisisdue toalackofinterdisciplinarity.Whichinstead,inaworldof easycommunication,astoday,shouldbetheemblemofmost earthscienceresearchprojects.

Moreovernotsufficientattentionwasgiventothelarge displacementsoftheorderofhalfameteroveradistance oftheorderofakmobservedinPozzuoliandtothe causesofthisdisplacementandtotheneedofextendingthe observationsatdepth.

Inotherwordssinceasurfacenetworkisnotsufficient foradetailedandsignificantstudyoftheinelasticfield, additionaldataatdepthisneededtomodeltheimportant anomaliesofthestressfieldatdepth.

Thesamemayberepeatedforthenetworksofmeteorologicalstationsusingtheessentialfactthattemperatureisnot aphysicallymeaningfulparameterunlessoneassociatestoit thehumidityandthepressure,thatisatleasttheelevationof thestation.

Thematter,tosayitconcisely,istoallowthedescription ofphysicalobjectswithalimitednumberofparameters which,inturn,permitsadirect,numerical,andtherefore quantitative,comparisonwithitspreviousconditionandwith similarcases.Theclassicexampleisthatofradioactivitywherethemostimportantpropertyofthematerialis describedbyasinglenumber:theaveragelifedirectlyrelated totheverysimplelineardifferentialequationofMalthus.

Iamnotsuggestingtointroduce“abstractequations”but thoserightlydefinedasphenomenological.

Thesephenomenologicalequations,whenadequatelyverifiedwithexperimentaldata,representastepforwardin respecttotheusualempiricalequationswhicharestillvery usefulinmanybranchesofappliedscienceandtechnology. Howeversomescientistsresentthefactthattheyarenot logicallyobtainedfromfirstprinciplesignoringthatifwe stickverystrictlytofirstprinciplessometypesofprogress aremadeveryslowanddifficult.

Also,whenpossibleandwhenbettermethodsarenot available,wemayusethemethodbasedonthenumerical solutionoftheCauchyproblemforastochasticdifferential equations(e.g.Caputoetal. 2000).

MoreovernotonlyGPSdatabutalsoearthquake’sdata resultfromobservationwithmany,sometimesdifferent, instrumentsand,mostimportant,atdifferentplaceswhere thesignalsarriveafterpathswhicharedifferentinlength butalsotravelledthroughdifferentmedia.This,inturn,may causeremarkableeffectsonseismicriskandcausegreat difficultiesinearthquakeprediction.Hereagaininterdisciplinarityisinorderthroughthemoderngeodetictheories.

Rheologyisoneofthemostefficientandlessdamaging mechanismsusedbynatureforitsevolutiontotheinevitable attractor:theequilibrium.Somemoryisimportantforthe planetarysystemsanddowntoincludewhatisgenerically understoodas economy.

Howevermathematicalmemoryformalismofrheology anditsirreversibility,changedalltheequationsofclassical physics,fromthoseofMaxwelltothoseofNavier-Stokes andofFourierandgeneralmechanics(BaleanuandTrujillo 2008).

Concerninganelasticityallattemptstousetheequipartitionofenergyaresubjecttotherestrictionthattheenergyis subjecttoadecaywithafrequencydependentmechanism. Itwouldbesurprisingtofindthat,forallmaterials,thefrequencydependentdissipationofenergybesuchtodissipate theenergyofeachdegreeoffreedominawaytopreserve theequipartition.Finallymostreciprocitytheoremshadto

Thetimeisripeforasystematic,numericallyquantitative,descriptionofthestateofhealthofregionsbymeans ofappropriateparameterswhichwouldallowcomparisons betweendifferentregionsandatdifferenttimes.Adescriptionessentialfortheestablishmentoftheprioritiesofthe interventions,ofthedetailedandoftheglobalrisk.Itisthe mostappropriatetaskforGeomatics,whichhowevershould havesomesortofinterdisciplinarity.

bechangedbecauseofthesecondprinciple;anexample athandisthatduetoGraffi(1946)whichwasrecently adjourned.

Iwishtoallofusgoodworkingdays,pleasantweather andgoodstayinRoma.

References

AgrawalOP(2002)Solutionsforafractionaldiffusionwaveequation definedinaboundeddomain.NonlinearDynam29(1–4):145–165. doi:10.1023/A:1016539022492

BaleanuD,TrujilloJJ(2008)FractionalEuler-LagrangeandHamilton equationswithinCaputo’sfractionalderivatives:anewperspective. In:3rdIFACworkshoponfractionaldifferentiationanditsapplications,5–7November.Ankara

BaleanuD,GolmankhanehAK,NigmatullinRR (2009)Newtonianlaw withmemory.NonlinearDyn.doi:10.1007/s11071-009-9581-1

BocchioF(1974)Fromdifferentialgeodesytodifferentialgeophysics. GeophysJRAstronSoc39(1):1–10.doi:10.1111/j.1365-246X. 1974.tb05435.x

CaputoM(1979a)Topologyanddetectioncapabilityofseismicnetworks.In:VogelA(ed)Proceedingsoftheinternationalworkshopon monitoringcrustaldynamicsinearthquakezones,Strasbourg1978. FriedrichVieweg&SohnVerlag,Wiesbaden/Braunschweig,FRG

CaputoM(1979b)Moderntechniquesandproblemsinmonitoring horizontalstrain.In:VogelA(ed)Proceedingsoftheinternationalworkshoponmonitoringcrustaldynamicsinearthquake zones,Strasbourg1978.FriedrichVieweg&SohnVerlag,Wiesbaden/Braunschweig,FRG

CaputoM(1979c)2000Yearsofgeodeticandgeophysicalobservations inthePhlegr.FieldsnearNaples.GeophysJRAstronSoc56:319–328

CaputoM(2012a)Reciprocityinelasticmediawithrheology.Meccanica.doi:10.1007/s11012-013-9693-z

CaputoM(2012b)Theconvergenceofeconomicdevelopments.NonlinearDynEconometrics16(2):22.doi:10.1515/1558-3708

CaputoM,CaputoR(2013)Massanomaliesandmomentsofinertia intheoutershellsoftheEarth.Terranova1:1–10.doi:10.1111/terr. 12003

CaputoM,RuggieroV,SuteraA,ZirilliF(2000)Ontheretrievalof watervapourprofilesfromasingleGPSstation.IlNuovoCimento 23(6):611–620

CesaroneF,CaputoM,CamettiC(2004)Memoryformalismin thepassivediffusionacrossabiologicalmembrane.JMembrSci 250:79–84

DelCastilloNegreteD,CarrerasBA,LynchVE(2004)Fractional diffusioninplasmaturbulence.PhysPlasmas11(8):3854–3864

DemariaG(1976)Iteoremidelpuntofissonell’analisienellasintesi economica.Applicationsofthefixedpointtheoremtoeconomy,vol 43.AccademiaNazionaledeiLincei,CentroLinceoInterdisciplinare diScienzeMatematicheeloroapplicazioni,pp11–29

ElShahedM(2003)Afractionalcalculusmodelofsemilunarheart valvevibrations.In:InternationalMathematicasymposium.Imperial College,London

GrafarendEW(1988)Thegeometryoftheearth’ssurfaceandthe correspondingfunctionspaceoftheterrestrialgravitationalfield. Deutsch.Geod.Komm.Bayer.Akad.Wiss.,ReiheB,HeftNr.287, pp76–94

GrafarendE,SansòF(eds)(1985)Optimizationanddesignofgeodetic networks.SpringerVerlag,Berlin

GraffiD(1946)Sulteoremadireciprocitànelladinamicadeicorpielastici.Memoriedell’Accademiadellescienzedell’Istitutodi Bologna 10(IV):103–109(InItalian)

HarrisonJC(1976)Cavityandtopographiceffectsintiltand strainmeasurement.JGeophysRes81(2):319–328.doi:10.1029/ JB081i002p00319

IaffaldanoG,CaputoM,MartinoS(2006)Experimentalandtheoretical memorydiffusionofwaterinsand.HydrolEarthSystSci10:93–100 MainardiF(1996)Fractionalrelaxation-oscillationandfractional diffusion-wavephenomena.ChaosSolitonsFractals7(9):1461–1477 MartinellJJ,DelCastilloNegreteD,RagaAC, Williams DA(2006) Non-localdiffusionandthechemicalstructureofmolecularclouds. MonNotRAstronSoc372:213–225

MilaniN(1961)Sulsignificatointrinsecodell’equazionecuisoddisfano lefamigliediLamé.In:TesidilaureainScienzematematiche, relatoriMarussiA.andCaputoM.,UniversitàdegliStudidiTrieste, AnnoAccademico1960–61(InItalian)

NaberM(2004)TimefractionalSchrödingerequation.JMathPhys 8(45):3339–3352

NeuberH(1937)Kerbespannungslehre.SpringerVerlag,Berlin SansòF,CaputoM(2008)OntheexistenceofLamèorthogonal familiesadaptedtoagivenpotentialfunction.BollGeodesiaScienze AffiniLXVII(2):77–86

SlichterL,MacDonaldGJF,CaputoM,HagerCL(1964)Report ofearthtidesresultsandofothergravityobservationsatUcla, Communicationsdel’ObservatoireRoyaldeBelgique,69,Série Geoph.,VI.MaréesTerrestres,pp124–131

TanniL(1940)Onthecontinentalondulationsofthegeoidasdeterminedbythepresentgravitymaterial.PublIsostInstIntAssocGeod, No.18 WeylH(1983)Symmetry.PrincetonUniversityPress,Princeton

FernandoSansòLaudation

MicheleCaputo

Youmustknowthatinthelastcenturythechairsofthe ItalianUniversity Prof wereloadedwithdustbecausethe Italian Prof wasalwaysaway:chairingmeetings,attending symposia,goingtothecapitalcitytoseeksubsidiesforthe researchofhisgroup,orattendingthefacultymeeting;some, asMarussiandDesio,wentinscientificexpeditionsloaded withcuriosityandofriskfortheirlives,Ineverheardthat theyhadaninsuranceforthis.The Profs wereveryrarelyon theirchair.

Fernandosurvivedallthis.

Inthis,asateacher,Fernandohasbeenverygoodand Iliketoquoteaninterestingresult,whichisnotinhis vita.Itconcernsoneofhisformerstudentswho,witha coupleofcolleagues,foundthatintheGPSisimbedded newtypeofseismograph,inotherwordstheyexploredthe GPSinadifferentfrequencybandandgavelighttoanew seismograph,theycalledit VADASE.Andthecreditforthis isperhaps,inoneofthosemysteriouswaysofnature,tobe addedtothemeritsofFernando.

Andletmeaddthatinthesedayswearegratifiedalso bytheinventionsofanewshoe-boxsizeseismographcalled the TREMINO whichmayperformasthosecostingtensof thousandofdollars.

Theinventionofthe TREMINO (S.Castellaro,M.Mucciarelli,F.Mulargia),andofthe VADASE (G.Colosimo,M. Crespi,A.Mazzoni),letmetellyou,areemblematicofthe evolutioninItalianGeodesyandGeophysics.

Itisaculturalchangeasthatfromlaserrangingforwhich wasneededabigtruck,asthatofContraves,totheGPS,from seismographsneedingroomsizespacetothe TREMINO and ofthe VADASE whichmaystayinshoesizeboxes.

TheoldgenerationofItaliangeodesistsacrossnineteenth andtwentiethcentury,IamreferringtothatofPizzettiSomiglianatheory,wasmostlytheoreticalwithlimited

M.Caputo( ) DepartmentofGeologyandGeophysics,TexasA&MUniversity, CollegeStation,TX,USA

e-mail: mic.caput27@gmail.com

interestedintheindustry.Inthemiddleofthetwentieth therewassomeinterestinthephotogrammetricindustry butapparentlytheItalianindustry missedthebus and disappeared.Thenwehadtherevolutionofthesatellites, theGPS;buttheItaliancontributioninthisfield,althoughof goodandrecognisedquality,couldnotgoveryfar.

Thenewgenerationofthiscenturyinsteadproduces instruments.AndpartofthisisalsoduetoFernando.

Heassuresusthathewillcontinuetobebusy.ButIwarn himfrommyexperience,itmaybebecauseoftheage,it maybebecausetheysaythattheclimateischangingor becauseoftheelectromagneticpollutionandthattheglaciers aremelting,weareboundtobemoreandmorebusy,simply becauseitistheonlychoicewehave.

SimplybecauseIdonotknowwhatelsetodoorIamnot interestedinanythingbutwhatIhopetobeabletodowell orIliketosatisfymycuriosity.Forinstanceaskingwhere arethoseblessedrootswhichkeeptheAlps,wheretheyare nowforourgamesofallsortsandpleasures,orwonderingif gravitonsmaybethecousinsofneutrinos

IfyoubelievethatretirementmeansfreedomIwarnyou since,beginning30yearsago,Iformallyretired3times andtheworseonewaswhenitwasnotformalbutpractical whenallsortsof rasthaus disappeared.Iwarnyoubecause youwillloosetherecreationofthefacultymeetings,the recreationofdrivingtotheofficeandback,therecreation givenbysomecolleaguesmomentarilyfreecometoyour officewarningyouthat itisforfewminutesonly, you willloosetherecreationofstudentsaskingquestions,the recreationofhopingthatonedaywefinallygivesomeorder toallthepapersspreadonthetables,theshelvesonthewalls, thechairsandflooroftheoffice.

Oftentheretiredpersondoesnotneedtogototheoffice andfinallymaystayhometoworkwhereallescapesor recreationsaregonebecauseeventhewife,whorespectshim somuch,doesnotdaretointerrupthisworkanddoesnot allowhimtospreadthepapersalloverthestudio;becauseat thattimetheofficeisa studio tobekeptinrigorousconstant order.

Afterretirementitwillbeendlessworkalmostanobsession,withthefearthatthecuriositywillnothavetimeto unravelallsecretsofnature.

AfterthisjocularperspectiveIcongratulateFernando forhislifeandhisexceptionalandinnovativescientificproduction,forhisdevotiontoscience,forgrowing

andguidingsuchanumerousgroupofverygoodstudents.I alsothankhimforhispurposetostayaroundandworkand Iwishhimlongbusylifeingeodesywithus.

GlobalReferenceSystems:TheoryandOpen Questions

AthanasiosDermanis

Abstract

Inthisreviewpapertheoreticalaspectsofglobalreferencesystemsarecriticallydiscussed inrelationtotheirpracticalimplementationthroughreferenceframes.Theseincludethe problemofthemathematicalmodelingofaspatiotemporalreferencesystemforthedeformingEarth,therelationofgeodeticdiscrete-networkreferencesystemstogeophysicalones fortheEarthmasscontinuum,thecontributionofthevariousgeodeticspacetechniques, theestimationissuesrelatedtothecombinationofthevariousdatatypes,andissues relatingtothecompatibilityofearthrotationrepresentation.Finallyissuesrelatedtofuture developmentoftheInternationalTerrestrialReferenceFramearediscussed,concerning theadditionofnon-linearquasi-periodicterms incoordinatevariationandtheirproper geophysicalinterpretation.

Keywords

Earthrotationrepresentation•Geodeticdatumproblem•Globalgeodeticnetworks• ITRF•Nonlinearstationmotions•Referencesystems

1Introduction

Areferencesystemismerelyamathematicaldevicewithin theframeworkofNewtonianmechanicsthatisconveniently usedforthedescriptionofshapeanditstemporaldeformation.Itconsistsofalocalbasis ! e D h! e 1 ! e 2 ! e 3 i ata particularorigin O andprovidesCartesiancoordinates x D x 1 x 2 x 3 T ofpoints P asthecomponentsoftheirposition vectors ! x D ! OP D ! ex.InEarthrelatedapplicationsa (usuallygeocentric)globalreferencesystemseparatesthe motionofEarthmassesinspaceintothetranslationalmotion oftheorigin,therotationofitsaxesaroundtheorigin (Earthrotation)andtheapparentmotionofEarthmasses withrespecttothereferencesystem(Earthdeformation).

A.Dermanis( )

DepartmentofGeodesyandSurveying,AristotleUniversityof Thessaloniki,UniversityBox503,54124Thessaloniki,Greece e-mail: dermanis@topo.auth.gr

Earthdeformationforcesgeodesytointroduceakinematics spatiotemporalconceptofreferencesystem,definedatevery timeepoch,whichismuchricherthanthestaticspatial conceptofclassicalmechanics.TruesdellandNoll(1965) whogavetheaxiomaticfoundationof“rational”mechanics givesuchalimitedconcept:

::: Thepositionofaneventcanbespecifiedonlyifaframeof reference,orobserver,isgiven.Physically,aframeofreference isasetofobjectswhosemutualdistanceschangecomparatively littleintime,likethewallsofanobservatory,thefixedstars,or thewoodenhorsesonamerry-go-round.

TheFrenchastronomerFelixTisserand(1845–1896) hasrealizedtheneedofaspatiotemporalreferencesystem andintroducedtheconceptofwhatwenowcallTisserand axes(MunkandMacDonald 1960).Inhischoicethe temporalevolutionoftheorientationofthereference systemaxesisdefinedbyminimizingtheapparentmotion ofthepointmasses,quantifiedbytherelativekinetic energy TR D 1 2 R E P xT P x dm D min,whichissecuredby vanishingoftherelativeangularmomentumcomponents hR D R E Œx x dm D 0 (heredotsdenotedifferentiation

N.Sneeuwetal.(eds.), VIIIHotine-MarussiSymposiumonMathematicalGeodesy,InternationalAssociation ofGeodesySymposia142,DOI10.1007/1345_2015_9 9 ©SpringerInternationalPublishingSwitzerland2015

withrespecttotime, Œa denotestheantisymmetric matrixwithaxialvector a, dm isthemasselementand integrationiscarriedoverthewholeEarth).Theorigin ofTisserand’sreferencesystemisthegeocenter G with vanishingcoordinates xG D 1 M R E x dm D 0 (M D Earth mass).

Theproblemofthedefinitionofareferencesystemfor aglobalgeodeticnetworkshowsupintheformulationof a“referenceframe”atermwhichingeodesymeansthe realizationofareferencesystembymeansofthecoordinatefunctions xi (t)ofaselectedsetofnetworkpoints Pi .Coordinatetimeseriesprovidedbyfourfundamental spacetechniquesVLBI,SLR,GPSandDORISareutilized bytheInternationalEarthRotationandReferenceSystems Service(IERS)inordertorealizetheofficialInternational TerrestrialReferenceFrame(ITRF)(Altamimietal. 2002, 2004, 2007, 2011)andprovideEarthOrientationParameters (EOPs)describingtherotationoftheEarth(Bizouardand Gambis 2009).Althoughtheoperationalproceduresforthe ITRFformulationarenowamatterofroutine,therearestill somerecentadvancesaswellasopenproblemsinthetheory ofreferencesystemsthatmaycontributetotheimprovement oftheexistingtechniques.Thepresentworkisashortreview ofrelevantresultsandadiscussionofremainingproblemsfor futureinvestigations.

2AReferenceSystemModel forGeodeticNetworks

Thechoiceofreferencesystemfora N -pointthreedimensionalgeodeticnetworkassignstoitavector x D xT i T of3N coordinates,whichrepresentsa pointin R3N .Thesecoordinatesarenottheonlyones thatdescribethenetworkshape.Anyarbitraryrigid transformation xi D R .™ / xi C d (™ beingtherotationand d thetranslationparameters)providesavector Q x 2 R3N thatdescribesthesamenetworkshape.Thesubmanifold Mx D nx ˇ ˇ ˇ xi D R .™ / xi C d; 8™ ; do R3N generatedas ™ and d takeallpermissiblevalues,isthesetofallpoints correspondingtothesamenetworkshape.Thus Mx isthe shapemanifoldgeneratedby x (Dermanis 2000).Shape manifoldsarenaturallydisjointandthrougheachpoint in R3N passesonlyonemanifold.Hencetheyconstitutea fibering ofanopensubsetof R3N .Thesixparameters ™ , d mayserveasasetofcoordinatesonthesix-dimensional shapemanifold.Foradeformablenetwork, x(t)isacurve in R3N whichrepresentsthecontinuoustimesequenceof shapemanifolds Mx(t) correspondingtotheshapesofthe networkatvarioustimeepochs t.Wemaydefineareference systemasa section oftheshapemanifoldfibering,i.e.acurve

intersectingeachmanifoldatonepoint.Eachsuchcurve x.t/ canbegeneratedfromtheoriginalcurve x(t)bymeansofsix functions ™ (t), d(t)through xi .t/ D R .™ .t// xi .t/ C d.t/. Differentoptimalreferencesystemsarepossible,depending onthearbitrarychoiceof Q x .t0 /.Theoptimalchoice Q x.t/, istheshortestgeodesicthrough x .t0 / connectingtheinitial manifold Mx.t0 / withthefinalone Mx.tF / foratimeinterval ofinterest t 2 Œt0 ;tF .Suchshortestgeodesicsareknownto beperpendiculartoboth Mx.t0 / and Mx.tF / .Sincethechoice ofinitialandfinalepochisratherarbitrary, Q x.t/ shouldbe perpendiculartoanyofthemanifolds Mx.t/ thatitcrosses. Thereforethetangentvector x shouldbeperpendicularto thetangenthyperplaneof M x.t/ at x.t/,whichisspannedby thecoordinatebasevectors @ Q x.t/=@q i , qi beentheelements of ™ (t), d(t),i.e.,bythecolumnsofthematrix Œ@™ Q x @d Q x Thereforethedifferentialequationsdefiningtheoptimal referencesystem x.t/ onthebasisofagivenarbitrary referencesystem x(t)are Œ@™ x @d x T x D 0.With x(t)chosen tobebarycentric . 1 N †i xi .t/ D 0/ wearriveatthedifferential equations

where W D Œw1 w2 w3 with wk beingtheaxialvectors oftheantisymmetricmatrices Œwk D @ k RT R, h D Pi Œxi P xi isthediscreterelativeangularmomentumin theinitialreferencesystemand C D Pi Œxi 2 isthe discreteinertiamatrix.Thesolutiontotheaboveequations isnotuniquebutdependsonintegrationconstants ™ (t0 ), d(t0 )or Q x .t0 /.Anytwosolutionsgeneratecorresponding referencesystemcurves x.t/, x0 .t/ whichare“parallel”in thesensethattheyareconnectedbyatime-independentrigid transformation x0 i .t/ D Qxi .t/ C c, .Q; c/ D constant.

Passingfroman N -pointdiscretenetworktothecontinuousEarthbody,callsforthereplacementof R3N withthe infinite-dimensionalspaceofthecoordinatesofallmaterial pointsoftheEarth,buttheintricaciesofacorresponding rigorousmathematicalmodelarefarfromtrivial.Inanycase thefiberingbysix-dimensionalshapemanifoldshavingthe transformationparameters ™ , d ascoordinatesispreserved.

Itisinterestingtoseehowagivengeocentric(i.e. barycentricforallEarthpoints)referencesystem x(t) canbetransformedbyapoint-wiserigidtransformation Q x.t/ D R .™ .t// x.t/ C d.t/ intoageocentricTisserand referencesystem Q x.t/,satisfying 1 M R E Q x dm D 0 and Q h D R E Œ Q x Q x dm D 0.Carryingoutthenecessarycomputations wearriveat d D 0 (whichisoneofthesolutionsof P d D 0)while ™ satisfiesagainthedifferentialequation W .™ / P ™ D C 1 h,withtheonlydifferencethattherelative angularmomentumandinertiamatrixaregiveninthiscase by h D R E Œx P x dm and C D R E Œx 2 dm

3DefinitionoftheReferenceSystem intheITRFFormulation

Thestaticversionoftheadoptionofareferencesystemfor non-deformingnetworksisanoldgeodeticproblemmostly knownasthe“geodeticdatumproblem”thatemergedin thesocalled“freenetworks”,i.e.localnetworkswhich donotinherittheirreferencesystemfromapre-existing higherordernetwork.Ithasalsogivenrisetogeodeticcontributionstothestatisticallinearestimationtheorywithout fullrank,inrelationtothelinear(ized)model b D Ax C v, v 0; 2 P 1 with n observations b, m unknowns x and rankA D r<m.Therankdeficiencyandthecorrespondinginfinityofleastsquares vT Pv D min solutionsfor theunknownparametersisduetotheuseofcoordinates asunknowns,whileobservationscanonlydeterminethe geometricfigureofthenetwork.Howeverallleast-squares solutionsleadtothesamevaluesforobservablequantitiesas wellasallthefunctionsoftheobservableswhicharestatisticallycharacterizedasestimablequantities.Auniquesolution isobtainedbyposingadditionalconstraints CT x D d onthe unknowns,whichareminimali.e.theyresolvethecoordinate indeterminacywithoutaffectingtheestimatedgeometricfigureofthenetwork.Thecoordinateestimatesandtheirsingularcovariancematricesmerelyserveasadepositoryofinformationforthefurthercomputationofestimatesofestimable quantitiesandtheircovariancematrices.Theroleofminimal constraintsisthatofassigninganarbitraryreferencesystem sothatcoordinatescanbecomputed.Particularlypopular havebeenthesocalledinnerconstraints ET x D 0,which satisfy xT x D minamongallleastsquaressolutions.The matrix E resultsfromthecoordinatetransformation x ! x D T.x; p/ underachangeofthereferencesystem,where p aretransformationparameterssuchthat T.x; 0/ D x.The linearizedformofthetransformation Q x D x C Ep allowsthe determinationofthedesiredmatrix E.Sometimesthetotal innerconstraints ET x D ET 1 ET 2 x1 x2 D 0 arereplacedby partialconstraints ET 1 x1 D 0 involvingonlyasubset x1 ofthe parametersandsatisfying xT 1 x1 D min,instead.

FordeformableITRFnetworkthechoiceofreference systemisdominatedbyoneextension,thatofdefining itstemporalevolutionandtworestrictions.Thefirst istherestrictiontocoordinatetransformations“close totheidentity”i.e.withverysmalltransformation parameters p(t),whichallowthereplacementof x.t/ D .1 C s.t// R .™ .t// x.t/ C d.t/,withthelinearapproximation Q x.t/ x.t/ C s.t/x.t/ C Œx.t/ ™ .t/ C d.t/ realized by R .™ / I Œ™ andneglectionofsecondandhigher orderterms.Moresevereisthesecondrestrictionto

transformationswhichpreservethelinear-in-timeformof theITRFcoordinatemodel xi D xi0 C .t t0 / vi xi0 C t vi ,where xi0 , vi aretheinitialcoordinatesandconstant velocitiesofthenetworkpoint Pi .Thisnecessitatestheuse oftransformationparametersthatarealsolinearintime,i.e., s.t/ D s0 C t P s; ™

whicheffectivelyrestrictsthetransformationparametersto the14parameterset p

.Theuseof minimalorinnerconstraintsontheparameters x,whichdue tothelinearizationarecorrectionstoapproximatevalues oftheunknowns,havethedisadvantagethattheydepend onthechoiceoftheapproximatevalues.Anevenmore seriousdisadvantageofsuch“algebraic”constraintsisthat theyhavenoclearphysicalmeaningandtheydonotleadtoa choiceofreferencesystemwhichis“optimal”inaphysically meaningfulway.Adifferenttypeofphysicallymeaningful kinematicconstraintshavebeenproposedbyAltamimiand Dermanis(2009),whichminimizetheapparentmotionof networkpointswithrespecttothereferencesystem.They arebasedonadiscreteversionofTisserand’sideaswhere networkpointsaretreatedasmasspointsofunitmass.The mainideaistominimizethenetwork’sdiscreterelative kineticenergy T.t/ D 1 2 Pi xT i .t/xi .t/ ateveryepoch t orequivalentlytonullifytherelativeangularmomentum h.t/ D Pi Œxi .t/ xi .t/ D 0,arelationwhichestablishesthe temporalevolutionoftheorientationofthereferencesystem. Theoriginofthesystemisdefinedbysettingconstantthe coordinatesofitsbarycenter xB .t/ D 1 N Pi xi .t/ D xB .t0 /, aparticularchoicebeing xB .t0 / D 0 (barycentricsystem). Scaleistakencarebysettingconstantthemeanquadratic scaleofthenetwork S.t/ D Q.t/1=2 definedby Q.t/ D 1 N †i Œxi .t/ xB .t/ T Œxi .t/ xB .t/ D Q.t0 / D const AppliedtotheITRFmodel xi D xi0 C t vi thekinematic constraintsbecome

where x0 D 1 N †i xi0 and v D 1 N †i vi .Operationalexpressionsintermsofcorrections ı xi0 D xi0 xap i0 , ı vi D vi vap i toapproximatevalues xap i0 , vap i oftheinitialcoordinates andvelocitiescanbefoundinAltamimiandDermanis (2009).

Kinematicconstraintsdefineonlytheevolutionofthe referencesystemwithrespecttoorientation(2),origin(4) andscale(6).Initialepochconstraintsareeithermissing (fororientation)ordependonthearbitraryconstants xB (t0 ) fororigin(3)and Q(t0 )forscale(5).Thearbitrarychoice oftheinitialepochorientation,originandscaleleadsto differentbutequivalent“parallel”referencesystems,i.e., realizedbycoordinatefunctions xi (t), Q xi .t/,whichareat anyepochrelatedbyatime-independentsimilaritytransformation xi .t/ D .1 C / Qxi .t/ C t withconstant , Q and t.ThelackofinitialorientationisinherentinTisserand referencesystems;initialoriginmayresultbyselectinga barycentric xB .t/ D xB .t0 / D 0 orageocentriconesince thegeocenterisanadditionalnetworkpointinSLRobservations.Thepresenceofthescaleparameterinthecoordinate transformationsdoesnotactuallycorrespondtoadeficiency butrathertothefactthatdifferentspacetechniqueshave adifferentunitoflength.Thisistheoreticallyduetothe useofdifferentunitsoftimerealizedbydifferentsets ofclocks,whileadditionaleffectscomefromsystematic errors(troposphericcorrections,phasecentercorrectionsfor satelliteandgroundantennas, etc.).Inthepast(ITRF2005) ITRFscalewasbasedonVLBIonly,currently(ITRF2008) aweightedcombinationofVLBIandSLRisused,with expectedfuturecontributionsfromGNSS.

4ITRFFormulation:TheOne-Step andtheTwo-StepApproach

Therearetwobasicapproach esfortheformulationofthe ITRFtheone-step(Angermannetal. 2004;Rothacheretal. 2011;Seitzetal. 2012)andthetwo-stepapproach(Altamimi etal. 2002, 2004, 2007, 2011).Theybothuseaspseudoobservationscoordinatesseriesestimatesfromspacetechniques

withweightmatrix Pc areutilized,whichconnectstationsof differenttechniquesatco-locationpoints.

Theone-stepapproachproceedstotheformulationofthe normalequationsforalldata

(V D VLBI,S D SLR,G D GPS,D D DORIS)usingthe standardassumptionsoftheGauss–Markovmodelforzero meanerrorswithknowncovariancematricesuptoascalar factor.Sincethenetworksofdifferenttechniquesaredistinct, additionalobservations

thereisaninherentrankdeficiencydueto theirinabilitytodetermineareferencesystemtowhichthe unknowncoordinatesrefer.Thisisexpressedasadeficiency inthe(column)rankofthedesignmatrix AT .If Q xT D xT C ET pT istheresultofacoordinatetransformationwithtransformationparameters pT ,duetoachangeofthereference system,thenboth xT and xT yieldthesamevalueforthe invariantobservables yT D AT xT D AT xT .Thismeansthat AT ET D 0 andconsequently

Nomatterhowtheone-stepapproachisoperationallyrealized,itcanbeshownforthesakeofcomparisonwiththe two-stepapproachtobeequivalenttoamodifiedtwo-part approach(Dermanis 2011).Thefirstpartisidenticalto thefirststepofthetwo-stepapproachbutitisalsoused intheone-stepapproachforpreprocessing(identification ofoutliersanddiscontinuities).Itinvolvestheformulationoftheseparatenormalequationsforeachtechnique

and estimatecomputationusingseparateminimalconstraints. Thesecondpartshouldreplacethesecondstepofthetwostepapproachinordertosecu reidenticalresultswiththe straightforwardone-stepapproach.Itinvolvesaleastsquares adjustmentofthefollowingsetofuncorrelatedobservation equations

withcorrespondingweightmatrices: NV , NS , NG , ND and Pc .

Thecorrespondingnormalequations 2 6 6 6 4

D 2 6 6 4

arereadilyseentobeidenticaltothoseinasinglestep (Eq.(9))sincetheseparateestimatessatisfy

Inthetwo-stepapproachthefirstone(stackingpertechnique)isidenticaltotheabovefirstpartoftheone-step approach.Inthesecondstephoweveritisrecognizedthat eachoftheseparateestimates b x s V , b x s S , b x s G , b x s D refersto separatereferencesystemswhichalsodifferfromthefinal ITRFreferencesystemofthesoughtestimates

G , b xD .Forthisreasontransformationparametersareincluded inthemodelwhichbecomes b D 2

Howeveraccordingto(10)itholdsthat

Thecorrespondingnormalequationstakeinthiscasethe extendedform

,where

Consequentlythenormalequationsdegenerateinto

i.e.intothenormalequations Nb x D u ofthesecondpartof theone-stepapproachand 0b p D 0,whichdoesnotallowthe determinationofthetransformationparameterestimate b p.In conclusion,thesecondstepofthetwo-stepapproachneeds tobereplacedbyapropercombinationatsolutionlevel(as opposedtothecombinationatthenormalequationlevelof theone-stepapproach),inordertosecureidenticalresults. Foracomparisonofthesetwoapproachesascurrently appliedseeAppendixBofAngermannetal.(2004).

5RelatingtheReferenceSystem ofaGlobalGeodeticNetwork toaTisserandReferenceSystem

Fromageophysicalpointofviewan“optimal”reference systemforaglobalnetworkfallsshortinrepresentingthe deformationoftheEarth,orjustthelithosphere,evenwhen establishedbykinematicconstraintswhichminimizethe apparentmotionofstationpoints.Forthispurposethe motionofthemassesofthelithospheremustbeapproximatelyinferredwiththehelpofageophysicalmodel. Suchawidelyacceptedmodelisthatofrotatingtectonic plateswherewithinplatedeformationsplayaminorrole andmaybeignoredinafirstapproximation.Ifoneach plateorsubplate PK liesasubnetwork DK oftheglobal geocentricnetwork,informationofthecoordinatevariation xi (t), i 2 DK canbeusedtodeducetherotationvector ¨K ofeachplateanditscontribution Q hPK totherelativeangular momentumofthelithosphere Q h D PK Q hPK .Setting Q h D 0 allowsthedeterminationoftherotationvector ¨ which transformstheoriginalgeocentricreferencesystemintoa Tisserandgeocentriconeforthelithosphere.Suchasystem isappropriateforcomparingitsobservedrotationwiththat predictedbytheoriesofEarthrotation.Thecomputation algorithmconsistsofthefollowing:Foreachsubnetwork DK thediscretematrixofinertia CDK D Pi 2DK Œxi 2 and therelativeangularmomentum hDK D Pi 2DK Œxi P xi with

respecttothegeocenterareusedtocomputetherotation vector ¨K D C 1 DK hDK ofthecorrespondingplate PK .The contributionofeachplate PK tothematrixofinertiaofthe lithosphereiscomputedas CPK D R PK Œx 2 dm inorder tocomputetherotationvector ¨ D XK CPK 1 XK CPK ¨K (18)

fromtheoriginalreferencesystem fxi g totheTisserand referencesystemofthelithosphere fxi g.SolvingthegeneralizedEulerdifferentialequations Œ¨ D RT P R D P RT R,the parameters ™ oftherotationmatrix R(™)aredeterminedand finallythecoordinatesoftheglobalnetworkareconverted accordingto Q xi D Rxi .ThespecificTisserandsystem,outof infinite“parallel”oneshavingthesametemporalevolution, dependsonthechoseninitialvalues ™ (t0 ).Itisalsopossible touseamodelwitharbitraryrigidplatemotioninsteadof simplerotationoreventoincorporateinternalplatedeformationsdeducedfromthestationmotionofthecorresponding subnetwork.Itmustbenotedthatthecomputationofthe inertiamatrices CPK requiresknowledgeofthegeometric boundariesofeachplateorsubplateaswellasofitsinternal densitydistribution.

6CompatibilityofEarthRotation Representation

TheofficialIERSrepresentation(PetitandLuzum 2010) oftherotationmatrix R convertingcelestialtoterrestrial coordinates(xT D RxC )hastheform R D WDQ D R3 . F/ R2 . g/ R3 .F

. / R3 . E s/ R2 .d/R3 .E/ (19) where Q D R3 . E s/ R2 .d/R3 .E/ D R3 . s/ G .X;Y/ istheprecession–nutationmatrix, D D R3 . / isthediurnalrotationmatrix, W D R3 . F/ R2 . g/ R3 .F C s 0 / D GT . ; / R3 .s 0 / R1 . yP / R2 . xP / R3 .s 0 / isthepolar motionmatrix,while xP and yP arethecoordinatesofthepole.Twointermediatereferencesystemsarethe celestialintermediatesystem ! e IC D ! e C QT andtheintermediateterrestrialone ! e IT D ! e T W,whicharerelatedby ! e IT D ! e IC DT D ! e IC R3 . / havingacommon3rdaxis along ! p D ! e IC 3 D ! e IT 3 D ! e C pC D ! e T pT theunitvectorinthedirectionofthecelestialintermediatepole(CIP), withcelestialcomponents pC D ŒXYZ T andterrestrial ones pT D Œ T .Theoriginal7parameters(functionsof time)arereducedto5bymeansofthe2NRO(NonRotating Origin)conditions(Capitaineetal. 1986) s D s.d;E/ D s.X;Y/ and s 0 D s 0 .g;F/ D s. ; / D s.xP ; yP / which

definethedirectionsoftheTIO(TerrestrialIntermediateOrigin) ! e IT 1 andtheCIO(CelestialIntermediateOrigin) ! e IC 1 . ForaprecisedefinitionoftheNROconditionsweneedthe conceptoftherelativerotationvector ! ! A!B D ! e A ¨A D ! e B ¨B betweentworeferencesystemsconnectedby ! e B D ! e A RT A!B havingcomponentsdeterminedfromthegeneralizedEulerkinematicequations Œ¨A D RA!B P RT A!B and ¨B D RA!B ¨A .TheNROconditionsaretheperpendicularityconditions ! ! T !IT ? ! e IT 3 D ! p and ! ! C !IC ? ! e IC 3 D ! p ,orintermsofthecomponentsintheintermediatesystems .! T !IT /3IT D 0, .! C !IC /3IC D 0.They producethedifferentialequations

Anyorthogonalrotationmatrix R dependsononlythree parameters,sothattheoriginalsevenparameters(E, d, s, , F, g, s0 )or(X, Y, s, , xP , yP , s0 )must fulfillfourconditions.Suchconditionscannotbeestablished fortheCIP ! p D ! e IC 3 D ! e IT 3 ,becausethelatterlacksa clearandrigorousdefinitioneitherphysicalormathematical. Roughlyspeakingitisasmoothedversionofthedirection ! n D ! 1! ! (!

)oftherotationvectoroftheEarth ! ! (! ! C !T ),where“unobserved”highfrequenciesof ! ! predictedbytheoryhavebeenremovedfromprecession–nutationandincludedinpolarmotionsothatthesame rotationmatrixismaintained.TheCIPisanevolutionof theCEPwhichreplacedtheinstantaneousrotationvector ! ! afterAtkinson(1973)andothersremarkedthathigherthan diurnalfrequenciescannotbe observedasaconsequenceof therelatedtemporalresolutionofthethenavailableobservations.Althoughtheideaofreplacinginamodelaconcept withitssmoothedversionduetoobservationalresolution problemsmayseemstrangetoanoutsider,astronomers developedthefirmbeliefthattheCEPis“observable”while thedirectionofinstantaneousrotationvectorisnot.Nevertheless,today’sobservationshavehigherthandiurnalresolutionandwillcertainlyimproveinthefuture(seee.g.Hefty etal. 2000;Artzetal. 2012)andevenastronomersrecognize thepossibilityofobservingtheinstantaneousrotationvector (seee.g.,Bolotinetal. 1997).Wewillthereforeseekcompatibilityconditionsforthecasewheretherotationmatrix hasasimilartotheIERSrepresentation,withthethirdaxes arealignedtotherotationvectordirection ! e IC 3 D ! e IT 3 D ! n D ! 1! ! insteadoftheCIP ! p .Thisallowstheuseof therigorousdefinitionoftherotationvector ! ! D ! e T ¨T ! e T ¨ throughthegeneralizedkinematicEulerequations Œ¨ D R P RT .Assuringthatthediurnalrotation D D R3 . / representstherotationoftheterrestrialsystem,bothwith

respecttothedirectionofitsrotation ! e IC 3 D ! e IT 3 D ! n and itsrotationrate ! D P yieldsthethreeconditions ! ! D P! e IT 3 D P! e IC 3 (21)

orintermsofcomponents ¨IT D P i3 D ¨IC .Theexplicit computationofeitherofthelastrelationsisrather complicatedbutsimpleinprinciple.With R D WDQ in Œ¨ D R P RT itfollowsthat ¨

¨D C ¨

where

W D W P WT .Setting

¨

.Finallywearriveatthethree conditions

beappliedto inordertoobtaintheangularvelocityofEarth rotation ! andtherelatedcorrectUniversalTime(UT1).

Nomatterwhatthechosendirectionof ! e IT 3 D ! e IC 3 (axisofdiurnalrotation)theresultingrotationmatrix R impliesamathematicallycompatibleinstantaneousrotation axis ! ! whichcanbecomputedandcomparedtotheCIP direction ! p .Theseparationbetween ! n D

ˇ

! !

ˇ ˇ 1! ! and ! p arethesocalledOppolzerterms(oratleastonethe possibledefinitions,seeMoritzandMueller 1987).Dermanis andTsoulis(2007)havecomputedthesedifferencesbytwo independentmethodsandfoundthatalthoughtheyhavethe samespectralcharacteristicsastheOppolzerterms,their amplitudesaretoolarge,risingtotheorderoftensofmeters ontheEarthsurface.

7OpenIssuesforFurtherResearch

where R(˛ )denotesrotationintheplane.Insteadofthefour conditionsrequiredtoreducethesevenrotationparameters tothethree,wehaveonlythree,whichfixtheorientationof ! n andtherotationrate ! D ,buttheyleaveundefinedthe positionsof ! e IT 1 (TIO)and ! e IC 1 (CIO),regulatedbythe valuesof s and s0 .Ifbothsidesof(23)aresetequalto f , where f isanarbitraryfunction,then s and s0 aredetermined from s 0 D F.cos g 1/ C P f , s D E.cos d 1/ C P f and TIOandCIOarebothdisplacedbythesameamount f,leavingtheangle between ! e IT 1 and ! e IC 1 unaltered.Toresolve thisindeterminacywemustresorttotheNROconditions whichcorrespondtotheparticularchoice f D 0!Thus(23) splitsintothetwoNROconditions

s 0 F cos g C F D 0; s E cos d C E D 0; (24) whichtogetherwiththetwoconditionsin(22)reducethe sevenparameterstotherequiredthreeindependentones. Theessenceofconditions(22)isthatwhenprecession–nutation(E, d )anddiurnalrotation areknownthenpolar motion(F, g)isuniquelydetermined!Andtheotherway aroundwhenpolarmotionanddiurnalrotationareknown precession–nutationisuniquelydetermined!Theaboveconditionsshouldallbesatisfiedtoassurethealignmentof ! e IT 3 D ! e IC 3 with ! ! andtheproperrate ! D .Thecondition(23)ortheNROconditions(24)alonedonotguarantee that ! D .ForthecurrentIERSrepresentationwith ! e IT 3 D ! e IC 3 alignedtotheCIP ! ¤ andspecificcorrectionsmust

Thereareofcoursemanyopenproblemswithintheanalysis ofdatainthevariousspacetechniquesdeservingseparate reviews,herehoweverwewillconcentratetotheoretical issuesrelatingtotheexploitationofdatacomingfromthese techniques.

CurrentapproachestotheformulationoftheITRFare basedonthefalseassumptionofzeromeanrandomerrors andknowncovariancematrices.Ananalysisandcomparison isneededontheeffectofbothbiasessuchasquasi-periodic termsandofincorrectcovariances,which,fortheGPScase atleast,areknowntobetoooptimistic.

Anotherissueofgreatpracticalimportanceistheoptimal mergingoflocalorregionalnetworkstotheITRF.This istheoldnetworkdensificationproblem,wherethemain ruleisthattheITRFcoordinatesandvelocitiescannotbe altered.Someresearchers(seee.g.Altamimi 2003;Kotsakis 2013)suggesttheuseofminimalconstraintsbasedonITRF parameters.Thequestioniswhichminimalconstraintstouse inordertobestreferencelocalnetworkstothe“official” ITRFreferencesystem.Ananswertothisproblemhasbeen givenbythegeneralizedinnerconstraintsofKotsakis(2013) whereheminimizesthetraceof thecovariancematrixofthe localnetworkwhentheeffectoftheuncertaintyintheITRF parametersusedintheconstraintsistakenintoaccount.This howeverisonlyoptimalityinappearance,whileoptimality oftheconnectiontotheITRFreferencesystemisrather desired.Inanycasemergingthroughminimalconstraintshas animportantdisadvantage:highqualityITRFinformation ontheshapeofthecommonsubnetworkanditstemporal variationiscompletelyignored.

TheexaminationoftheresidualsofITRFcoordinates afterthefittingofthelinear-in-timemodeldemonstrates theexistenceofmostlyannualandalsosemiannualsignals,