Table Of Content

3.01 Geodesy: An Introduction and Overview TAHerring,MassachusettsInstituteofTechnology,Cambridge,MA,USA ã2015ElsevierB.V.Allrightsreserved. 3.01.1 Introduction 1 3.01.2 CoordinateSystems 1 3.01.3 GeodeticMethods 3 3.01.3.1 Ground-BasedPositioningSystems 3 3.01.3.2 SatelliteSystems 6 3.01.3.3 Gravimeters 6 3.01.4 ErrorSources,Signals,andNoise 8 3.01.5 Conclusions 8 References 8 3.01.1 Introduction errors in measurements, signals, and noise. In this overview, weexaminethesethreethemes. Modern geodesyasdiscussed inthisvolumestarted withthe developmentofdistancemeasurementusingpropagatingelec- tromagneticsignalsandthelaunchofEarth-orbitingsatellites. 3.01.2 CoordinateSystems Withthesedevelopments,space-basedgeodesyallowedglobal measurements of positions, changes in the rotation of the Acommonaspectofgeodesythatpermeatestheliteratureof Earth,andtheEarth’sgravityfield.Thesethreeareas,position- thesubjectiscoordinatesystemdefinition.Withthedevelop- ing,Earthrotation,andgravityfield,areconsideredthethree ment of space-based methods that allow global measure- pillars of geodesy. The accuracy of current measurement sys- ments to be made, the subject has in some sense been temsallows time variationstobeobserved inallthree areas. greatly simplified in recent years while at the same time Also,thecomplexityofproblemsissuchthateachofthepillars complicatedbytheincreasedaccuracyneeds.Thesimplifica- interacts with each other and with many other branches of tion comes from being able to use a Cartesian coordinate Earth science. This interaction is most apparent in the role systemwithanoriginatthecenterofmassoftheEarthand that water plays in modern geodetic measurements. Every axesalignedinawell-definedmannertotheoutersurfaceof chapterinthisvolumementionstheroleofwater.Itiscritical theEarth.Thecomplicationsarisefromneedingacoordinate because it can move rapidly and over large distances; it can systemdefinitionthataccountsfordeformationsfromplate exist in all three phases, gas, fluid, and solid; and modern tectonics, tides, tectonic events, and other time-variable geodeticmethodsareaccurateenoughthattheirmeasurements deformations.Themeasurementandunderstandingofthese aresensitivetoitseffects.Initsvaporform,itsrefractiveprop- deformations are themes that are common to many of the ertiesdelaymicrowavesignalspropagatingthroughtheEarth’s chaptersinthisvolume. atmosphere.Forgeodeticpositioning,thisisanoisesourcebut Historically,coordinatesystemsingeodesyaredividedinto itisasignalformeteorologicapplications.Asaliquid,itforms two parts: horizontal coordinates such as latitude and oceansthataffectboththetidalsignalandtherotationofthe longitude and a vertical coordinate called height. These two Earth.Alsoinliquidform,itsmasschangesthegravityfieldas systemsarefundamentallydifferentinthatthehorizontalone itmovesthroughthehydrologiccycle.Insolidform,ithasa is geometric and based on the direction of the normal to an gravitationalanddeformationalsignalthatchangesifmelting ellipsoidalbodythatrepresentstheaverageshapeoftheEarth. the ice unloads the surface of the Earth. The interactions Theheightsystem,calledorthometricheight,isbasedonthe betweenthepillarsincludetheelasticloadingeffectsofchang- gravitational potential field of the Earth. In this system, sur- ingmassloadsthatcanbeseeninthegravityfieldandinthe facesofconstantheightareequipotentialsurfaces,andhence, positions ofground stations. The movement ofwater toand fluid will not flow along these surfaces. One special height from the oceans can be seen with altimeter satellites whose surface is called the geoid and is associated with the surface orbital information is derived from measurements from representing mean sea level. In Chapters 3.02 and 3.05, the groundstationswhosepositionsareaffectedbythechanging subtleproblemswiththesedefinitionsfromdefininganequi- massload.Inmodern,time-dependentgeodeticdataanalysis, potentialsurfacetothemeaningofmeansealevelinanocean these interactions need to be accounted for. The common withcurrentsarediscussed.Inconventionalgeodeticsystems, interfacebetweenthegeodeticmethodsisthecoordinatesys- thereisablendofpotential-basedandgeometricsystems.The temsandreferenceframeusedtoanalyzedata. determinationofthegeodeticlatitudeandlongitudeiscompli- Coordinate systems and the associated reference frames catedbymeasurementsbeingmadeintheEarth’sgravityfield. formacorethemeinthechaptersinthisvolume.Twoother Specificallywiththedevelopmentofelectronicdistancemea- unifyingthemesarethemeasurementsystemsofgeodesythat surements,thedistancemeasurementitselfdoesnotdependon are used again throughout the volume and the interplay of the gravity field (except for small relativistic effects), but the TreatiseonGeophysics,SecondEdition http://dx.doi.org/10.1016/B978-0-444-53802-4.00055-5 1 2 Geodesy:AnIntroductionandOverview projectionofthedirectmeasurementtoahorizontalmeasure- where f is the flattening and the centrifugal component of mentdoesdependontheslopeofthemeasuredline.Theangle acceleration from the rotation of the Earth (rotation rate o) betweenmeasureddirectionandlocalvertical(whichdepends hasbeenincluded.Theratiooftheequatorialgravitationalto onthelocaldirectionofgravity)canbeeasilymeasuredbutis rotationalforceisgivenbym.Fromeqns[1]to[3],weseethe strictly not the correct measurement. The angle to the local first-orderrelationshipsbetweenthegravityfieldoftheEarth, normal to the ellipsoid is needed. The difference between themomentsofinertiaoftheEarth,andtheshapeoftheEarth. thesedirectionsiscalledthe‘deflectionofthevertical’andas Withmodernspace-basedmeasurements,allofthesequanti- discussed inJekelicanbe determined frommeasurements of tiescanbemeasuredwithgreatprecision.Chapter3.02details gravityandsolvingtheappropriateboundaryvalueproblem. thetheoryofgravitationalpotential,Chapters3.09and3.10 Deflection of vertical is also the difference between geodetic examine the rotational consequences of moments of inertia, andastronomiclatitudeandlongitude. and Chapter 3.05 examines the detailed shape of the geoid Theblendofpotential-basedandgeometry-basedsystems basedonaltimetermeasurementsfromspace. in geodetic coordinate systems is a consequence of the With the development of space-based geodetic systems, it methodsavailableformakingmeasurements.Thelocalnature becamepossibletodefineapurelygeometricglobalcoordinate of these measurements resulted in large differences among system.ThesenewsystemsareCartesianandhavetheirorigin systemsadoptedbydifferentcountries.Thereferenceellipsoi- approximatelyatthecenterofmassoftheEarth.Theaxesare dalshapefortheEarthcouldhavedifferencesinthesemimajor alignedapproximatelyalongtherotationaxisandtheGreen- axis of over a kilometer (e.g., Clarke 1866 6378206.0m and wich meridian. The alignments here are only approximate Everest6377276.0m(Smith,1996)),anditwasnotuncom- because,withtheaccuracyofmodernmeasurements,thecen- monatboundariesofcountriestohavecoordinatedifferences terofmassandrotationaxismovewithrespectthefigureofthe ofseveralhundredmeters.Theadventofspace-basedmeasure- Earth.Therotationaxismotionsarelarge,oforder10movera mentsallowsthedeterminationofabest-fitellipsoidalshape year, and have been known about for over a century. The fortheEarthbasedonglobaldata.Therearesimplerelation- center-of-mass movements are much smaller, of order shipsbetweenthegravityoftheEarth,expressedinspherical 1–2cm, and have been measured for the past few decades. harmonics,themomentsofinertiaoftheEarth,andanappro- Even now, center-of-mass variations are not fully accounted priateflatteningfortheEarth.Afterthecentralforceterminthe for in modern coordinate systems. The current international gravity field, the flattening term is the largest, being at least terrestrial reference frames (ITRF) (Altamini, 2012; Altamini 1000timeslargerthananyotherterminthefield.Theleading etal.,2002,2011)haveanoriginatthecenterofthefigureof termsinthegravitationalpotentialfield,V,oftheEarthcanbe theEarth.MassmovementsonthesurfaceoftheEarth,mainly writteninsphericalharmonicsas(e.g.,Stacey,1992) atmosphericandhydrospheric,resultinthemovementofthe centerofmassrelativetothecenteroffigurethatneedstobe GM a a 2 V r,y J P J P cosy J P cosy [1] accounted for in accurate representations of the gravity field. ð Þ¼(cid:2) r 0 0(cid:2) 1r 1ð Þ(cid:2) 2 r 2ð Þ (cid:4) (cid:1) (cid:3) (cid:5) When center-of-figure reference frames are used, the gravity whereGisthegravitationalconstant;Mistheadoptedmassof fieldsintheseframesshouldhavetime-dependentfirst-degree theEarth;yandrarecolatitudeandradialdistancestothepoint spherical harmonic terms included. Currently, these terms wherethepotentialisdetermined;aistheequatorialradiusof need to be observationally determined and the routine pro- theEarth;J ,J ,andJ arethecoefficientsofthegravityfield;and ductionofsuchvaluesisnotyetavailable.Chapter3.08dis- 0 1 2 P ,P ,andP areLegendrefunctions.Whentheadoptedmassof cusses in detail the measurement of time-variable gravity. To 0 1 2 theEarthmatchesthatoftheEarth,theJ P termis1,andifthe furthercomplicateaccountingforcenter-of-massmovements, 0 0 center of mass of the Earth corresponds to the origin of the the mass movements that cause the center of mass to move, coordinate system used to measure latitude, J is zero. When also typically load the surface, and hence deform the surface 1 only the largest terms in the gravity field are considered, the theEarth.Thedeconvolutionoftheeffectsofsurfacedeforma- Earth is axially symmetrical, and thus, there is no longitude tionsaffectingthecenteroffigureandthemasscontributions dependence.Thesecond-degreeharmonicterm,J ,isrelatedto thataffectcenter-of-masspositionsisnottrivial.Thissubjectis 2 boththemomentsofinertiaoftheEarth,whichwillcontrolthe explored in Lavalle´e and Blewitt (2002) and Blewitt (2003) Earth’srotationalbehaviorandtheflatteningoftheshapeofthe andisdiscussedinChapter3.11. Earth.ThemomentsofinertiaoftheEarthcanbeexpressedin ThetreatmentofEarthrotationvariationsiscommonplace termsofJ throughMacCullagh’sformulayielding inmodernsystemsbecausetheeffectsaresolarge.However, 2 evenwiththistopic,therearesubtleproblemsassociatedwith J C A = Ma2 [2] 2¼ð (cid:2) Þ deformationsoftheEarth.Themotionoftherotationaxiswith whereAandCaretheminimuma(cid:6)ndm(cid:7)aximummomentsof respecttothecrustoftheEarthisdominatedbyaresonance term called the Chandler wobble, named after the person to inertiaoftheaxiallysymmetricalEarth,respectively.Bysetting first observe the effect, and an annual signal due mainly to thepotentialtobethesameattheequatorandthepole,the semimajor and semiminor axes of an ellipsoid, a and c, that seasonalmassmovements.TheChandlerwobblehasaperiod ofabout430dayswiththefrequencyrelatedtothemoments matchthesevaluesallowtheflatteningoftheellipsoidtobe ofinertiaoftheEarth,theelasticpropertiesofthemantle,and derivedfromthegravityfield.Neglectinghigher-ordertermsin theeffectsoftheoceansandthefluidcore.Theannualsignal thegravityfield,wehave(Stacey,1992) and Chandler wobble cause motions of the pole with an a c C A a2 c 1a2co2 3 1 amplitudeofabout10m.Superimposedontheselargequasi- f¼ (cid:2)c ¼ M(cid:2)a2 c2 +2a +2 GM (cid:3)2J2+2m [3] periodic signals are other broadband signals that arise from (cid:4) (cid:5) Geodesy:AnIntroductionandOverview 3 movements in the atmosphere, oceans, and fluid core. The theinterioritselfmovesduetomantleconvection.Todefine details of these motions are discussed in Chapter 3.09. In rotation changes, the deformation component must be addition to these periodic-type variations, there is a secular accountedfor.Forsecularmotionsduetoplatetectonics,the driftofthepoleduetochangesofmomentsofinertiaresulting separationismade,makingtheaveragemotionhavingnonet from glacial isostatic adjustment. This area is discussed in rotation, the so-called no-net-rotation or NNR frame. The Chapter3.07. currentdefinitions ofrotationanglesfortheEarthasdistrib- In addition tochanges in the position ofthe rotation axis utedbytheInternationalEarthRotationServiceusethisdefi- withrespecttothecrustoftheEarth,therotationratealsovaries nition.Withmoderngeodeticaccuracy,nonsecularmotionsof withbothasecularterm,duetoenergydissipationintheEarth– sitesarenowevidentduemainlytoloadingeffects.Thegrav- Moonsystem,andshorter-periodvariationsduetomomentum itationalconsequencesofthemassmovementsandloadsare exchangesbetweentheatmosphere,oceans,andfluidcoreand discussedinChapter3.08buttherearealsoconsequencesfor thesolidEarth.Forperiodslessthanafewyears,theatmosphere Earthrotationmeasurementswhensmallnumbersofsitesare isthedominantsourceofchangesinangularmomentum.For used.ForEarthrotationmeasurementsusingalargenumberof longer-period terms, the fluid core is the most likely source. sites, some of these nonsecular site movements can be aver- RotationratechangesarediscussedinChapters3.09and3.10. aged. However, the spatial correlation of loading effects sug- The concept of rotation changes introduces another coor- geststhatthereisanupperlimittothenumberofsitesthatare dinate system that is needed in modern geodesy. This addi- neededtoeffectivelymeasureEarthrotationchangesandaver- tionalcoordinatesystemisonethatdoesnotrotateininertial age out the loading effects and other nonsecular motions. space. Rotation rate variations are measured relative to this Ultimately,inclosedsystemanalysis,nonseculardeformations nonrotating system. Historically, this external, nonrotating would be accounted for, for example, from the analysis of framehasbeendefinedbymeasurementstoopticalstarsand gravitychanges,beforetherotationanglesarecomputed. to bodies in our solar system. Both of these systems have problemswhenveryaccuratemeasurementsareneeded.Easily 3.01.3 GeodeticMethods seenopticalstarsarequiteclosetousandexhibittheso-called proper motions, meaning that they move relative to a non- Moderngeodeticmethodshaveinrecentdecadespushedthe rotatingframe.Solarsystemmeasurementssufferfromlackof precision of measurements and the completeness of models accurateobservationsandtheirinterpretationrequiresthatthe neededtoexplainthesemeasurements.Ingeodesy,itiscom- equationsofmotionofobjectsinthesolarsystemcontainall mon to talk about the fractional precision of measurement. ofthecorrectforcemodels.Soneitheropticalobjectsnorsolar A1kmdistancemeasuredwithaprecisionof1mmissaidto systemdynamicsprovideastablenonrotatingcoordinatesys- be precise to 1 part per million or 1ppm. On large scales, tem. Such a coordinate system is provided by extragalactic modern geodetic measurements are precise to better than 1 radio sources whose directions can be measured using very partperbillion(1ppb)inmanycases.Forglobalheightmea- long-baselineinterferometry(VLBI)(Maetal.,1998).Changes surement,1ppbcorrespondsto6mmuncertaintiesinheights. in rotation rate are measured relative to this nonrotating Forgravitymeasurements,1ppbis10nms(cid:2)2or1mgalwhere system, and time as measured by the rotation of the Earth, galisthecgsunitforgravitynamedafterGalileo.TheEarth’s UniversalTime1orUT1,istheintegrationoftherotationrate. gravityisabout980gals.Thereareavarietyofgeodeticsystems Withthedefinitionsofaninertialcoordinatesystemandone that achieve these precisions. In this treatise volume, all the thatisattachedtotheEarth,instantaneousrotationanglescan modern geodetic measurement methods are discussed. They bedefinedthatrotateonesystemintotheother.Thespectrum fallintothreebasicclasses:ground-basedpositioningsystems oftheserotationanglesshowsstrongpeaksinthenearlydiurnal that provide geometric positioning and tracking of external band and broad spectral response outside the diurnal band. objects, satellite systems that sense the Earth’s gravity field Historically, the rotation vector has been divided into two and/ormakemeasurementsdirectlyfromspace,andground- parts.WhentherotationaxisisviewedfromtherotatingEarth, based instrumentation that measures the gravity field on the thenearlydiurnalretrogrademotionoftherotationaxisrepre- Earth’s surface. The interpretation of geodetic measurements sentsamotionthatisslowininertialspaceandiscallednuta- oftenrequiresinputsfromallthesesystems. tion. Thismotion has beenknown about since ancient Greek times. The word nutation is Greek for nodding. The nodding 3.01.3.1 Ground-BasedPositioningSystems iscausedbythetime-dependenttorquesappliedtotheequato- rial bulge of the Earth by the gravitational forces from the Fourmajorgeodeticmeasurementsfallintothiscategory:VLBI, Moon, Sun,and planets.The average torque applied by these satellitelaserranging(SLR),globalpositioningsystem(GPS),and bodies causes a secular motion of the pole in inertial space the Doppler Orbitography and Radiopositioning Integrated by called precession. The longer-period motion is called polar Satellite (DORIS) system. These systems use either microwave motion.Inanapproximatesense,nutationiscausedbyexternal (VLBI, GPS, and DORIS) or optical (SLR) frequency signals to torques applied to the Earth and pole motion due to mass providethe carrier fortheir measurements. The basic measure- andangularmomentumredistributionsintheEarth.Thesepa- ment types include group and phase delay and Doppler shift ration and interpretation of these motions are discussed in measurements.Estimatesofthepositionsandmotionsofloca- Chapters3.09and3.10. tionson theEarth’ssurfaceobtainedfromthe analysisof data Thefinalaspectofmeasuringchangesintherotationofthe fromthesesystemsareused in theformationoftheITRF.The Earth is the definition of rotation angles. All points on latest version of the ITRF is ITRF2008 (Altimini et al., 2011; the surface of the Earth move relative to the interior just as Altimini,2012). 4 Geodesy:AnIntroductionandOverview VLBIisamicrowave-basedmeasurementsystemthatmea- Chapter3.10.VLBIalsoprovidesaccuratepositionandatmo- suresthedifferenceinarrivaltimesofincoherentsignalsfrom sphericdelaymeasurements. radiosourcesbycrosscorrelation.Mostcommonly,theradio SLR isanoptical-based systemthatusesshort pulsedlaser sources are extragalactic objects but beacons from satellites andaccuratetimingequipmenttomeasuretheroundtripflight have also been used. Group and phase delays along with time between a ground system and a satellite equipped with Doppler shift are measured. The phase delays are difficult to special corner cube retroreflectors. Some laser systems have use in geodetic measurements with this system because they enough power to make range measurements to corner cube are measured modulo 2p, and reconstructing the number of arrayson the Moon. Specially equippedoptical telescopesare cyclesofphaseisnontrivialingeodeticmeasurements.Inthe used for these measurements. Measurement accuracies of astronomicalapplicationsofVLBI,theuseofthephasedelays 1–10mm are common for modern systems. The optical fre- iscommon.Individualdelaymeasurementshaveprecisionsof quencies used inthese systems are not affected by the Earth’s 1–10mm.Generally,radiotelescopeswithdiametersbetween ionosphere,andthecontributionfromatmosphericwatervapor 10and30mareusedforVLBImeasurements,andbecauseof ismuchlessthanformicrowavesystems(theopticalfrequencies theseparationofthetelescopes(thousandsofkilometers),the are too high to efficiently excite the dipole water vapor reso- delaymeasurementsaremaderelativetoindependenthydro- nance). Figure 2 shows the distribution of SLR sites used in genmaserclocksattheobservatories.Theradiosignalsprop- ITRF2008. Of these sites, about 35 are currently active. SLR agatethroughtheEarth’satmosphereandaredelayedthrough measurements do not need to be closely coordinated, and therefractiveindexoftheatmosphere’sgasconstituents.VLBI hence,SLRstationscanoperatesemiautonomously.Somesta- measurements are made at two relatively high frequencies tionsoperatecontinuously,whileothersoperateon8hshifts. (usually 2.3and8GHz)andaredelayedbythepropagation Thereareprioriessetforwhichofthecurrent26satelliteswith (cid:4) intheionosphere(removedwithadual-frequencycorrection corner cubes should be tracked when multiple satellites are sincethismediumisdispersive)andtheyarestronglyaffected visible.ThecoordinationofSLRmeasurementsisprovidedby by the dipole component of the refractivity of atmospheric theInternationalLaserRangingService(Pearlmanetal.,2002). water vapor. Figure 1 shows the distribution of VLBI sites SLRprovidesmeasurementstoabout26orbitingspacecraft included in ITRF2008. Of the sites shown, about 40 stations andtheMoon.Thesemeasurementsareusedtodeterminethe are currently operating. VLBI requires coordinated measure- positions of the SLR sites and the orbits of the spacecraft. ments so that distant telescopes look at the same objects at Perturbations to the spacecraft orbits are used to study the thesametime.MeasurementswithVLBIareusually24hdura- Earth’s gravity field and its temporal variations. For geodetic tion observing sessions using four to eight radio telescopes positioning, the most commonly observed satellites are the aroundtheworld.VLBIactivitiesarenowcoordinatedthrough pair of Laser Geodynamics Satellites (LAGEOS). These are theInternationalVLBIService(Schlueteretal.,2002). high area-to-mass ratio satellites orbiting at about 6000km VLBIistheonlymoderntechniquewithdirectreferencetoa altitude.Atthisaltitude,theorbitsaresensitivetothelower- stableinertialreferenceframe.Ituniquelycontributestomain- degree terms in the Earth’s gravity and measurements to tainingtimeasmeasuredbytherotationoftheEarthandto LAGEOS provided the first measurements of secular changes monitoringthemotionoftheEarth’srotationaxisininertial inthelower-degreegravityfieldcoefficients.Thealtitudeisalso space.Theselattermeasurementsprovideuniqueinsightsinto highenoughtobeonlyslightlyaffectedbyatmosphericdrag, the interaction of the fluid-outer and solid-inner cores and althoughtherearesomenotwell-understoodelectromagnetic the mantle of the Earth. This topic is discussed in detail in perturbationstotheorbits.Thehighaltitude oftheLAGEOS VLBI sites (113) 60(cid:2) 30(cid:2) 0(cid:2) -30(cid:2) -60(cid:2) 180(cid:2)270(cid:2) 0(cid:2) 90(cid:2)180(cid:2) Figure1 Locationsofthe113VLBIsitesthatareincludedintheITRF2008referenceframe.Ofthesitesshown,about40arestillactivelymaking measurements.MeasurementswithVLBIareorganizedintosessions,usuallyof24hduration,usingfourtoeightradiotelescopes. Geodesy:AnIntroductionandOverview 5 SLR sites (125) 60(cid:2) 30(cid:2) 0(cid:2) -30(cid:2) -60(cid:2) 180(cid:2)270(cid:2) 0(cid:2) 90(cid:2)180(cid:2) Figure2 Locationsofthe125SLRsitesthatareincludedintheITRF2008referenceframe.Ofthesitesshown,about35arestillactivelymaking measurements.Thesestationsobservecontinuouslywithpriorities,butnotdetailedscheduling,setbytheinternationallaserrangingservice. satellitesalsomeansthatthecenterofmassoftheEarthcanbe thisfiguretheadditional1000GPSsiteswhosedataarerou- welldeterminedfromthemeasurements.Theprimaryaimof tinely available in the international GPS data archives. GPS muchoftheSLRtrackingispreciseorbitdetermination(POD). stations track satellites continuously, are low power, and are For the altimeter missions discussed in Chapter 3.05, the suited to autonomous operation. Many thousands of GPS determinationoforbitswithsub-centimeteraccuracyiscritical. stations operate continuously around the world. The coordi- The GPS is a microwave group and phase delay measure- nation of station standards and data analysis is through the ment system using relatively low L-band signals ( 1.2 and InternationalGlobalNavigationSatelliteSystemService(IGS) (cid:4) 1.5GHz).Thesystemwasinitiallydesignedasamilitarynav- (Beutleretal,1999).InadditiontoGPS,theIGSalsoincludes igationsystemthatusedgroupdelays.Timetagsareencoded measurementsfromreceiversthattracktheRussianGLONASS intothetransmittedsignalandagroundreceivercanmeasure satellitesandinthefuturewillincludemeasurementsfromthe thedifferenceintimebetweenwhenasignalwastransmitted European Galileo system. All these systems use similar fre- fromasatellitebasedonthesatellite’sclockandwhenitwas quency bands and their satellites are in medium Earth orbit receivedonthegroundbasedonthegroundreceiver’sclock.If withaltitudesnear20000km(and12horbitalperiods). theclocksweresynchronized,thetimedifferencewouldbea GPSandtheotherGNSSsystemsprovideinexpensive,very measure of range. In the navigation application, accurate precise positioning capabilities on both static and moving clocks are placed on the satellites, and because the ground receivers.Thesystemsareoftendeployedindensenetworksto receiverscansimultaneouslymakemeasurementstomultiple monitortectonicdeformationswithcontinuousandoccasional satellites,thegroundreceiver’sclockerrorcanbeestimatedor occupations. The large numbers of stations available provide eliminated by differencing measurements. The ability ofGPS robustmeasurementsofpolarmotionandshort-periodvaria- receivers to make measurements simultaneously to multiple tionsinlengthofday.DensenetworksofGPSstationsarealso satellitesandthenarrowbandwidthofthetransmittedsignals usedtomonitoratmosphericwatervapor,andinmanycases, meansrelativelyinexpensiveelectronicscanbeusedtomake results are used in operational weather forecasting. The high very precise measurements. In the geodetic applications of dataratesavailablewithGPSwithsamplingfrequenciesupto GPS, the phase measurements are used, which is possible 50Hz make the system ideal for tracking fast-moving objects because multiple ground stations can see the same satellites, suchasaircraftandformonitoringseismicevents.Chapter3.11 andbyeitherestimationordifferencing,thecontributionsof discussesmanyoftheapplicationsofGPSalongwiththeother localoscillatorphasesinthereceiversandtransmitterscanbe spacegeodeticpositioningsystems. eliminated.Phasemeasurementsofrangechangesaccuratetoa The DORIS system is also a microwave system that uses few millimeters are possible. Dual-frequency measurements two-wayDopplertracking.Unliketheothergeodeticsystems, are used to eliminate the ionospheric delays, but because of DORISstationshaveactivecommunicationwiththesatellites the low frequency, the dual-frequency corrections are not beingtracked.Thesatellitesinitiatecommunicationswiththe always adequate. The second-order effects from the Earth’s groundstations.Thissystemallowstwo-waycommunications magneticfieldneedtobeaccountedforinthemostaccurate thatcancancelerrorsduetodifferentfrequenciesintheoscil- analyses. Continuous tracking of signals from the satellites latorsinthespacecraftandgroundstations.DORISisprimarily allowsthenumberofcyclesinthephasetobecounted,and usedforPODandplaysacriticalroleintrackingsatellitesused onlywhenthesignalfromthesatelliteisinterruptedorwhena forradaraltimetrydiscussedinChapter3.05.Earthorientation satelliteis firstseen isitnecessarytoestimatethenumber of measurements,center-of-massvariations,andstationlocations integer cycles in the phase. Figure 3 shows the GPS sites arealsodeterminedintheanalysisofDORISdata.Theactivi- included in the ITRF2008 reference frame. We also show on tiesofDORISarecoordinatedundertheInternationalDORIS 6 Geodesy:AnIntroductionandOverview GNSS sites (560) 60(cid:2) 30(cid:2) 0(cid:2) -30(cid:2) -60(cid:2) 180(cid:2)270(cid:2) 0(cid:2) 90(cid:2) 180(cid:2) Figure3 Locationsofthe560GPSsitesthatareincludedintheITRF2008referenceframe(redsquares).Therearemanyadditionalsiteswhosedata aremadefreelyavailablethroughinternationaldataarchives.Thesesitesoperatecontinuously. DORIS sites (130) 60(cid:2) 30(cid:2) 0(cid:2) -30(cid:2) -60(cid:2) 180(cid:2)270(cid:2) 0(cid:2) 90(cid:2)180(cid:2) Figure4 Locationsofthe139DORISsitesthatareincludedintheITRF2008referenceframe.Thesesiteshaveactivecommunicationbetweenthe groundstationsandthesatellitesthataretrackedwithDORIS. service (Tavernier et al., 2006). In Figure 4, the DORIS sites withJason-1usedprimarilyforoceanstudiesandICESatused usedintheITRF2008areshown. primarily for ice sheet studies. Chapter 3.05 discusses results from these and earlier altimeter satellites that are no longer operating.TheCHAMPandGRACEsatellitesareusedforgravity fieldstudieswithGRACEbeingapairofsatelliteswithaccurate 3.01.3.2 SatelliteSystems rangeandrange-ratemeasurementsbetweenthem.Thesemea- Awidevarietyofsatellitesystemsareusedinmoderngeodetic surementsprovide veryaccurate determinationsofchanges in measurements.Thosethatarecurrentlybeingtrackedwithlaser theEarth’sgravityfieldasdetailedinChapter3.08.TheERSand rangingsystemsarelistedinTable1.Theapplicationsofsatel- Envisat satellites carry synthetic aperture radars, which when litesfallintodifferentcategories.Fiveoftheentriesarereflector usedinaninterferometricmode,asdiscussedinChapter3.12, arrayson the Moon, three from the Apollo program and two allow high-spatial-resolution maps of deformation fields or fromtheRussianLunaprogram.AlloftheRussianGLONASS topographytobemade. satelliteshavecornercubearrays,asdotwooftheGPSsatellites. TheGPSarraysareusedforaccuracyevaluation.TheLAGEOS, Starlette,andStellaspacecraftarepassive,sphericalbodiescov- 3.01.3.3 Gravimeters ered with corner cube reflectors and are primarily used for gravity field determination and positioning. The Jason-1 and Gravimeter instrumentation for precise geodesy is classed as ICESat satellites are microwave and laser altimeter satellites absolute gravimeters that normally use the acceleration of an Geodesy:AnIntroductionandOverview 7 Table1 ActivegeodeticsatellitesbeingtrackedwithSLR Satellitename SatelliteID Altitude(km) Inclination(deg) Firsttrackeddate Ajisai 8606101 1485 50 13August1986 Apollo11SeaofTranquility 0000100 356400 5 20August1969 Apollo14FraMauro 0000102 356400 5 07February1971 Apollo15HadleyRille 0000103 356400 5 01September1971 Beacon-C 6503201 927 41 02January1976 COMPASS-G1 1000101 42164 55.5 28April2012 COMPASS-I3 1101301 42161 55.5 27April2012 COMPASS-I5 1107301 42161 55.5 06July2012 COMPASS-M3 1201801 21528 55.0 11July2012 CryoSat-2 1001301 720 92 20April2010 Envisat 0200901 800 98 10April2002 Etalon-1 8900103 19105 65 26January1989 Etalon-2 8903903 19135 65 13July1989 GIOVE-A 0505101 23916 56 11May2006 GLONASS-102 0606201 19140 65 04May2007 GLONASS-109 0706503 19140 65 04May2007 GLONASS-110 0804601 19140 65 10December2009 GLONASS-118 0907003 19140 65 04January2010 GLONASS-129 1106402 19140 65 02January2012 GLONASS-130 1107101 19140 65 01January2012 GOCE 0901301 295 96.7 01April2009 GPS-36 9401601 20030 55 21April1994 GRACE-A 0201201 485–500 89 18March2002 GRACE-B 0201202 485–500 89 18March2002 Galileo-101 1106001 23220 56 29November2011 Galileo-102 1106002 23220 56 29November2011 Galileo-103 1205501 23220 56 07November2012 Galileo-104 1205502 23220 56 07November2012 HY-2A 1104301 971 99.35 02October2011 IRNSS-1A 1303401 42164 29 05September2013 Jason-2 0803201 1336 66 24June2008 KOMPSAT-5 1304201 550 97.6 09September2013 LAGEOS-1 7603901 5850 110 10May1976 LAGEOS-2 9207002 5625 53 24October1992 LARES 1200601 1450 69.5 17February2012 LRO-LR 0903101 50(lunar) 90(wrtlunarequator) 30June2009 (cid:4) (cid:4) Larets 0304206 691 98.204 04November2003 Luna17SeaofRains 0000101 356400 5 21May1975 Luna21SeaofSerenity 0000104 356400 5 16November1973 QZS-1 1004501 32000–40000 45 11September2010 RadioAstron 1103701 500–350000 51.4 15November2011 SARAL 1300901 814 98.55 04March2013 STPSat-2 1006201 650 72 02April2013 STSAT-2C 1300301 300–1500 80 29March2013 Starlette 7501001 815 50 03January1976 Stella 9306102 815 99 30September1993 TanDEM-X 1003001 514 97.44 21June2010 TerraSAR-X 0702601 514 97.44 16June2007 Listavailablefromandupdatedathttp://ilrs.gsfc.nasa.gov/missions/satellite_missions/current_missions/index.html. Source:InternationalLaserRangingService. object falling in vacuum or superconducting instruments that uplift or secular mass movements. These instruments are nor- measurechangesingravityonatestmassthatiscooledtovery mallymovedfromplacetoplacetomakerepeatedmeasurements low temperatures to minimize the random accelerations from oftenovermanyyears.Thesuperconductinggravimetersprovide thermalnoise.Theapplicationsanddevelopmentoftheseclasses continuoushigh-precisiongravitymeasurementsatobservatory of gravimeters are discussed in Chapters 3.03 and 3.04. The locations. They are suitable for tidal studies, particularly the absolute gravimeters provide absolute measurements that are effectsofthefluidcoreontidalamplitudes,seismicmodesfrom suitableformonitoringlong-termchangesingravityduetoeither largeearthquakes,andatmosphericmasschangeexcitations. 8 Geodesy:AnIntroductionandOverview 3.01.4 ErrorSources,Signals,andNoise chaptersprogressthroughthedifferentareasofgeodesystart- ingwiththetheoryoftheEarth’sgravityfield,followedthenby Severalcommonthemesrunthroughthechaptersinthisvol- ground-based measurements of gravity and space-based alti- ume.Thefirstoftheseisthatgeodesyisthecombinationofan metric measurements. The accuracy of the modern measure- observationalscienceandatheoreticalscience.Duringvarious ments allows temporal variations in gravity to be measured, stages of its history, either new theories or new observations andthistopicisdiscussedinthreechaptersdealingwithtidal determinethedirectionofthescience.Duringthelast20years, variations in gravity, short-term variations that can now be observationalaccuracieshaveoftenbeendrivingnewtheoret- measured from space, and long-period (secular) variations ical developments. The other critical aspect of modern mea- that are related to the viscoelastic response of the Earth to surements is the errors in these measurements. At times, unloading of ice mass that occurred at the end of the last assessingerrorlevelshasbeendifficult,especiallywhenaccu- Pleistoceneiceage.Ofgreatinterestcurrentlyistheseparation racy is considered, because there have been no standards to ofthissecularsignalfromthesignalduetocurrentmeltingof comparewith.Manyofthechaptersinthisvolumediscussthe ice sheets and glaciers. The combination of gravity measure- errorspectraofgeodeticmeasurementsinthetimeandspace ments and surface uplift measurements shows promise for domains.Inmanycases,acontributionthatcanbeconsidered allowingthisseparation.Thesamemassmovementsthateffect noise for one user may be a signal for others. The effects of time-dependent gravity also change the rotation of Earth on water vapor and liquid water fall into the category of being manytimescales.TwochaptersaddressEarthrotationchanges bothnoiseandsignal. in inertial space and relative to the crust of the Earth. The An example of how theoretical and observational drivers volumeconcludeswiththetheory,instrumentation,andappli- havechangedovertheyearsisthestudyofnutation.Formuch cationsofmodernpositionmethodsandsurfacedeformation ofgeodetichistory,themotionoftherotationaxisinspacewas monitoring. computedtheoreticallyfromtheephemeridesoftheMoonand SunandknowledgeofthemomentsofinertiaoftheEarth.At thetimethatVLBIsystemswerebeingdeployedworldwideand References usedonaregularbasis,anewnutationtheorywasadoptedby the International Astronomical Union (IAU) called the IAU AltamimiZ,CollilieuxX,andMe´tivierL(2011)ITRF2008:Animprovedsolutionofthe 1980 theory of nutation. This new theory included effects of InternationalTerrestrialReferenceFrame.JournalofGeodesy85(8):457–473. elasticityoftheEarth(derivedfromseismicobservations)and http://dx.doi.org/10.1007/s00190-011-0444-4. afluidcorewhoseshapewascomputedfromtheassumption AltamimiZ,SillardP,andBoucherC(2002)ITRF2000:Anewreleaseofthe InternationalTerrestrialReferenceFrameforearthscienceapplications. of hydrostatic equilibrium throughout the Earth. Just a few JournalofGeophysicalResearch107(B10):2214.http://dx.doi.org/ years after the adoption of this theory, VLBI observations 10.1029/2001JB000561. revealed discrepancies between the predictions of the theory AltiminiZ(2012)http://itrf.ensg.ign.fr/ITRF_solutions/2008/more_ITRF2008.php. andmeasurements,whichresultedinanewassessmentofthe BeutlerG,RothacherM,SchaerS,SpringerTA,KoubaJ,andNeilanRE(1999)The InternationalGPSService(IGS):Aninterdisciplinaryserviceinsupportofearth theory.Somepartsoftheassessmentshowedthattheoriginal sciences.AdvancesinSpaceResearch23(4):631–635.http://igscb.jpl.nasa.gov. theories were truncated at too coarse a level for the modern BlewittG(2003)Self-consistencyinreferenceframes,geocenterdefinition,andsurface VLBIaccuracies,andotherpartsshowedthatthetheoryneeded loadingofthesolidEarth.JournalofGeophysicalResearch108(B2):2103.http:// toextendtoincludetheeffectsofnonhydrostaticforces,mag- dx.doi.org/10.1029/2002JB002082. neticcoupling,andthepresenceofthesolid-innercore.These Lavalle´eDandBlewittG(2002)Degree-1Earthdeformationfromverylongbaseline interferometrymeasurements.GeophysicalResearchLetters29(20):1967.http://dx. developments are discussed in Chapter 3.10. Chapter 3.04 doi.org/10.1029/2002GL015883. discussesaverydifferent measurementtypethatcan beused MaC,AriasEF,EubanksTM,etal.(1998)TheInternationalcelestialreferenceframe to address the theory. Currently, there are theoretical predic- asrealizedbyverylongbaselineinterferometry.AstronomicalJournal tions about the conductivity structure at the core–mantle 116:516–546. PearlmanMR,DegnanJJ,andBosworthJM(2002)TheInternationalLaserRanging boundarythatgenerateresultsconsistentwithVLBImeasure- Service.AdvancesinSpaceResearch30(2):135–143.http://ilrs.gsfc.nasa.gov. mentsthatneedtobeassessedbyanindependentmethod. SchlueterW,HimwichE,NothnagelA,VandenbergN,andWhitneyA(2002)IVSandits importantroleinthemaintenanceoftheglobalreferencesystems.Advancesin SpaceResearch30(2):145–150.http://ivscc.gsfc.nasa.gov. 3.01.5 Conclusions SmithJR(1996)IntroductiontoGeodesy:TheHistoryandConceptsofModern Geodesy.WileySeriesinSurveyingandBoundaryControl,p.224.NewYork:Wiley. StaceyFD(1992)PhysicsoftheEarth,3rdedn.Brisbane:BrookfieldPress,pp.513. ModerngeodesyinteractswithnearlyallfieldsofEarthscience, TavernierG,FagardH,FeisselM,etal.(2006)TheInternationalDORISService:Genesis andthechaptersinthisvolumeshowtheseinteractions.The andearlyachievements.JournalofGeodesy80(8–11):403–428.http://ids.cls.fr/. 3.02 Potential Theory and the Static Gravity Field of the Earth CJekeli,TheOhioStateUniversity,Columbus,OH,USA ã2015ElsevierB.V.Allrightsreserved. 3.02.1 Introduction 9 3.02.1.1 HistoricalNotes 9 3.02.1.2 CoordinateSystems 10 3.02.1.3 PreliminaryDefinitionsandConcepts 11 3.02.2 Newton’sLawofGravitation 12 3.02.3 Boundary-ValueProblems 14 3.02.3.1 Green’sIdentities 15 3.02.3.2 UniquenessTheorems 16 3.02.3.3 SolutionsbyIntegralEquation 17 3.02.4 SolutionstotheSphericalBVP 17 3.02.4.1 SphericalHarmonicsandGreen’sFunctions 17 3.02.4.2 InverseStokes’andHotineIntegrals 20 3.02.4.3 Vening-MeineszIntegralandItsInverse 21 3.02.4.4 ConcludingRemarks 22 3.02.5 Low-DegreeHarmonics:InterpretationandReference 23 3.02.5.1 Low-DegreeHarmonicsasDensityMoments 23 3.02.5.2 NormalEllipsoidalField 24 3.02.6 MethodsofDetermination 26 3.02.6.1 MeasurementSystemsandTechniques 26 3.02.6.2 Models 30 3.02.7 TheGeoidandHeights 31 References 33 3.02.1 Introduction the mass of the falling object – all bodies fall with the same acceleration. This truly monumental contribution to physics Classicalgravitationalpotentialtheoryhasitsrootsinthelate was,however,onlyalocalexplanationofhowbodiesbehaved renaissanceperiodwhenthepositionoftheEarthinthecosmos undergravitationalinfluence.JohannesKepler’s(1571–1630) was established on modern scientific, observation-based observationsofplanetaryorbitspointedtoothertypesoflaws, grounds. A study of Earth’s gravitational field is a study of principally an inverse-square law according to which bodies Earth’s mass including its transport in time and its influence are attracted by forces that vary with the inverse square of onnearobjects;and,fundamentally,itisageodeticstudyof distance.ThegeniusofIsaacNewton(1642–1727)broughtit Earth’sshape,whichisdescribedlargely(70%)bythesurface alltogetherinhisPhilosophiæNaturalisPrincipiaMathematicaof oftheoceans.Thisinitialsectionprovidesahistoricalbackdrop 1687with a single andsimple all-embracing lawthatinone toNewtonianpotentialtheoryandintroducessomeconcepts bold stroke explained the dynamics of the entire universe inphysicalgeodesythatsetthestageforlaterformulations. (today, there is more to understanding the dynamics of the cosmos, but Newton’s law remarkably holds its own). The mass of a body was again an essential aspect, not as a self- 3.02.1.1 HistoricalNotes attributeasAristotlehadimplied,butasthesourceofattraction Gravitation is aphysical phenomenon so pervasive and inci- forotherbodies:eachmaterialbodyattractseveryothermate- dentalthathumankindgenerallyhastakenitforgrantedwith rial body according to a very specific rule (Newton’s law of scarcely a second thought. The Greek philosopher Aristotle gravitation; see Section 3.02.2). Newton regretted that he (384–322BC)allowednomorethantoassertthatgravitation couldnotexplainexactlywhymasshasthisproperty(asone isanaturalpropertyofmaterialthingsthatcausesthemtofall stillyearnstoknowtodaywithinthestandardmodelsofparti- (orrise,inthecaseofsomegases)andthemorematerial,the cleandquantumtheories).EvenAlbertEinstein(1879–1955) greaterthetendencytodoso.Itwasenoughofaself-evident indevelopinghisgeneraltheoryofrelativity(i.e.,thetheoryof explanation that it was not yet to receive the scrutiny of the gravitation)couldonlyimproveonNewton’stheorybyincor- scientific method, the beginnings of which, ironically, are poratingandexplainingactionatadistance(gravitationalforce creditedtoAristotle.Almosttwothousandyearslater,Galileo acts with the speed of light as a fundamental tenet of the Galilei (1564–1642) finally took up the challenge to under- theory).Ofcourse,Einsteindidmuchmorebyreinterpreting standgravitationthroughobservationandscientificinvestiga- gravitationalattractionasawarpingofspaceandtime,which tion.Hisexperimentallyderivedlawoffallingbodiescorrected thenallowsconsiderationofgravitationeffectsathighvelocity theAristotelianviewanddivorcedtheeffectofgravitationfrom andduetogreatquantitiesofmass(e.g.,blackholes).However, TreatiseonGeophysics,SecondEdition http://dx.doi.org/10.1016/B978-0-444-53802-4.00056-7 9

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3.01 Geodesy: An Introduction and Overview TAHerring,MassachusettsInstituteofTechnology,Cambridge,MA,USA ã2015ElsevierB.V.Allrightsreserved. 3.01.1 Introduction 1 3.01.2 CoordinateSystems 1 3.01.3 GeodeticMethods 3 3.01.3.1 Ground-BasedPositioningSystems 3 3.01.3.2 SatelliteSystems 6 3.01.3.3 Gravimeters 6 3.01.4 ErrorSources,Signals,andNoise 8 3.01.5 Conclusions 8 References 8 3.01.1 Introduction errors in measurements, signals, and noise. In this overview, weexaminethesethreethemes. Modern geodesyasdiscussed inthisvolumestarted withthe developmentofdistancemeasurementusingpropagatingelec- tromagneticsignalsandthelaunchofEarth-orbitingsatellites. 3.01.2 CoordinateSystems Withthesedevelopments,space-basedgeodesyallowedglobal measurements of positions, changes in the rotation of the Acommonaspectofgeodesythatpermeatestheliteratureof Earth,andtheEarth’sgravityfield.Thesethreeareas,position- thesubjectiscoordinatesystemdefinition.Withthedevelop- ing,Earthrotation,andgravityfield,areconsideredthethree ment of space-based methods that allow global measure- pillars of geodesy. The accuracy of current measurement sys- ments to be made, the subject has in some sense been temsallows time variationstobeobserved inallthree areas. greatly simplified in recent years while at the same time Also,thecomplexityofproblemsissuchthateachofthepillars complicatedbytheincreasedaccuracyneeds.Thesimplifica- interacts with each other and with many other branches of tion comes from being able to use a Cartesian coordinate Earth science. This interaction is most apparent in the role systemwithanoriginatthecenterofmassoftheEarthand that water plays in modern geodetic measurements. Every axesalignedinawell-definedmannertotheoutersurfaceof chapterinthisvolumementionstheroleofwater.Itiscritical theEarth.Thecomplicationsarisefromneedingacoordinate because it can move rapidly and over large distances; it can systemdefinitionthataccountsfordeformationsfromplate exist in all three phases, gas, fluid, and solid; and modern tectonics, tides, tectonic events, and other time-variable geodeticmethodsareaccurateenoughthattheirmeasurements deformations.Themeasurementandunderstandingofthese aresensitivetoitseffects.Initsvaporform,itsrefractiveprop- deformations are themes that are common to many of the ertiesdelaymicrowavesignalspropagatingthroughtheEarth’s chaptersinthisvolume. atmosphere.Forgeodeticpositioning,thisisanoisesourcebut Historically,coordinatesystemsingeodesyaredividedinto itisasignalformeteorologicapplications.Asaliquid,itforms two parts: horizontal coordinates such as latitude and oceansthataffectboththetidalsignalandtherotationofthe longitude and a vertical coordinate called height. These two Earth.Alsoinliquidform,itsmasschangesthegravityfieldas systemsarefundamentallydifferentinthatthehorizontalone itmovesthroughthehydrologiccycle.Insolidform,ithasa is geometric and based on the direction of the normal to an gravitationalanddeformationalsignalthatchangesifmelting ellipsoidalbodythatrepresentstheaverageshapeoftheEarth. the ice unloads the surface of the Earth. The interactions Theheightsystem,calledorthometricheight,isbasedonthe betweenthepillarsincludetheelasticloadingeffectsofchang- gravitational potential field of the Earth. In this system, sur- ingmassloadsthatcanbeseeninthegravityfieldandinthe facesofconstantheightareequipotentialsurfaces,andhence, positions ofground stations. The movement ofwater toand fluid will not flow along these surfaces. One special height from the oceans can be seen with altimeter satellites whose surface is called the geoid and is associated with the surface orbital information is derived from measurements from representing mean sea level. In Chapters 3.02 and 3.05, the groundstationswhosepositionsareaffectedbythechanging subtleproblemswiththesedefinitionsfromdefininganequi- massload.Inmodern,time-dependentgeodeticdataanalysis, potentialsurfacetothemeaningofmeansealevelinanocean these interactions need to be accounted for. The common withcurrentsarediscussed.Inconventionalgeodeticsystems, interfacebetweenthegeodeticmethodsisthecoordinatesys- thereisablendofpotential-basedandgeometricsystems.The temsandreferenceframeusedtoanalyzedata. determinationofthegeodeticlatitudeandlongitudeiscompli- Coordinate systems and the associated reference frames catedbymeasurementsbeingmadeintheEarth’sgravityfield. formacorethemeinthechaptersinthisvolume.Twoother Specificallywiththedevelopmentofelectronicdistancemea- unifyingthemesarethemeasurementsystemsofgeodesythat surements,thedistancemeasurementitselfdoesnotdependon are used again throughout the volume and the interplay of the gravity field (except for small relativistic effects), but the TreatiseonGeophysics,SecondEdition http://dx.doi.org/10.1016/B978-0-444-53802-4.00055-5 1 2 Geodesy:AnIntroductionandOverview projectionofthedirectmeasurementtoahorizontalmeasure- where f is the flattening and the centrifugal component of mentdoesdependontheslopeofthemeasuredline.Theangle acceleration from the rotation of the Earth (rotation rate o) betweenmeasureddirectionandlocalvertical(whichdepends hasbeenincluded.Theratiooftheequatorialgravitationalto onthelocaldirectionofgravity)canbeeasilymeasuredbutis rotationalforceisgivenbym.Fromeqns[1]to[3],weseethe strictly not the correct measurement. The angle to the local first-orderrelationshipsbetweenthegravityfieldoftheEarth, normal to the ellipsoid is needed. The difference between themomentsofinertiaoftheEarth,andtheshapeoftheEarth. thesedirectionsiscalledthe‘deflectionofthevertical’andas Withmodernspace-basedmeasurements,allofthesequanti- discussed inJekelicanbe determined frommeasurements of tiescanbemeasuredwithgreatprecision.Chapter3.02details gravityandsolvingtheappropriateboundaryvalueproblem. thetheoryofgravitationalpotential,Chapters3.09and3.10 Deflection of vertical is also the difference between geodetic examine the rotational consequences of moments of inertia, andastronomiclatitudeandlongitude. and Chapter 3.05 examines the detailed shape of the geoid Theblendofpotential-basedandgeometry-basedsystems basedonaltimetermeasurementsfromspace. in geodetic coordinate systems is a consequence of the With the development of space-based geodetic systems, it methodsavailableformakingmeasurements.Thelocalnature becamepossibletodefineapurelygeometricglobalcoordinate of these measurements resulted in large differences among system.ThesenewsystemsareCartesianandhavetheirorigin systemsadoptedbydifferentcountries.Thereferenceellipsoi- approximatelyatthecenterofmassoftheEarth.Theaxesare dalshapefortheEarthcouldhavedifferencesinthesemimajor alignedapproximatelyalongtherotationaxisandtheGreen- axis of over a kilometer (e.g., Clarke 1866 6378206.0m and wich meridian. The alignments here are only approximate Everest6377276.0m(Smith,1996)),anditwasnotuncom- because,withtheaccuracyofmodernmeasurements,thecen- monatboundariesofcountriestohavecoordinatedifferences terofmassandrotationaxismovewithrespectthefigureofthe ofseveralhundredmeters.Theadventofspace-basedmeasure- Earth.Therotationaxismotionsarelarge,oforder10movera mentsallowsthedeterminationofabest-fitellipsoidalshape year, and have been known about for over a century. The fortheEarthbasedonglobaldata.Therearesimplerelation- center-of-mass movements are much smaller, of order shipsbetweenthegravityoftheEarth,expressedinspherical 1–2cm, and have been measured for the past few decades. harmonics,themomentsofinertiaoftheEarth,andanappro- Even now, center-of-mass variations are not fully accounted priateflatteningfortheEarth.Afterthecentralforceterminthe for in modern coordinate systems. The current international gravity field, the flattening term is the largest, being at least terrestrial reference frames (ITRF) (Altamini, 2012; Altamini 1000timeslargerthananyotherterminthefield.Theleading etal.,2002,2011)haveanoriginatthecenterofthefigureof termsinthegravitationalpotentialfield,V,oftheEarthcanbe theEarth.MassmovementsonthesurfaceoftheEarth,mainly writteninsphericalharmonicsas(e.g.,Stacey,1992) atmosphericandhydrospheric,resultinthemovementofthe centerofmassrelativetothecenteroffigurethatneedstobe GM a a 2 V r,y J P J P cosy J P cosy [1] accounted for in accurate representations of the gravity field. ð Þ¼(cid:2) r 0 0(cid:2) 1r 1ð Þ(cid:2) 2 r 2ð Þ (cid:4) (cid:1) (cid:3) (cid:5) When center-of-figure reference frames are used, the gravity whereGisthegravitationalconstant;Mistheadoptedmassof fieldsintheseframesshouldhavetime-dependentfirst-degree theEarth;yandrarecolatitudeandradialdistancestothepoint spherical harmonic terms included. Currently, these terms wherethepotentialisdetermined;aistheequatorialradiusof need to be observationally determined and the routine pro- theEarth;J ,J ,andJ arethecoefficientsofthegravityfield;and ductionofsuchvaluesisnotyetavailable.Chapter3.08dis- 0 1 2 P ,P ,andP areLegendrefunctions.Whentheadoptedmassof cusses in detail the measurement of time-variable gravity. To 0 1 2 theEarthmatchesthatoftheEarth,theJ P termis1,andifthe furthercomplicateaccountingforcenter-of-massmovements, 0 0 center of mass of the Earth corresponds to the origin of the the mass movements that cause the center of mass to move, coordinate system used to measure latitude, J is zero. When also typically load the surface, and hence deform the surface 1 only the largest terms in the gravity field are considered, the theEarth.Thedeconvolutionoftheeffectsofsurfacedeforma- Earth is axially symmetrical, and thus, there is no longitude tionsaffectingthecenteroffigureandthemasscontributions dependence.Thesecond-degreeharmonicterm,J ,isrelatedto thataffectcenter-of-masspositionsisnottrivial.Thissubjectis 2 boththemomentsofinertiaoftheEarth,whichwillcontrolthe explored in Lavalle´e and Blewitt (2002) and Blewitt (2003) Earth’srotationalbehaviorandtheflatteningoftheshapeofthe andisdiscussedinChapter3.11. Earth.ThemomentsofinertiaoftheEarthcanbeexpressedin ThetreatmentofEarthrotationvariationsiscommonplace termsofJ throughMacCullagh’sformulayielding inmodernsystemsbecausetheeffectsaresolarge.However, 2 evenwiththistopic,therearesubtleproblemsassociatedwith J C A = Ma2 [2] 2¼ð (cid:2) Þ deformationsoftheEarth.Themotionoftherotationaxiswith whereAandCaretheminimuma(cid:6)ndm(cid:7)aximummomentsof respecttothecrustoftheEarthisdominatedbyaresonance term called the Chandler wobble, named after the person to inertiaoftheaxiallysymmetricalEarth,respectively.Bysetting first observe the effect, and an annual signal due mainly to thepotentialtobethesameattheequatorandthepole,the semimajor and semiminor axes of an ellipsoid, a and c, that seasonalmassmovements.TheChandlerwobblehasaperiod ofabout430dayswiththefrequencyrelatedtothemoments matchthesevaluesallowtheflatteningoftheellipsoidtobe ofinertiaoftheEarth,theelasticpropertiesofthemantle,and derivedfromthegravityfield.Neglectinghigher-ordertermsin theeffectsoftheoceansandthefluidcore.Theannualsignal thegravityfield,wehave(Stacey,1992) and Chandler wobble cause motions of the pole with an a c C A a2 c 1a2co2 3 1 amplitudeofabout10m.Superimposedontheselargequasi- f¼ (cid:2)c ¼ M(cid:2)a2 c2 +2a +2 GM (cid:3)2J2+2m [3] periodic signals are other broadband signals that arise from (cid:4) (cid:5) Geodesy:AnIntroductionandOverview 3 movements in the atmosphere, oceans, and fluid core. The theinterioritselfmovesduetomantleconvection.Todefine details of these motions are discussed in Chapter 3.09. In rotation changes, the deformation component must be addition to these periodic-type variations, there is a secular accountedfor.Forsecularmotionsduetoplatetectonics,the driftofthepoleduetochangesofmomentsofinertiaresulting separationismade,makingtheaveragemotionhavingnonet from glacial isostatic adjustment. This area is discussed in rotation, the so-called no-net-rotation or NNR frame. The Chapter3.07. currentdefinitions ofrotationanglesfortheEarthasdistrib- In addition tochanges in the position ofthe rotation axis utedbytheInternationalEarthRotationServiceusethisdefi- withrespecttothecrustoftheEarth,therotationratealsovaries nition.Withmoderngeodeticaccuracy,nonsecularmotionsof withbothasecularterm,duetoenergydissipationintheEarth– sitesarenowevidentduemainlytoloadingeffects.Thegrav- Moonsystem,andshorter-periodvariationsduetomomentum itationalconsequencesofthemassmovementsandloadsare exchangesbetweentheatmosphere,oceans,andfluidcoreand discussedinChapter3.08buttherearealsoconsequencesfor thesolidEarth.Forperiodslessthanafewyears,theatmosphere Earthrotationmeasurementswhensmallnumbersofsitesare isthedominantsourceofchangesinangularmomentum.For used.ForEarthrotationmeasurementsusingalargenumberof longer-period terms, the fluid core is the most likely source. sites, some of these nonsecular site movements can be aver- RotationratechangesarediscussedinChapters3.09and3.10. aged. However, the spatial correlation of loading effects sug- The concept of rotation changes introduces another coor- geststhatthereisanupperlimittothenumberofsitesthatare dinate system that is needed in modern geodesy. This addi- neededtoeffectivelymeasureEarthrotationchangesandaver- tionalcoordinatesystemisonethatdoesnotrotateininertial age out the loading effects and other nonsecular motions. space. Rotation rate variations are measured relative to this Ultimately,inclosedsystemanalysis,nonseculardeformations nonrotating system. Historically, this external, nonrotating would be accounted for, for example, from the analysis of framehasbeendefinedbymeasurementstoopticalstarsand gravitychanges,beforetherotationanglesarecomputed. to bodies in our solar system. Both of these systems have problemswhenveryaccuratemeasurementsareneeded.Easily 3.01.3 GeodeticMethods seenopticalstarsarequiteclosetousandexhibittheso-called proper motions, meaning that they move relative to a non- Moderngeodeticmethodshaveinrecentdecadespushedthe rotatingframe.Solarsystemmeasurementssufferfromlackof precision of measurements and the completeness of models accurateobservationsandtheirinterpretationrequiresthatthe neededtoexplainthesemeasurements.Ingeodesy,itiscom- equationsofmotionofobjectsinthesolarsystemcontainall mon to talk about the fractional precision of measurement. ofthecorrectforcemodels.Soneitheropticalobjectsnorsolar A1kmdistancemeasuredwithaprecisionof1mmissaidto systemdynamicsprovideastablenonrotatingcoordinatesys- be precise to 1 part per million or 1ppm. On large scales, tem. Such a coordinate system is provided by extragalactic modern geodetic measurements are precise to better than 1 radio sources whose directions can be measured using very partperbillion(1ppb)inmanycases.Forglobalheightmea- long-baselineinterferometry(VLBI)(Maetal.,1998).Changes surement,1ppbcorrespondsto6mmuncertaintiesinheights. in rotation rate are measured relative to this nonrotating Forgravitymeasurements,1ppbis10nms(cid:2)2or1mgalwhere system, and time as measured by the rotation of the Earth, galisthecgsunitforgravitynamedafterGalileo.TheEarth’s UniversalTime1orUT1,istheintegrationoftherotationrate. gravityisabout980gals.Thereareavarietyofgeodeticsystems Withthedefinitionsofaninertialcoordinatesystemandone that achieve these precisions. In this treatise volume, all the thatisattachedtotheEarth,instantaneousrotationanglescan modern geodetic measurement methods are discussed. They bedefinedthatrotateonesystemintotheother.Thespectrum fallintothreebasicclasses:ground-basedpositioningsystems oftheserotationanglesshowsstrongpeaksinthenearlydiurnal that provide geometric positioning and tracking of external band and broad spectral response outside the diurnal band. objects, satellite systems that sense the Earth’s gravity field Historically, the rotation vector has been divided into two and/ormakemeasurementsdirectlyfromspace,andground- parts.WhentherotationaxisisviewedfromtherotatingEarth, based instrumentation that measures the gravity field on the thenearlydiurnalretrogrademotionoftherotationaxisrepre- Earth’s surface. The interpretation of geodetic measurements sentsamotionthatisslowininertialspaceandiscallednuta- oftenrequiresinputsfromallthesesystems. tion. Thismotion has beenknown about since ancient Greek times. The word nutation is Greek for nodding. The nodding 3.01.3.1 Ground-BasedPositioningSystems iscausedbythetime-dependenttorquesappliedtotheequato- rial bulge of the Earth by the gravitational forces from the Fourmajorgeodeticmeasurementsfallintothiscategory:VLBI, Moon, Sun,and planets.The average torque applied by these satellitelaserranging(SLR),globalpositioningsystem(GPS),and bodies causes a secular motion of the pole in inertial space the Doppler Orbitography and Radiopositioning Integrated by called precession. The longer-period motion is called polar Satellite (DORIS) system. These systems use either microwave motion.Inanapproximatesense,nutationiscausedbyexternal (VLBI, GPS, and DORIS) or optical (SLR) frequency signals to torques applied to the Earth and pole motion due to mass providethe carrier fortheir measurements. The basic measure- andangularmomentumredistributionsintheEarth.Thesepa- ment types include group and phase delay and Doppler shift ration and interpretation of these motions are discussed in measurements.Estimatesofthepositionsandmotionsofloca- Chapters3.09and3.10. tionson theEarth’ssurfaceobtainedfromthe analysisof data Thefinalaspectofmeasuringchangesintherotationofthe fromthesesystemsareused in theformationoftheITRF.The Earth is the definition of rotation angles. All points on latest version of the ITRF is ITRF2008 (Altimini et al., 2011; the surface of the Earth move relative to the interior just as Altimini,2012). 4 Geodesy:AnIntroductionandOverview VLBIisamicrowave-basedmeasurementsystemthatmea- Chapter3.10.VLBIalsoprovidesaccuratepositionandatmo- suresthedifferenceinarrivaltimesofincoherentsignalsfrom sphericdelaymeasurements. radiosourcesbycrosscorrelation.Mostcommonly,theradio SLR isanoptical-based systemthatusesshort pulsedlaser sources are extragalactic objects but beacons from satellites andaccuratetimingequipmenttomeasuretheroundtripflight have also been used. Group and phase delays along with time between a ground system and a satellite equipped with Doppler shift are measured. The phase delays are difficult to special corner cube retroreflectors. Some laser systems have use in geodetic measurements with this system because they enough power to make range measurements to corner cube are measured modulo 2p, and reconstructing the number of arrayson the Moon. Specially equippedoptical telescopesare cyclesofphaseisnontrivialingeodeticmeasurements.Inthe used for these measurements. Measurement accuracies of astronomicalapplicationsofVLBI,theuseofthephasedelays 1–10mm are common for modern systems. The optical fre- iscommon.Individualdelaymeasurementshaveprecisionsof quencies used inthese systems are not affected by the Earth’s 1–10mm.Generally,radiotelescopeswithdiametersbetween ionosphere,andthecontributionfromatmosphericwatervapor 10and30mareusedforVLBImeasurements,andbecauseof ismuchlessthanformicrowavesystems(theopticalfrequencies theseparationofthetelescopes(thousandsofkilometers),the are too high to efficiently excite the dipole water vapor reso- delaymeasurementsaremaderelativetoindependenthydro- nance). Figure 2 shows the distribution of SLR sites used in genmaserclocksattheobservatories.Theradiosignalsprop- ITRF2008. Of these sites, about 35 are currently active. SLR agatethroughtheEarth’satmosphereandaredelayedthrough measurements do not need to be closely coordinated, and therefractiveindexoftheatmosphere’sgasconstituents.VLBI hence,SLRstationscanoperatesemiautonomously.Somesta- measurements are made at two relatively high frequencies tionsoperatecontinuously,whileothersoperateon8hshifts. (usually 2.3and8GHz)andaredelayedbythepropagation Thereareprioriessetforwhichofthecurrent26satelliteswith (cid:4) intheionosphere(removedwithadual-frequencycorrection corner cubes should be tracked when multiple satellites are sincethismediumisdispersive)andtheyarestronglyaffected visible.ThecoordinationofSLRmeasurementsisprovidedby by the dipole component of the refractivity of atmospheric theInternationalLaserRangingService(Pearlmanetal.,2002). water vapor. Figure 1 shows the distribution of VLBI sites SLRprovidesmeasurementstoabout26orbitingspacecraft included in ITRF2008. Of the sites shown, about 40 stations andtheMoon.Thesemeasurementsareusedtodeterminethe are currently operating. VLBI requires coordinated measure- positions of the SLR sites and the orbits of the spacecraft. ments so that distant telescopes look at the same objects at Perturbations to the spacecraft orbits are used to study the thesametime.MeasurementswithVLBIareusually24hdura- Earth’s gravity field and its temporal variations. For geodetic tion observing sessions using four to eight radio telescopes positioning, the most commonly observed satellites are the aroundtheworld.VLBIactivitiesarenowcoordinatedthrough pair of Laser Geodynamics Satellites (LAGEOS). These are theInternationalVLBIService(Schlueteretal.,2002). high area-to-mass ratio satellites orbiting at about 6000km VLBIistheonlymoderntechniquewithdirectreferencetoa altitude.Atthisaltitude,theorbitsaresensitivetothelower- stableinertialreferenceframe.Ituniquelycontributestomain- degree terms in the Earth’s gravity and measurements to tainingtimeasmeasuredbytherotationoftheEarthandto LAGEOS provided the first measurements of secular changes monitoringthemotionoftheEarth’srotationaxisininertial inthelower-degreegravityfieldcoefficients.Thealtitudeisalso space.Theselattermeasurementsprovideuniqueinsightsinto highenoughtobeonlyslightlyaffectedbyatmosphericdrag, the interaction of the fluid-outer and solid-inner cores and althoughtherearesomenotwell-understoodelectromagnetic the mantle of the Earth. This topic is discussed in detail in perturbationstotheorbits.Thehighaltitude oftheLAGEOS VLBI sites (113) 60(cid:2) 30(cid:2) 0(cid:2) -30(cid:2) -60(cid:2) 180(cid:2)270(cid:2) 0(cid:2) 90(cid:2)180(cid:2) Figure1 Locationsofthe113VLBIsitesthatareincludedintheITRF2008referenceframe.Ofthesitesshown,about40arestillactivelymaking measurements.MeasurementswithVLBIareorganizedintosessions,usuallyof24hduration,usingfourtoeightradiotelescopes. Geodesy:AnIntroductionandOverview 5 SLR sites (125) 60(cid:2) 30(cid:2) 0(cid:2) -30(cid:2) -60(cid:2) 180(cid:2)270(cid:2) 0(cid:2) 90(cid:2)180(cid:2) Figure2 Locationsofthe125SLRsitesthatareincludedintheITRF2008referenceframe.Ofthesitesshown,about35arestillactivelymaking measurements.Thesestationsobservecontinuouslywithpriorities,butnotdetailedscheduling,setbytheinternationallaserrangingservice. satellitesalsomeansthatthecenterofmassoftheEarthcanbe thisfiguretheadditional1000GPSsiteswhosedataarerou- welldeterminedfromthemeasurements.Theprimaryaimof tinely available in the international GPS data archives. GPS muchoftheSLRtrackingispreciseorbitdetermination(POD). stations track satellites continuously, are low power, and are For the altimeter missions discussed in Chapter 3.05, the suited to autonomous operation. Many thousands of GPS determinationoforbitswithsub-centimeteraccuracyiscritical. stations operate continuously around the world. The coordi- The GPS is a microwave group and phase delay measure- nation of station standards and data analysis is through the ment system using relatively low L-band signals ( 1.2 and InternationalGlobalNavigationSatelliteSystemService(IGS) (cid:4) 1.5GHz).Thesystemwasinitiallydesignedasamilitarynav- (Beutleretal,1999).InadditiontoGPS,theIGSalsoincludes igationsystemthatusedgroupdelays.Timetagsareencoded measurementsfromreceiversthattracktheRussianGLONASS intothetransmittedsignalandagroundreceivercanmeasure satellitesandinthefuturewillincludemeasurementsfromthe thedifferenceintimebetweenwhenasignalwastransmitted European Galileo system. All these systems use similar fre- fromasatellitebasedonthesatellite’sclockandwhenitwas quency bands and their satellites are in medium Earth orbit receivedonthegroundbasedonthegroundreceiver’sclock.If withaltitudesnear20000km(and12horbitalperiods). theclocksweresynchronized,thetimedifferencewouldbea GPSandtheotherGNSSsystemsprovideinexpensive,very measure of range. In the navigation application, accurate precise positioning capabilities on both static and moving clocks are placed on the satellites, and because the ground receivers.Thesystemsareoftendeployedindensenetworksto receiverscansimultaneouslymakemeasurementstomultiple monitortectonicdeformationswithcontinuousandoccasional satellites,thegroundreceiver’sclockerrorcanbeestimatedor occupations. The large numbers of stations available provide eliminated by differencing measurements. The ability ofGPS robustmeasurementsofpolarmotionandshort-periodvaria- receivers to make measurements simultaneously to multiple tionsinlengthofday.DensenetworksofGPSstationsarealso satellitesandthenarrowbandwidthofthetransmittedsignals usedtomonitoratmosphericwatervapor,andinmanycases, meansrelativelyinexpensiveelectronicscanbeusedtomake results are used in operational weather forecasting. The high very precise measurements. In the geodetic applications of dataratesavailablewithGPSwithsamplingfrequenciesupto GPS, the phase measurements are used, which is possible 50Hz make the system ideal for tracking fast-moving objects because multiple ground stations can see the same satellites, suchasaircraftandformonitoringseismicevents.Chapter3.11 andbyeitherestimationordifferencing,thecontributionsof discussesmanyoftheapplicationsofGPSalongwiththeother localoscillatorphasesinthereceiversandtransmitterscanbe spacegeodeticpositioningsystems. eliminated.Phasemeasurementsofrangechangesaccuratetoa The DORIS system is also a microwave system that uses few millimeters are possible. Dual-frequency measurements two-wayDopplertracking.Unliketheothergeodeticsystems, are used to eliminate the ionospheric delays, but because of DORISstationshaveactivecommunicationwiththesatellites the low frequency, the dual-frequency corrections are not beingtracked.Thesatellitesinitiatecommunicationswiththe always adequate. The second-order effects from the Earth’s groundstations.Thissystemallowstwo-waycommunications magneticfieldneedtobeaccountedforinthemostaccurate thatcancancelerrorsduetodifferentfrequenciesintheoscil- analyses. Continuous tracking of signals from the satellites latorsinthespacecraftandgroundstations.DORISisprimarily allowsthenumberofcyclesinthephasetobecounted,and usedforPODandplaysacriticalroleintrackingsatellitesused onlywhenthesignalfromthesatelliteisinterruptedorwhena forradaraltimetrydiscussedinChapter3.05.Earthorientation satelliteis firstseen isitnecessarytoestimatethenumber of measurements,center-of-massvariations,andstationlocations integer cycles in the phase. Figure 3 shows the GPS sites arealsodeterminedintheanalysisofDORISdata.Theactivi- included in the ITRF2008 reference frame. We also show on tiesofDORISarecoordinatedundertheInternationalDORIS 6 Geodesy:AnIntroductionandOverview GNSS sites (560) 60(cid:2) 30(cid:2) 0(cid:2) -30(cid:2) -60(cid:2) 180(cid:2)270(cid:2) 0(cid:2) 90(cid:2) 180(cid:2) Figure3 Locationsofthe560GPSsitesthatareincludedintheITRF2008referenceframe(redsquares).Therearemanyadditionalsiteswhosedata aremadefreelyavailablethroughinternationaldataarchives.Thesesitesoperatecontinuously. DORIS sites (130) 60(cid:2) 30(cid:2) 0(cid:2) -30(cid:2) -60(cid:2) 180(cid:2)270(cid:2) 0(cid:2) 90(cid:2)180(cid:2) Figure4 Locationsofthe139DORISsitesthatareincludedintheITRF2008referenceframe.Thesesiteshaveactivecommunicationbetweenthe groundstationsandthesatellitesthataretrackedwithDORIS. service (Tavernier et al., 2006). In Figure 4, the DORIS sites withJason-1usedprimarilyforoceanstudiesandICESatused usedintheITRF2008areshown. primarily for ice sheet studies. Chapter 3.05 discusses results from these and earlier altimeter satellites that are no longer operating.TheCHAMPandGRACEsatellitesareusedforgravity fieldstudieswithGRACEbeingapairofsatelliteswithaccurate 3.01.3.2 SatelliteSystems rangeandrange-ratemeasurementsbetweenthem.Thesemea- Awidevarietyofsatellitesystemsareusedinmoderngeodetic surementsprovide veryaccurate determinationsofchanges in measurements.Thosethatarecurrentlybeingtrackedwithlaser theEarth’sgravityfieldasdetailedinChapter3.08.TheERSand rangingsystemsarelistedinTable1.Theapplicationsofsatel- Envisat satellites carry synthetic aperture radars, which when litesfallintodifferentcategories.Fiveoftheentriesarereflector usedinaninterferometricmode,asdiscussedinChapter3.12, arrayson the Moon, three from the Apollo program and two allow high-spatial-resolution maps of deformation fields or fromtheRussianLunaprogram.AlloftheRussianGLONASS topographytobemade. satelliteshavecornercubearrays,asdotwooftheGPSsatellites. TheGPSarraysareusedforaccuracyevaluation.TheLAGEOS, Starlette,andStellaspacecraftarepassive,sphericalbodiescov- 3.01.3.3 Gravimeters ered with corner cube reflectors and are primarily used for gravity field determination and positioning. The Jason-1 and Gravimeter instrumentation for precise geodesy is classed as ICESat satellites are microwave and laser altimeter satellites absolute gravimeters that normally use the acceleration of an Geodesy:AnIntroductionandOverview 7 Table1 ActivegeodeticsatellitesbeingtrackedwithSLR Satellitename SatelliteID Altitude(km) Inclination(deg) Firsttrackeddate Ajisai 8606101 1485 50 13August1986 Apollo11SeaofTranquility 0000100 356400 5 20August1969 Apollo14FraMauro 0000102 356400 5 07February1971 Apollo15HadleyRille 0000103 356400 5 01September1971 Beacon-C 6503201 927 41 02January1976 COMPASS-G1 1000101 42164 55.5 28April2012 COMPASS-I3 1101301 42161 55.5 27April2012 COMPASS-I5 1107301 42161 55.5 06July2012 COMPASS-M3 1201801 21528 55.0 11July2012 CryoSat-2 1001301 720 92 20April2010 Envisat 0200901 800 98 10April2002 Etalon-1 8900103 19105 65 26January1989 Etalon-2 8903903 19135 65 13July1989 GIOVE-A 0505101 23916 56 11May2006 GLONASS-102 0606201 19140 65 04May2007 GLONASS-109 0706503 19140 65 04May2007 GLONASS-110 0804601 19140 65 10December2009 GLONASS-118 0907003 19140 65 04January2010 GLONASS-129 1106402 19140 65 02January2012 GLONASS-130 1107101 19140 65 01January2012 GOCE 0901301 295 96.7 01April2009 GPS-36 9401601 20030 55 21April1994 GRACE-A 0201201 485–500 89 18March2002 GRACE-B 0201202 485–500 89 18March2002 Galileo-101 1106001 23220 56 29November2011 Galileo-102 1106002 23220 56 29November2011 Galileo-103 1205501 23220 56 07November2012 Galileo-104 1205502 23220 56 07November2012 HY-2A 1104301 971 99.35 02October2011 IRNSS-1A 1303401 42164 29 05September2013 Jason-2 0803201 1336 66 24June2008 KOMPSAT-5 1304201 550 97.6 09September2013 LAGEOS-1 7603901 5850 110 10May1976 LAGEOS-2 9207002 5625 53 24October1992 LARES 1200601 1450 69.5 17February2012 LRO-LR 0903101 50(lunar) 90(wrtlunarequator) 30June2009 (cid:4) (cid:4) Larets 0304206 691 98.204 04November2003 Luna17SeaofRains 0000101 356400 5 21May1975 Luna21SeaofSerenity 0000104 356400 5 16November1973 QZS-1 1004501 32000–40000 45 11September2010 RadioAstron 1103701 500–350000 51.4 15November2011 SARAL 1300901 814 98.55 04March2013 STPSat-2 1006201 650 72 02April2013 STSAT-2C 1300301 300–1500 80 29March2013 Starlette 7501001 815 50 03January1976 Stella 9306102 815 99 30September1993 TanDEM-X 1003001 514 97.44 21June2010 TerraSAR-X 0702601 514 97.44 16June2007 Listavailablefromandupdatedathttp://ilrs.gsfc.nasa.gov/missions/satellite_missions/current_missions/index.html. Source:InternationalLaserRangingService. object falling in vacuum or superconducting instruments that uplift or secular mass movements. These instruments are nor- measurechangesingravityonatestmassthatiscooledtovery mallymovedfromplacetoplacetomakerepeatedmeasurements low temperatures to minimize the random accelerations from oftenovermanyyears.Thesuperconductinggravimetersprovide thermalnoise.Theapplicationsanddevelopmentoftheseclasses continuoushigh-precisiongravitymeasurementsatobservatory of gravimeters are discussed in Chapters 3.03 and 3.04. The locations. They are suitable for tidal studies, particularly the absolute gravimeters provide absolute measurements that are effectsofthefluidcoreontidalamplitudes,seismicmodesfrom suitableformonitoringlong-termchangesingravityduetoeither largeearthquakes,andatmosphericmasschangeexcitations. 8 Geodesy:AnIntroductionandOverview 3.01.4 ErrorSources,Signals,andNoise chaptersprogressthroughthedifferentareasofgeodesystart- ingwiththetheoryoftheEarth’sgravityfield,followedthenby Severalcommonthemesrunthroughthechaptersinthisvol- ground-based measurements of gravity and space-based alti- ume.Thefirstoftheseisthatgeodesyisthecombinationofan metric measurements. The accuracy of the modern measure- observationalscienceandatheoreticalscience.Duringvarious ments allows temporal variations in gravity to be measured, stages of its history, either new theories or new observations andthistopicisdiscussedinthreechaptersdealingwithtidal determinethedirectionofthescience.Duringthelast20years, variations in gravity, short-term variations that can now be observationalaccuracieshaveoftenbeendrivingnewtheoret- measured from space, and long-period (secular) variations ical developments. The other critical aspect of modern mea- that are related to the viscoelastic response of the Earth to surements is the errors in these measurements. At times, unloading of ice mass that occurred at the end of the last assessingerrorlevelshasbeendifficult,especiallywhenaccu- Pleistoceneiceage.Ofgreatinterestcurrentlyistheseparation racy is considered, because there have been no standards to ofthissecularsignalfromthesignalduetocurrentmeltingof comparewith.Manyofthechaptersinthisvolumediscussthe ice sheets and glaciers. The combination of gravity measure- errorspectraofgeodeticmeasurementsinthetimeandspace ments and surface uplift measurements shows promise for domains.Inmanycases,acontributionthatcanbeconsidered allowingthisseparation.Thesamemassmovementsthateffect noise for one user may be a signal for others. The effects of time-dependent gravity also change the rotation of Earth on water vapor and liquid water fall into the category of being manytimescales.TwochaptersaddressEarthrotationchanges bothnoiseandsignal. in inertial space and relative to the crust of the Earth. The An example of how theoretical and observational drivers volumeconcludeswiththetheory,instrumentation,andappli- havechangedovertheyearsisthestudyofnutation.Formuch cationsofmodernpositionmethodsandsurfacedeformation ofgeodetichistory,themotionoftherotationaxisinspacewas monitoring. computedtheoreticallyfromtheephemeridesoftheMoonand SunandknowledgeofthemomentsofinertiaoftheEarth.At thetimethatVLBIsystemswerebeingdeployedworldwideand References usedonaregularbasis,anewnutationtheorywasadoptedby the International Astronomical Union (IAU) called the IAU AltamimiZ,CollilieuxX,andMe´tivierL(2011)ITRF2008:Animprovedsolutionofthe 1980 theory of nutation. This new theory included effects of InternationalTerrestrialReferenceFrame.JournalofGeodesy85(8):457–473. elasticityoftheEarth(derivedfromseismicobservations)and http://dx.doi.org/10.1007/s00190-011-0444-4. afluidcorewhoseshapewascomputedfromtheassumption AltamimiZ,SillardP,andBoucherC(2002)ITRF2000:Anewreleaseofthe InternationalTerrestrialReferenceFrameforearthscienceapplications. of hydrostatic equilibrium throughout the Earth. Just a few JournalofGeophysicalResearch107(B10):2214.http://dx.doi.org/ years after the adoption of this theory, VLBI observations 10.1029/2001JB000561. revealed discrepancies between the predictions of the theory AltiminiZ(2012)http://itrf.ensg.ign.fr/ITRF_solutions/2008/more_ITRF2008.php. andmeasurements,whichresultedinanewassessmentofthe BeutlerG,RothacherM,SchaerS,SpringerTA,KoubaJ,andNeilanRE(1999)The InternationalGPSService(IGS):Aninterdisciplinaryserviceinsupportofearth theory.Somepartsoftheassessmentshowedthattheoriginal sciences.AdvancesinSpaceResearch23(4):631–635.http://igscb.jpl.nasa.gov. theories were truncated at too coarse a level for the modern BlewittG(2003)Self-consistencyinreferenceframes,geocenterdefinition,andsurface VLBIaccuracies,andotherpartsshowedthatthetheoryneeded loadingofthesolidEarth.JournalofGeophysicalResearch108(B2):2103.http:// toextendtoincludetheeffectsofnonhydrostaticforces,mag- dx.doi.org/10.1029/2002JB002082. neticcoupling,andthepresenceofthesolid-innercore.These Lavalle´eDandBlewittG(2002)Degree-1Earthdeformationfromverylongbaseline interferometrymeasurements.GeophysicalResearchLetters29(20):1967.http://dx. developments are discussed in Chapter 3.10. Chapter 3.04 doi.org/10.1029/2002GL015883. discussesaverydifferent measurementtypethatcan beused MaC,AriasEF,EubanksTM,etal.(1998)TheInternationalcelestialreferenceframe to address the theory. Currently, there are theoretical predic- asrealizedbyverylongbaselineinterferometry.AstronomicalJournal tions about the conductivity structure at the core–mantle 116:516–546. PearlmanMR,DegnanJJ,andBosworthJM(2002)TheInternationalLaserRanging boundarythatgenerateresultsconsistentwithVLBImeasure- Service.AdvancesinSpaceResearch30(2):135–143.http://ilrs.gsfc.nasa.gov. mentsthatneedtobeassessedbyanindependentmethod. SchlueterW,HimwichE,NothnagelA,VandenbergN,andWhitneyA(2002)IVSandits importantroleinthemaintenanceoftheglobalreferencesystems.Advancesin SpaceResearch30(2):145–150.http://ivscc.gsfc.nasa.gov. 3.01.5 Conclusions SmithJR(1996)IntroductiontoGeodesy:TheHistoryandConceptsofModern Geodesy.WileySeriesinSurveyingandBoundaryControl,p.224.NewYork:Wiley. StaceyFD(1992)PhysicsoftheEarth,3rdedn.Brisbane:BrookfieldPress,pp.513. ModerngeodesyinteractswithnearlyallfieldsofEarthscience, TavernierG,FagardH,FeisselM,etal.(2006)TheInternationalDORISService:Genesis andthechaptersinthisvolumeshowtheseinteractions.The andearlyachievements.JournalofGeodesy80(8–11):403–428.http://ids.cls.fr/. 3.02 Potential Theory and the Static Gravity Field of the Earth CJekeli,TheOhioStateUniversity,Columbus,OH,USA ã2015ElsevierB.V.Allrightsreserved. 3.02.1 Introduction 9 3.02.1.1 HistoricalNotes 9 3.02.1.2 CoordinateSystems 10 3.02.1.3 PreliminaryDefinitionsandConcepts 11 3.02.2 Newton’sLawofGravitation 12 3.02.3 Boundary-ValueProblems 14 3.02.3.1 Green’sIdentities 15 3.02.3.2 UniquenessTheorems 16 3.02.3.3 SolutionsbyIntegralEquation 17 3.02.4 SolutionstotheSphericalBVP 17 3.02.4.1 SphericalHarmonicsandGreen’sFunctions 17 3.02.4.2 InverseStokes’andHotineIntegrals 20 3.02.4.3 Vening-MeineszIntegralandItsInverse 21 3.02.4.4 ConcludingRemarks 22 3.02.5 Low-DegreeHarmonics:InterpretationandReference 23 3.02.5.1 Low-DegreeHarmonicsasDensityMoments 23 3.02.5.2 NormalEllipsoidalField 24 3.02.6 MethodsofDetermination 26 3.02.6.1 MeasurementSystemsandTechniques 26 3.02.6.2 Models 30 3.02.7 TheGeoidandHeights 31 References 33 3.02.1 Introduction the mass of the falling object – all bodies fall with the same acceleration. This truly monumental contribution to physics Classicalgravitationalpotentialtheoryhasitsrootsinthelate was,however,onlyalocalexplanationofhowbodiesbehaved renaissanceperiodwhenthepositionoftheEarthinthecosmos undergravitationalinfluence.JohannesKepler’s(1571–1630) was established on modern scientific, observation-based observationsofplanetaryorbitspointedtoothertypesoflaws, grounds. A study of Earth’s gravitational field is a study of principally an inverse-square law according to which bodies Earth’s mass including its transport in time and its influence are attracted by forces that vary with the inverse square of onnearobjects;and,fundamentally,itisageodeticstudyof distance.ThegeniusofIsaacNewton(1642–1727)broughtit Earth’sshape,whichisdescribedlargely(70%)bythesurface alltogetherinhisPhilosophiæNaturalisPrincipiaMathematicaof oftheoceans.Thisinitialsectionprovidesahistoricalbackdrop 1687with a single andsimple all-embracing lawthatinone toNewtonianpotentialtheoryandintroducessomeconcepts bold stroke explained the dynamics of the entire universe inphysicalgeodesythatsetthestageforlaterformulations. (today, there is more to understanding the dynamics of the cosmos, but Newton’s law remarkably holds its own). The mass of a body was again an essential aspect, not as a self- 3.02.1.1 HistoricalNotes attributeasAristotlehadimplied,butasthesourceofattraction Gravitation is aphysical phenomenon so pervasive and inci- forotherbodies:eachmaterialbodyattractseveryothermate- dentalthathumankindgenerallyhastakenitforgrantedwith rial body according to a very specific rule (Newton’s law of scarcely a second thought. The Greek philosopher Aristotle gravitation; see Section 3.02.2). Newton regretted that he (384–322BC)allowednomorethantoassertthatgravitation couldnotexplainexactlywhymasshasthisproperty(asone isanaturalpropertyofmaterialthingsthatcausesthemtofall stillyearnstoknowtodaywithinthestandardmodelsofparti- (orrise,inthecaseofsomegases)andthemorematerial,the cleandquantumtheories).EvenAlbertEinstein(1879–1955) greaterthetendencytodoso.Itwasenoughofaself-evident indevelopinghisgeneraltheoryofrelativity(i.e.,thetheoryof explanation that it was not yet to receive the scrutiny of the gravitation)couldonlyimproveonNewton’stheorybyincor- scientific method, the beginnings of which, ironically, are poratingandexplainingactionatadistance(gravitationalforce creditedtoAristotle.Almosttwothousandyearslater,Galileo acts with the speed of light as a fundamental tenet of the Galilei (1564–1642) finally took up the challenge to under- theory).Ofcourse,Einsteindidmuchmorebyreinterpreting standgravitationthroughobservationandscientificinvestiga- gravitationalattractionasawarpingofspaceandtime,which tion.Hisexperimentallyderivedlawoffallingbodiescorrected thenallowsconsiderationofgravitationeffectsathighvelocity theAristotelianviewanddivorcedtheeffectofgravitationfrom andduetogreatquantitiesofmass(e.g.,blackholes).However, TreatiseonGeophysics,SecondEdition http://dx.doi.org/10.1016/B978-0-444-53802-4.00056-7 9

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