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Vsub1(r,,)=`GMsub1overr```  Ѐ{stack{#scalesym150SIGMA#n=0}}``LEFT(asub1overrRIGHT)supn`  {stack{n#scalesym150SIGMA#m=n}}{OVERLINEC}sub{nm}{OVERLINEY}sub{nm}(,)((3$ !      0  (#$  0  <09Z .Courier New Regular N=Rover{4pi}{vert50stack{scalesym200intscalesym200int#%}}(gsub{TB}+`0.1119H)`S(1)`d%{piG#H  sup2}over{}XX#NXXX-XXRXXC4XX!XX#XVLXV)LXXG%XX(XXXXmgBTBXXmXX0XX_.XX1119XX!HXX)XX SXXm (XX 1XX7 )XX dXX %XXc  X XX !XXy GXX #XXw H'@2XX  ##Xd# N=`GMover{r}```  Ѐ{stack{360#scalesym150SIGMA#n=2}}``LEFT(aoverrRIGHT)supn`{stack{n#scalesym150SIGMA#m=n}}{OVERLINEF}sub{nm}{OVERLINEY}sub{nm}(,) 6?7?{8?9?E:?;??E??PXXP XXPNXX P8X~XXGMXXrXXEXXU[360:$XXKFnXXFXX'F2XXhXXOjXXiXXoXXOqXXpX2~XXHaXXSrLnXXf[n3:$XXFmXX6FXXFXX&FnXXXPF nm]X XX PY nmXX P(XX PXXr P,XX PXX P) 0T(r,,)=`GMsub2overr```  Ѐ{stack{#scalesym150SIGMA#n=0}}``LEFT(asub2overrRIGHT)supn`  {stack{n#scalesym150SIGMA#m=n}}LEFT[``{OVERLINEE}sub{nm}`{OVERLINED}sub{nm}RIGHT]``{OVERLINEY}sub{nm}(,)XXPTXXP(XXPrXXP,XXOPXXP,XXPXXeP)XXPXA~XXWGM2XXrXX[b:$XX'FnXXFXXF0XXhXXjXXOjXXiXXoXXqXXOqXXp.X~XX$a2XXPrj]nXX [nQ :$XXFmXXT FXX FXXD FnXH tXH {X XX PEh nmXX P X XX PD* nm]XXXPYnmXX.P(XXpPXXP,XXPXXP) &N= +{gsubB}over`hXXNXXXX XX}tXXXXXgFBXX}GXXh ?{2SUBa}overa`[1+f+m+(3f+5over2m)`sinSUP2-]X/XXE2XXFaXXGaXX[XX1XX5XXfXXXX]mXXXXe(XXXX3XXfXXX?XXUV5XXUG2XXmXX_)XXsin52XX-XXi ] {OVERLINED}subnm X$XXxD7nm  N=`Rover{4pi}{vert50stack{scalesym300intscalesym300int#%}}```  ЀLEFT[GMoverr```{stack{360#scalesym150SIGMA#n=2}}LEFT(aoverrRIGHT)supn`LEFT({n1}overr  RIGHT){stack{n#scalesym150SIGMA#m=n}}{OVERLINEF}sub{nm}{OVERLINEY}sub{nm}(,)RIGHT]`  ЀLEFT[vert5{stack{#scalesym150SIGMA#n=2}}vert5{2n+1}over{n1}`Psubn(cos`1)RIGHT]``d%XX XXNXX XXX*`RXXQ4XX!Q!XXQ'LLXXQ%XXvXXxXXVxXXxXXxXXxXXwXX}XXXXVXXXXXXXX~8XXX+`GMXXQrXX360$XXnXXXX2XX5hXXjXX!iXX 5oXX qXX !pXtXX`aXXQr nXX 5hXX jXX !iXX 5oXX qXX !pnXc XXy `nXX `XXU `1XX QrXX n $XXD mXX XXL XX nX(XX(Fnm]XFXXFYnmXXZ(XXXX,XXBXX)XX<vXX<xXX<VxXX<xXX<xXX<xXX<wXX}XXXXVXXXXXXXX~XX$XXhnXXXXD2 XXX`2XX.`nXX`XX `1XXQnXX`QXXQ1XXPnXX](XXcosXX1XXG)XXdXXK%>       dXX]V 1XX](XX]rXXg],XX]XX ],XX?]XX])XX]XXX GM1XX6rXX{[$XXqXnXXXXXMX0XXhXXjXXjXXFjXXiXX*oXX*qXX*qXX*FqXX*p.XXXXn a1XXrsnXX {n [$XX XmXX XXX XXX XnX XX ]Cx nm]X XX ]Y nmXX0 ](XXr ]XX ],XX]XX]) \Vsub1(r,,)=`GMsub2overr```  Ѐ{stack{360#scalesym150SIGMA#n=0}}``LEFT(asub2overrRIGHT)supn`  {stack{n#scalesym150SIGMA#m=n}}`LEFT(GMsub1overGMsub2RIGHT)`LEFT(asub1overasub2RIGHT)supn`{OVERLINEC}sub{nm}{OVERLINEY}sub{nm}(,) Vsub2(r,,)=`GMsub2overr```  Ѐ{stack{#scalesym150SIGMA#n=0}}``LEFT(asub2overrRIGHT)supn`  {stack{n#scalesym150SIGMA#m=n}}{OVERLINED}sub{nm}{OVERLINEY}sub{nm}(,)XXPV2XXP(XXPrXXiP,XXPXXP,XXAPXXP)XXPX~XXGM2XX8rXX[:$XXsFnXXFXXOF0XXhXXjXXOjXXiXX,oXX,qXX,OqXX,p.XZ~XXpa2XXr]nXX [n :$XX FmXX FXX FXX Fn X XX PD nm]X( XX( PY nmXX< P(XX~ PXX P,XX$PXXP)*+,.M << deUULevel 1Level 2Level 3Level 4Level 5(3$ !  (n$ (  1  )  Vsub1(r,,)=`GMsub2overr```  Ѐ{stack{#scalesym150SIGMA#n=0}}``LEFT(asub2overrRIGHT)supn`  {stack{n#scalesym150SIGMA#m=n}}{OVERLINEE}sub{nm}{OVERLINEY}sub{nm}(,)'dxd+M -, dc$$$$'dxdXXPV1XXP(XXPrXXiP,XXPXXP,XXAPXXP)XXPX~XXGM2XX8rXX[:$XXsFnXXFXXOF0XXhXXjXXOjXXiXX,oXX,qXX,OqXX,p.XZ~XXpa2XXr]nXX [n :$XX FmXX FXX FXX FnX XX PEn nm]X XX PY nmXX& P(XXh PXX P,XXPXX~P) {OVERLINEC}subnmX$XXxC7nm {OVERLINEY}subnm]X$XXxY7nmXXPV1XXP(XXPrXXiP,XXPXXP,XXAPXXP)XXPX~XXGM2XX8rXX}[360:$XXsFnXXFXXOF0XXhXXjXXOjXXiXX,oXX,qXX,OqXX,p.XZ~XXpa2XXr]nXX [n :$XX FmXX FXX FXX FnXX hXX jXX cjXX jXX OiXX) oXX) qXX) cqXX) qXX) OpX ~XX GM 1XX GM o2XX hXX jXX cjXX jXX OiXXoXXqXXcqXXqXXOp.X/~XXEa1XXEao2qnXXXPCknm]XXXPYnmXX#P(XXePXXP,XX PXX{P) N``=``Rover{4pi}{vert50stack{scalesym200intscalesym200int#%}}g`S(1)`d%``=``Rover{4pi}{vert50  stack{scalesym200intscalesym200int#%}}(ggQsub360)`S(1)`d%``+``Rover{4pi}{vert50stack{scalesym  200intscalesym200int#%}}gQsub360`S(1)`d%XX)KNXXKXyXXRXX4XX!XXXV LXVLXXO%XXKXXKgXXKSXXgK(XXK1XX1K)XXKdXXK%XXKX9yXXRXXO4XX!XX/ XV LXV5 LXX %XX K(XX KXXy KgXX KXXU KXX KgXX? KQ{ 360XXAK)XXKSXXK(XX?K1XXK)XXKdXXK%XXKXyXXRRXX4XXI!XXXVOLXVLXX%XXGKXXKgXX1KQm 360XXIKSXXK(XXK1XXwK)XXKdXX3K%TABLE B   сLevel 1Level 2Level 3Level 4Level 5 3SUBaoverasup2XX3X5XXfaXXGa'2 shatXX2GXXGs Nsub+={R-}over{4!}`Fsup{1}[F(gsub{+``Q}`cos-subQ)`F(S(-subP-subQ`,`subPsubQ))]XXNXXXwXXVRXXVXXV-XXVXXVXX!G4XXG!XXGXX5F551XXa[XXFXX(XXaXXgKQXX.cosXX8 - QXX )XXv FXX (XX2 SXX (XX -^ PXX XX( - QXX$,XXlPXX.XXQXXv)XX)XX]   0$-% 0shat`=`CSUBst`(CSUBtt+`CSUBnn)SUP{1} S(1)=1over{sin1over2}46{sin1over2}+10{sinsup21over2}(36{sinsup21over2})ln({sin1over2}+{sin  sup21over2})XXSXX(XX1XXK)XX X$XXe1XX'sinXXX3V1XXEG2XXXX_4XXXX;6XXsinX$XXe1XXV2XXIXX10XXsins D2X $XX e1XX V2XXu XX (XX/ 3XX XX 6XXo sinY D2X $XX e1XX V2XX[)XXlnXX9(XX{sinXq$XXe1XXV2XX%XXsinD2X$XXe1XXV2XX)XX2xXXxsXX{xXX xC7stXXx(XXFxC7ttXXxXXxC67nnXXx)P1 )C=CSUB0`eSUP{(dSUP2`/`LSUP2)}XXxCXXxXXxC70XXxek(d,,@2/L,,2C)Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5A+T.&OLE 2.0 Box <=8C HKKKK( $ Figure  1   {OVERLINEE}subnmX$XXxE7nm B{OVERLINEF}subnm=`{OVERLINEE}subnm``{OVERLINED}subnmX$XXxF7nmXXJxX$XXxEg7nmXX x X$XXxDT7nm !  ݛXX   7XXdd71  HP@r#r"P EvaluationofPreliminaryModelsoftheGeopotentialintheUnitedStates #XX#  >  XX#XX#D.A.SmithandD.G.Milbert  , NationalGeodeticSurvey,NOAA,SSMC3,1315EastWestHwy.,SilverSpring,MD20910XX    #XXQ# v &  Introduction  |    AsmembersoftheSpecialWorkingGroupoftheInternationalGeoidService,wehaveevaluatedthepreliminarygeopotentialmodelsX01throughX05;computedbythejointNASAGSFC/DMAproject(RappandNerem1994)'| .OurfocushasbeenthebehaviorofthesemodelsintheconterminousUnitedStates.Inthiseffort,ourstrategyhasbeentocomparegeoidmodelsderivedfromgeopotentialcoefficientsagainstGPSellipsoidheightslocatedonopticallyleveledbenchmarks.Thiscomparisonisdoneintwoways.Inoneapproach,theGPSbenchmarksarecomparedagainstthegeoidvaluesobtainedfromwhatisessentiallydirectevaluationofthepreliminarymodels.Thisapproachsuffersfromtheomissionortruncationerrorinherentinann=360modelcomputation.Intheotherapproach,thepointgravityanddigitalterraininandaroundtheUnitedStatesareusedtocomputehighresolutiongeoidmodelsinremove,compute,andrestoreproceduresaboutthevariouspreliminarymodels.Whilethisapproachalleviatestheomissionerrorproblem,itisseentoremoveportionsofthegeopotentialmodelcommissionerror.  Inthisreport,webeginbyoutliningafewimportantpointsinthecomputationofgravimetricquantitiesfromgeopotentialcoefficients.Next,wereviewthefeaturesoftheGPSbenchmarkdatasetthatwasusedinevaluationofthepreliminarymodels.Wediscussthecomputationprocedureusedtoobtainthehighresolutiongeoidmodels,anddisplayresultswiththeGPSbenchmarkcomparison.Then,wecomparethegeopotentialmodelsdirectlyagainsttheGPSbenchmarks,withouttheremovalofomissionerror.Weclosethereportwithconclusionsandrecommendations. &  GeoidHeightComputation  "L    ItstandstoreasonthatifonechoosestoevaluategeopotentialcoefficientsbymeansofGPSonleveledbenchmarks,thenonemustusetheutmostrigorincomputationofthegeoidheightsfromthecoefficients.Towardsthisgoal,wehavedevelopedformulationstosupporttheuseofgeopotentialcoefficientscomputedwithanarbitrarysetofnormalellipsoidconstants'" ,andtosupportthecomputationofvariousgravimetricquantitiesrelativetoadifferent,arbitrarysetofnormalellipsoidconstants(whichwetakeasGRS80).Inaddition,itmustberecognizedthatthetheoryusedtoestablishthegeopotentialcoefficientsisrootedinasolutionofLaplacesequation.However,Laplacesequationonlyholdsinavacuum.ThegeoidoverlandistypicallyinteriortotheEarthstopography,wherePoissonsequationwouldoperate. L-(+ 17XXdXXd7    Webeginbyconsideringtheevaluationofgravimetricquantitiesfromspherical h harmoniccoefficientswitharbitraryvaluesofGManda.Thetotalexternalgravitational X potentialoftheEarthcanbeexpressedas:̜k)%5%!`|l `E lIO#8 k (#(#(#(# (#(#(#(#where:  r,,=geocentricradialdistance,colatitude,longitude p  Ѐ  GM1,a1=scalefactorsassociatedwiththeCnmvalues `  Ѐ  W/34! `EmRmRmRW=fullynormalizedcoefficientsofthetotalfield P  Ѐ  W/56! `EzWRWRzWRW=fullynormalizedLegendrefunctions @  ЀAnd,thenormalexternalgravitationalpotentialoftheEarthcanbeexpressedas:k)'(5%!`|l `ElBWxk (#(#(#(# r  (#(#(#(#where:GM2=GravitationMassconstantofthenormalfield  Ѐ  a2=semi-majoraxisoftheellipsoidandwhichis,ofitself,a"spheropotential"surface,  Ѐwithconstantvalueofnormalgravitypotential,Uo.  ЀW/! `EwRwRwRWߛ=fullynormalizedcoefficientsofthenormalfield    (=0ifnisnotpositiveeven,ormisnotzero) p ThetotalexternalgravitypotentialoftheEarthis:  W(r,,)=V1(r,,)+,(r,) !0 andthenormalexternalgravitypotentialoftheEarthis:  U(r,,)=V2(r,,)+,(r,) X%! Ѐwhere,(r,)isthecentrifugalpotential. 8' # Wemakethefollowingassumptions:  1)r,,arethesameinbothequations,thatis,theoriginsofthetwofieldscoincide,as *$' dotheirthreeCartesianco-ordinateaxes.  ,p&)   2)thevalueofnormalgravitypotentialontheellipsoid(Uo),isequivalent,numerically, h  tothevalueoftotalgravitypotentialonthegeoid(Wo) X   3)Thetotalandnormalcentrifugalpotentialsareequivalent,thusthedisturbingpotential,willbe:T=W-U=V1-V2 (    Uponevaluatingthedisturbingpotential,wepaycarefulattentiontothedifferingvaluesofGM1versusGM2anda1versusa2.WestartbyrewritingtheequationforV1as(andtruncating   theseriesatthegivenhighvalueof360):̜k)&95%!`| `EW<vPk (#(#(#(#   (#(#(#(#wherethescaledcoefficientswillbeexpressedas:  W/UV! `E*aRaR*aRWߛ=(GM1/GM2)*(a1/a2)nCnm= scale (n)Cnm X Thus:k)-15%!`| `EpW(Bk (#(#(#(# ^  (#(#(#(#Now,withacommonGM2anda2value"outfront",wecannowwritetheformulaofthe ` disturbingpotentialas:k)5%!`| `E?!lW Bk (#(#(#(# ^  (#(#(#(#  OfprimaryimportancehereisthefactthatthedegreezerotermisnonzeroifGM1cGM2. H&" InthecaseofGM1forOSU91A(Rappetal.1991)versusGM2ofGRS80(Moritz1992),this 8' # effectisontheorderof1meter,ingeoidundulation.  Ofsecondaryimportanceisthescalingofthecoefficients.Thescalefactorsareontheorderof10-4,andthiseffecthasbeenshowninsomeofourteststohaveanRMSof0.3mm, *$' withmin/maxvaluesof-0.3mmand+0.5mmforthegeoid.Whileanyoneeffectofsub-mmlevelwillnotadverselyeffectourresults,wefeelthatthenumberofsuchapproximationsthatcurrentlyexistinthetheoryneedtoberemovedtopreventthepossibilityoftheircombined -`'* effectsbeingnoticeableatthe1cmlevel.Itshouldbenotedthatthiseffectofproperlyscalingthecoefficientsisnotencompassedinasimplegeometrictransformation,butrepresentsafundamentalcorrectiontothewaythatasphericalharmonicexpansionisevaluated.  Ingeneralremarks,wearealsocarefulaboutthedifferencebetween,/,rand,/,h.The (  differencebetweenthetwoderivativeswillhavesmall(about5ppm),butcalculableeffects,onquantitiesofinterest(gravityanomalies,upwardderivativesoftheheightanomaly,etc...).And,whereverpossible,exact(closed)formulasoriterationsareperformed(forexample,computingtheflatteningfromJ2,3,GM,anda),ratherthanusinganyapproximateformulas.Considering   thespeedwithwhichiterationscanbeperformed,wefeltitwasbesttoremoveasmanyapproximateformulasaswasreasonablypossible.  Next,weconsidertheissuethatgeopotentialcoefficientsdonotaccuratelyrepresentthebehavioroftheactualgeopotentialinteriortotheEarthssurface.ThisproblemwasraisedinRapp(1992),andamplifiedinRapp(1996).Basically,onemustevaluatethegeopotentialonorabovetheEarthssurface,andthendevelopthegeoidundulationfromthatcomputation.Itisconvenient,therefore,tofirstcomputeheightanomalies, ,andthenconvertthemtogeoid h undulation,N,bymeansofEq.(810)ofHeiskanenandMoritz(1967) X k)5%!`|X `EX ?H  k o(#o(#(#(#&   (#(#o(#o(#wherègB  simple(nonterraincorrected)Bougueranomaly'+) p .    ApopularchoiceingeopotentialcoefficientevaluationistheClenshawsummationapproachdescribedinTscherning,etal(1983).Thisimplementation,however,mustreformcertainsumswheneverthegeocentricradius,r,varies.Toaidincomputationalefficiency,Rapp P (1996)computestheheightanomalyontheellipsoid, o,andthenupwardcontinuestothe  @ surfacebytheorthometricheight,H, !0   P= o+, /,rH. x# Inanenhancementtothisapproach,weperformtheupwardcontinuationby  P= o+, /,hh 8' # ̜where,aftersupressinganegligibledependence(0.1ppm), )"%   , /,h=,(T/)/,h=[(1/),T/,r,r/,h󀄀(T/2),/,h] *$'   +%( andwhereellipsoidalheight,h,isestablishedbyHandapreliminarygeoidheight,N.Notethat h thesecondterm,whichcontainsthenormalgravitygradient,,/,h,canbeofthesame X magnitudeasthefirstterm.   Asadditionalconfirmationthatsometypeof upward/downwardprocedureisneededtoestablishaccurategeoidheightsinteriortotheEarthssurface,wecomparethegeoidheightscomputedfrom2497GPSbenchmarkstothegeoidheightscomputedfromtheX01model.InFigure1a,theX01coefficientsweredirectlyevaluatedatageocentricradius,rN,whichis   continuallyreevaluatedtoconformtothegeoidsurface.InFigure1b,ourenhanced upward/downwardprocedureisfollowed.BothfiguresincorporatetheGMandavariations p  discussedearlier.ComparisonoftheplotsdemonstrateclearheightdependenterrorinFigure1awhenoneattemptstodirectlycomputethegeoidundulationsinteriortotheEarthssurface. TheGPSBenchmarkDataSet  0    NGSisengagedinaprojecttoestablishahighaccuracyFederalBaseNetwork(FBN),andanassociatedCooperativeBaseNetwork(CBN),throughnationwidemeasurementofGPSbaselinesto1ppmaccuracyorbetter.Inthecourseofthissurveyeffort,manyofthoseFBN/CBNpointsareestablishedonNAVD88opticallyleveledbenchmarks.Figure2displaysthelocationsof2497pointsthatareleveledbenchmarkswithNAVD88Helmertorthometricheights,andwhichhaveGPSmeasuredellipsoidalheightsintheNAD83(86)referencesystemasofMarch1996.Theirregulardistributionillustratesthestatebystateapproachtothesurveying,andthedifferentlevelsofstateparticipation.  Forthepurposesofthepreliminarymodelevaluation,theNAD83(86)coordinatesareconvertedintotheITRF93(1995.0)referenceframe.ThetransformationwasperformedusingparametersestablishedbySteveFrakes,NationalGeodeticSurvey.TheseparametersaresummarizedinTable1.Inthispaper,allellipsoidheightsareexpressedrelativetotheGRS+?+?80normalellipsoidintheITRF93(1995.0)frame.*}>$?Iddd Xdd Xdd X(#(#},dd ,dd ,dd , dd +  +! "  +'X[UXX'Table1TransformationfromNAD83(86)toITRF93(1995.0) ?5'""   ?X \Rp$ " B-0.9769B\Є0.9769 \RDp$!3 B-0.9769 B \0.0166 .$p$"3 .m .$p$# .Y 7-%$" 7+1.9392 7-%%3 70.0137 .$%&3 .m .$%' .Z 7-x'!(" 7+0.5461 7-x'!)3 70.0141 .$x'!*3 .m .$x'!+ .3X \R("," F%u-0.0264F%u\Є0.0264 \RD("-3 F%u-0.0264 F%u \0.0006 .$(".3 .arcsec .$("/ .3Y \R*$0" M O-0.0101M O\Є0.0101 \RD*$13 M O-0.0101 M O \0.0005 .$*$23 .arcsec .$*$3 .3Z \R,%4" 0*-0.01030*\Є0.0103 \RD,%53 0*-0.0103 0* \0.0004 .$,%63 .arcsec .$,%7 .scale \R- '8" _Q{-0.0068_Q{\Є0.0068 \RD- '93 _Q{-0.0068 _Q{ \0.0017 .$- ':3 .ppm&- ';   &HP    + XX[UX68+  TheNAVD88datumisexpressedinHelmertorthometricheights,andwascomputedin1991.Thenetworkcontainsover1millionkilometers(km)oflevelingatprecisionsrangingfrom0.7to3.0mm/km,andincorporatescorrectionsforrodscale,temperature,level (  collimation,astronomic,refraction,andmagneticeffects(Zilkoskietal.1992).TheNAVD88datumwasrealizedbyasingledatumpoint,FatherPoint/Rimouski,inQuebec,Canada.ItmustbementionedthatthereisnoguaranteethattheNAVD88datumcoincideswithagivennormalpotential,U0.Forexample,Rapp(1996)foundameanoffsetfortheNAVD88datumof34cm   relativetoaU0establishedbya=6378136.59.ForadditionaldetailsontheGPSbenchmarkdata p  set,pleaserefertoMilbert(1995). HighResolutionGeoidComputationandEvaluation  @    Thecomputationofthehighresolutiongeoidmodels&  Ԁisbasedontheuseof'DԀtheFastFourierTransform(FFT)toevaluateStokesequation.AsdescribedinSchwarzetal.(1990),abandwidthlimitedsignalisneededforinputtotheFFTconvolution.Thismandatestheuseofaremove,compute,andrestoreprocedure;wheregravityanomaliesfromaglobalgeopotentialmodelaresubtractedfromgravityanomalydata,followedbyconvolutionoftheresidualanomaliesintoresidualgeoidheight,andthenfollowedbyrestorationofgeoidheightsfromthatsamegeopotentialmodel.SuchanapproachwasusedintheGEOID90andGEOID93computationsfortheUnitedStates(Milbert1991,MilbertandSchultz1993).  Forthetestmodelevaluation,astandardgridofterraincorrected,Bougueranomalieswasgenerated.Now,anygriddingdatasetissubjecttoaliasinginthepresenceofhighfrequencysignal.Toremoveasmuchpredictable,highfrequencycontentaspossible,griddingisperformedonterraincorrected,Bougueranomalies,gTB.Foranomaliesonland ` gTB=g+W/! `E ? ? ? WH󀄀W/BC! `E___ WH2󀄀+A+C󀄀0.1119H x   @ whereA=0.86589.727105H+3.482109H2     p      x  h$  andg  surfacegravity,tidecorrected,IGSN71system(mgals) 8' # H  orthometricheight,NAVD88datum(meters) ((!$   normalgravityonellipsoid,GRS80(Somiglianasformula)(mgals) )"% A  atmosphericcorrection(Wichiencharoen1982)(mgals) *#& C  terraincorrectionfrom30"elevationgrid(mgals) *$' -  geodeticlatitude,NAD83(86)datum +%( a  semimajoraxis,GRS80(6378137meters) ,p&) a  normalgravityatequator,GRS80(978032.67715mgals) -`'* f  ellipsoidflattening,GRS80(0.00335281068118) h m  0.00344978600308(GRS80) X #  densityoftopographicmasses(2.67gm/cm3) H G  gravitationalconstant 8  R  meanradiusoftheearth (  1  sphericaldistance     &  Thegriddingalgorithmusesamethodofcontinuouscurvaturesplinesintension(SmithandWessel1990)withtensionparameter*?*?T*?*?B=0.75' O.Themethodisonewhichhonorsthedata   anddoesnotdisplaylargeoscillationsinareaswithoutdatacoverage.ThestandardBouguergridisa3'by3'regulargridextendingfrom24$Nto53$Nand230$Eto294$E(66$Wto `  130$W),andcontains581rowsand1281columns.Allanomaliesareprefilteredbycomputing P  meanvalueandmeanlocationoftheanomaliesin3'x3'cellscenteredovertheregular3'latitudeandlongitudeintersections.ThisprefilteringstepisrecommendedbySmithandWesseltoreducespatialaliasingeffectspriortogridding.  SinceweareusingStokestheorytocomputethegeoid,werequiregravityanomaliesonthegeoidsurface,ratherthanattheEarthssurface.WeadoptHelmertssecondmethodofcondensationtoregularizetheproblemandincludetheindirecteffectwithGrushinskysformulak) 5%!`|3 `EkG88P k (#(#(#(#     8    !  (#(#(#(#whereStokesfunctionisk)KL5%!`|$ `Ey$| k F(#F(#(#(# (#(#F(#F(#ThissetupissolvedinaremovecomputerestoreprocedureusingtheFFTformulationofStrangvanHees(1990),  g+=(gTB󛀜+0.1119H)g360 H&" k)FG5%!`|F3v `EFk'v4O~8'qk (#(#(#(#   J  (#(#(#(#  N=N++N360  *$' whereg+   ` highfrequencypartofgravityanomalies -`'* g360   ` geopotentialmodelgravityanomaliesatgeoidsurface h -, ` gridspacing X F,F1   ` directandinverse,twodimensional,Fouriertransforms H N+   ` highfrequencypartofgeoidundulations 8  N360   ` geopotentialmodelgeoidundulationsatgeoidsurface (  and,whereStokesfunction,S,isevaluatedwithameanlatitude,-m,andtheapproximation,     sin1[sin2(-P-Q)+sin2(PQ)(cos2-msin2(-P-Q))]   Theinputgrid,g+,(580rows,1280columns)had50%zeropaddingonallfouredgesto `  eliminatetheeffectofcyclicconvolution(Gleason1990).Notaperingofg+wasperformed, P  sincethelongwavelengthparthasalreadybeenremoved.TheFFTsubroutinehasanoptionwhichexploitstheHermitiansymmetryresultingfromrealvaluedgrids.Thus,doubledcomputationspeedandstorageefficiencywasobtainedwithoutresortingtoHartleytransforms.  Animportantissuemustbementionedatthispoint.Thegeopotentialmodelgeoidundulations,N360,areevaluatedatthegeoidsurface.FromtheresultsofFigures1aand1b,itis X seenthatthesegeoidheightsarebiased.However,theStokesproceduredescribedabovewillremovethatheightdependenterrorifthegravityanomalies,g360,arealsoevaluatedina 8 fashionconsistentwiththegeoidundulations.Sinceitisnotconvenienttoapplysomeformof upward/downwardproceduretothegravityanomalies,wechoosetoevaluatebothgeoidandanomaliesatthegeoidsurface.Thekeypointisthatweareconsistentinourchoiceofsurfaceforcoefficientevaluation.  Asaformaldemonstration,considerStokesequationwritteninaremove,compute,andrestoreprocedure:̜k):;5%!`|PF8 `EPE F8O.kߛ ^(#^(#(#(#̛̜ (#(#^(#^(#Thetermontherightformulatesthe N,whichmustberestored,asafunctionofthemodel X%! gravityanomalies,g360,thatwereremoved.ExpandingStokesfunctioninLegendre H&" polynomialsandthegravityanomaliesintermsoftheharmoniccoefficientsyields  ̜k)5%!`| Qn `E i)nB)kߛ (# (#(#(#̜-`'* (# (# ЇwhereW/WX! `E:RR:RW߀.Performthemultiplication,exchangetheorderofthesumsand h integrals,underastandardassumptionofconvergence,combineterms,andinvoketheorthogonalityrelationshipsofLegendrepolynomimals.Afteralgebra,theequationabovewillsimplifyintok)5%!`|d `Ed Wg (  k <(#<(#(#(# (#(#<(#<(#iftheleadingRbeforetheintegeralistakentober.TheequationaboveisequivalenttoN360. `  OtherthantheR=rrequirementoftheleadterm,thedevelopmentdidnotinvokeanyspecific P  surfacemandatedbyaremove,compute,andrestoreprocedure.ThedevelopmentdidshowtheconsistentN360andg360expressions. 0    Thehighresolutiongeoidmodelsarederivedfromfourdatasources:pointgravitymeasurements,satellitealtimetryderivedgravityanomalies,digitalterrain,andgeopotentialcoefficients.Afewremarksareappropriate.About1.8milliongravitypoints,bothshipandterrestrial,wentintothegridding.ThesedatawereacombinationofNGShelddataandqualitycontrolleddatafromtheDefenseMappingAgency.ThesatellitealtimetryderivedgravityanomalieswerecomputedbySandwellandSmith(1996).Thesewereincludedingravityvoidregionswheredepthsexceeded500metersandwhichwere200kmormorefromland.Thedigitalterraindatacameprimarilyfromthe30"pointtopographydatabase,TOPO30,distributedbytheNationalGeophysicalDataCenter(RowandKozleski1991).The30"elevationsetwasusedtocomputebothterraincorrections,and3'x3'meanelevations.And,ofcourse,thepreliminarygeopotentialmodels,X01throughX05,wereusedasthecoefficentsources.Wedenotetheresultanthighresolutionmodelsas9620through9624,respectively.  Weexplorethegeneralpropertiesofthesolutionsbycomputinggeoidheightresiduals,e,  @ inthesenseof̀e=highres.geoidheight(ITRF93(1995.0)ellipsoidheightNAVD88orthometricheight). x# Tiltedplanemodelsarefittothe2497residualsandaresummarizedinTable2.A$) xdE'(xA& }  ((!$  X\ XXX -! X X\ Model 󀀀 Offset (cm.) Tilt (ppm) RMSaboutplane (cm) Azimuth (deg)# X\ X! -m##XX X X\m# X\ XXX -! X X\ 7(!% 96204.580.0435.07293# X\ X! -n##XX X X\n# X\ XXX -! X X\ )"& 96215.900.0636.14317# X\ X! -o##XX X X\o# X\ XXX -! X X\ )e#' 96222.070.0536.14319#OV#! -p##XX#OVp# X\ XXX -! X X\ *-$( 96232.050.0636.56321# X\ X! -|q##XX X X\]q# X\ XXX -! X X\ ]+$) 96242.050.0536.52321#OV#! -9r##XX#OVXr#OV#XX -!#OV %,%* A$) xdE]-xA߀#OV#! -4s##XX#OVs# ,&+  Table2.Tiltedplanefitsto2497highresolutiongeoidmodelresiduals.'}((m  -M', Ї  Theseresultsbegintoindicatemodeldifferences.TheaverageoffsetssuggestthatX01andX02useddifferingdataand/orproceduresfromX03,X04,andX05.Thesmallmagnitudeoftheoffsetsmustbeconsideredinthecontextofthecomparisons.TheGRS80normalellipsoidisusedtoexpresstheellipsoidalheights,thenormalgravity,andthehighresolutiongeoidmodels.Thus,thesmalloffsetsarenotinconsistentwiththe34cmfoundbyRapp(1996).  Next,followingtheapproachofMilbert(1995),smoothedresidualsurfacesarecomputedbymeansofcollocationforall5highresolutionmodels.Empiricalcovariancefunctionswerefittothecovariancestatisticsofthedetrendedgeoidresiduals.Thefitsweremadeusingasimplefunctionoftheformk)NO5%!`|  `E Bp)" k K(#K(#(#(#     S    (#(#K(#K(#whered  =thesphericaldistancebetweenpoints(km)   L  =characteristiclength(km) x C0  =functionvariance(m2) h Thefunctionfitswerevirtuallyidentical,withonlyminorvariationsofL=485kmandC0= H (0.31)2m2.AswasseeninMilbert(1995),theresidualerrorissizableandcorrelatedoveralong 8 lengthscale.Thedetrendedresidualerror,W/DE! `E66W,ispredictedona30'x30'gridusingleastsquares ( collocationwithnoise(Moritz1980,p.102106).Thepredictionformulaisk)JM5%!`|~)  `E~1 CP k J(#J(#(#(# (#(#J(#J(#whereCtt  =signalcovariancebetweenobservations ` Cst  =signalcovariancebetweenpredictedsignalandobservations P Cnn  =covarianceofrandommeasuringerrors,takenasdiagonalandconstant:Cnn=so2I  @ Beforethegridsarecomputed,thepredictionprocessisiteratedtoestablishavalueofso2 "  consistentwiththeresidualmisfitaboutthepredictions.ItwasfoundtheRMSofresidualsfromthepredictionstepmatchedtheassignednoisewhenso2=(5.6)2cm2forthen=2497points.The h$  trendsreportedinTable2wererestoredtothedetrendedsignalgrids,resultinginthefinalcorrectorsurfacegrids(adjustedinsigntoprovideanadditivecorrection).  Figure3portraysatrendsurfacethat,whenaddedtothe9622geoidmodel,willdirectlyrelateITRF93(1995.0)ellipsoidheightsandNAVD88Helmertorthometricheights.InviewingthisfigureitmustberecalledthattheGPSbenchmarksusedtodevelopthisgridallliewithintheU.S.borders;andthathighsorlowsintheoceansorinothercountriesareextrapolations,andarenotreliable.Intheinterestsofconcision,thetrendsurfacesfortheother4modelsarenotdisplayed,sincetheyappearalmostidentical. -`'*   Aspartoftheevaluationprocess,wecomputedallpossiblecombinationsoftheresidualtrendsurfacedifferences,10inall.ThesearereproducedinFigures4athrough4j.Inspectionofthecombinationsindicatethatthe9623and9624modelswerevirtuallyidentical,andthat9622wasveryclosetothe9623model.(OnemustdiscountfeaturesintheGPSbenchmarkvoidareas.)Itisalsoseenthatmodel9621showssomedeparturesfrom9622,andthat9620isthemostdifferentofallthemodels.Theseresultsconfirmthesimilaritiesofthe9622,9623,and9624models,andalsosuggestthatboth9620and9621aredistinctfromthemandaredistinctfromoneanother.  Atthispointitishelpfultoreviewparticularfeaturesseeninthetrendsurfaces.AtiltisevidentinFlorida,anditisvirtuallythesameinallmodels.ThiseffecthasbeenobservedwithboththeGEOID90andGEOID93models,aswellaswithnumeroustestmodels.Giventhegreatcaretakenbythegeopotentialmodelcomputationteamintheintercomparisonsofmarinegravityandaltimetryderivedgravity,andgiventheveryflattopographyofFlorida,itisunlikelythatthegeoidmodelsarethesourceofthisfeature.Further,sincetheGPSnetworkisanchoredtotheRichmondVLBIsiteatthetipofFlorida,itisalsounlikelythattheellipsoidheightsarebiasedinthatregion.Thelevelingissuspect.Ontheotherhand,thetiltalongtheGulfCoastisprobablygravimetricinorigin;andwillbediscussedlaterinthissection.  ANorthSouthtiltisalsoevidentinWashingtonState.ThesourceofthisfeatureseemstobetheterraincorrectionscomputedfromdigitalterraindatainsouthernBritishColumbia.WeareworkingwiththeGeodeticSurveyofCanadatoresolvethisproblem.Itisthought,however,thatthemeangravityanomaliesusedtocomputethepreliminarygeopotentialmodelsareaccurate.ThetiltisseentoblendintoanelongatedfeatureoverthenorthernRockies.Thisraisessomeconcernoverpossibleheightdependenterror.  Asadifferentviewofthemodels,wemadescatterplotsofthegeoidheightresidualsrelativetoelevation.Asampleforthe9621modelisgiveninFigure5,sincethescatterplotslooksimilar.ThedispersionofvaluesnearthezeroelevationreflecttheissuesinFloridaandtheGulfCoastnotedearlier.NoobviousheightdependenceisevidentinFigure5,unliketheeffectseeninFigure1a.However,overlaysofthethe5differentscatterplotsdoshowsmallvariationsthatcouldhaveanelevationdependence.  Inpursuitofthispossibility,anexploratoryGPSbenchmarkdatasetwasformedbywitholdingthepointsinFloridasouthof30$N,andbywitholdingthepointsnorthof48$Nin H&" WashingtonState.Theaverageoffset,andtheRMSabouttheoffsetarethencomputed.Inaddition,theoffsetandRMSarecomputedwheredataarewitheldbelow1000m,below1500meters,andbelow2000metersinelevation.Thestatisticsaregroupedbymodel,orderedbyascendingelevation,anddisplayedinTable3.Inaddition,thedifferencesinaverageoffset,andseconddifferencesaredisplayed.  +%( A$) xdEgxA&    X\ XXX -! X X\ h  9620Model ` Cutoff (m) 򛀀(?(?n 󀀀 RMS (cm)# X\ X! -W# -! X X\#OV#! -8# -!#OVԀ Offset (cm.) 򀀀Diff2ndDiff # X\ X! -# -! X X\#OV#! -B# -!#OV w М (X01)  `  0219024.85  9.15 E    ` 100044823.99  6.58     ` 150026824.98  6.32  0.26 s    ` 200010724.79  6.70  0.38 x 0.64  ;  9621Model ` Cutoff (m)#OV#! -# -!#OV# X\ X! -# -! X X\ 򛀀(?(?+?+?(?񀛀(?n 󛀀 RMS (cm)# X\ X! -`# -! X X\#OV#! -# -!#OVԀ Offset (cm.) 򀀀Diff2ndDiff #OV#! -%# -!#OV# X\ X! -# -! X X\Ԁ# X\ X! -ה# -! X X\#OV#! -# -!#OV 3  М (X02)  `  0219024.59  9.82      ` 100044823.76  7.50  g     ` 150026824.71  7.15  0.35  /     ` 200010724.51  7.57  0.42 x 0.77 _  #OV#! -^# -!#OV# X\ X! -# -! X X\ 9622Model ` Cutoff (m)# X\ X! -# -! X X\#OV#! -_# -!#OV 򛀀'?'?(?(?'?񀛀'?n 󛀀 RMS (cm)# X\ X! - # -! X X\#OV#! -N# -!#OVԀ Offset (cm.) 򀀀Diff2ndDiff #OV#! -# -!#OV# X\ X! -ј# -! X X\Ԁ# X\ X! -ř# -! X X\#OV#! -# -!#OV   М (X03)  `  0219024.64  9.71 U     ` 100044823.77  7.07 #     ` 150026824.78  6.92  0.15 S     ` 200010724.50  7.46  0.54 x 0.69   # X\ X! - # -! X X\#OV#! -L# -!#OV 9623Model ` Cutoff#? (m)#OV#! -# -!#OV# X\ X! -K# -! X X\ #?񛀀&?&?'?'?&?񀛀&?n 󛀀 RMS (cm)# X\ X! -?# -! X X\#OV#! -# -!#OVԀ Offset (cm.) 򀀀Diff2ndDiff #OV#! - # -!#OV# X\ X! -ǝ# -! X X\Ԁ# X\ X! -# -! X X\#OV#! -y# -!#OV C М (X04)  `  0219024.66  9.92 y    ` 100044823.78  7.38 G    ` 150026824.77  7.26  0.12     ` 200010724.47  7.83  0.57 x 0.69 o #OV#! -B# -!#OV# X\ X! -# -! X X\ 9624Model ` Cutoff$? (m)# X\ X! -# -! X X\#OV#! -A# -!#OV $?񛀀%?%?&?&?%?񀛀%?n 󛀀 RMS (cm)# X\ X! -# -! X X\#OV#! -5# -!#OVԀ Offset (cm.) 򀀀Diff2ndDiff #OV#! -# -!#OV# X\ X! -# -! X X\Ԁ# X\ X! -# -! X X\#OV#! -o# -!#OV g М (X05)  `  0219024.73  9.84 5    ` 100044823.83  6.97     ` 150026824.80  6.88  0.09 c    ` 200010724.50  7.43  0.55 x 0.64# X\ X! -##XX X X\8# X\ XXX -! X X\ + A$) xdExA߀#OV#! -D##XX#OVc# [  Table3.Geoidresidualstatisticsorderedbyelevationcutofftolerance.'h"  #   Table3enablesustodiscernmoreaboutthecomputationofthepreliminarymodels.Thedifferencesintheaverageoffsetbetweenthe0melevationcutoffandthe1000mcutoffareprobablyindicativeofremainingsystematicerrorsintheGPSbenchmarkorinthedetailgravitydata,ratherthanreflectingcharacteristicsoftheglobalmodels.WeknowthattheaverageoffsetwillcontainNAVD88verticaldatumerror.Wedontknowwhatthaterroris,butwecanexpectthaterrortobeconstantwithrespecttoelevation.Thus,wecanexpectazerovalueforthedifferencesinaverageoffset.Heightdependenteffectswillcausevariationsintheaverage %;' offsetsastheelevationcutoffisvaried.  Lookingattheoffsetdifferencecolumn,weseethatX03,X04,andX05aredistinctfromX01andX02.Infact,theclosesimilaritiessuggestthatX03,X04,andX05aremembersofafamily,perhapscontrolledbysometuningparameter(s),suchasanomalyweighting.And,itseemsthatthemodelsX03,X04,andX05areprovidedinsequence;correspondingtomonotonicchangesinthehypotheticalparameter. #-&/ "P       Atthisstageitisseenthatmodel9624,computedfromX05,displaysthemostuniformaverageoffsets.  Toclosethissection,aresultwasobtainedthatsetsthestageforthenextsectionofthereport;wherethemodelsaredirectlycomparedagainstGPSbenchmarks.Recallthatinthissection,thegeopotentialmodelsareevaluatedthroughhighresolutiongeoidmodelscomputedinaremove,compute,andrestoreprocedure.OneoftheproductsofthisprocedureistheoutputgridfromtheFFTconvolution,whichwedenoted,N+.Thisgridcontainsthehighfrequency   partofthegeoid(truncationoromissionerror),butalsocontainsinformationaboutlowerfrequencycommissionerror.  OnemustinterpretanN+gridwithcaution.First,thegrid,whenaddedtothepreliminary P  modelgeoid,N360,willyieldthecogeoid.OnecanthinkoftheN+gridasbeingbiasedbythe @  indirecteffect,sinceFayeanomalieswereusedinitscomputation.Second,ourmodelgeoids,N360,werebiased,sincethe)?y)?)?re)?wereevaluatedinteriortotheEarth.Therefore,ourN+gridswill   alsocontainthatheightdependentbias.Despitethesedrawbacks,theN+gridcertainlycontains x usefulinformationatlowerelevations.  Figure6portraysacolorshadedreliefimageoftheN+gridassociatedwiththeX02 H model.Sixteenhueswereallocatedfromthelowof1.5m(magenta)tothehighof+1.5m(red).SixteenlevelsofgrayshadingwereaddedtogivetheeffectofilluminationfromtheEast.Themagnitudesofthegridactuallyrangedfrom4.1mto2.8m.Thus,valueswhichexceedthe1.5meterintervalwill saturateintothemagentaortheredcoloring.Thegridsfromtheotherpreliminarymodelslookverysimilar.  MostnoticableisthelargeblueandmagentastructureintheRockies.Becauseofthebiasissuesdiscussedabove,onecan,atmost,saythatithintsatsignificantcommissionerrorintheRockies.Thisissuewillbeexploredinthenextsection.  Otherfeaturesareinteresting.ThelowsinLakeSuperiormaybeduetoourtreatmentoftheelevationsoftheshipgravitydata.ThelowintheBermudasmayalsoberelatedtoourgravitydataset,andwillbeinvestigatedatalaterstage.WhiletheBermudasfeaturedoesinduceatiltacrossFlorida,itislocalizedtothesouth,anddoesnotexplainthecompleteamountoftiltwehaveseeninthepast.Inaddition,someportionofthelowseeninWashingtonStateandsouthernBritishColumbiaisduetocurrentlyunresolvedproblemswithterraincorrections,discussedearlier.  Ontheotherhand,the0.5mfeatureintheCentralU.S.,andthe1mfeaturesintheGulfofMexicoandofftheAtlanticseaboardarelesseasilyexplained.Theseareregionsoflowelevations,andareratherbroad.ThedataintheGulfandnearshoreAtlanticareshipgravitydata.ThealtimeterderivedgravityanomaliesfromSandwellandSmith(1996)wereonlyused inthesouthernpartoftheGulfandinthedeepAtlantic.Wesuspecttheseparticularfeatures ,p&) maybeduetopreliminarymodelcommissionerror;perhapscorrelatedwithsomeseasurface(dynamic)topographyeffectintheocean. LowResolutionGeoid ComputationandEvaluation  8    Inthissection,wecomparegeoidheightscomputeddirectlyfromthepreliminarygeopotentialmodelsagainsttheGPSbenchmarkdataset.Whilethesestatisticsareverynoisy,duetoomissionerror,theycanportraylongwavelengthcommissionerror.  Fivegeoidgridsat3'x3'spacingwerecomputedfromtheX01throughX05coefficients.Allgridswerecomputedwiththe up/downprocedure,wherethecoefficientsareevaluatedattheellipsoid,upwardcontinuedtothesurfacetoformaheightanomaly,andthenconverted downtoageoidheightbymeansofEq.(810)ofHeiskanenandMoritz(1967).Thislaststepisimplementedasa3'x3'gridwhichhasbeenlowpassfilteredtocorrespondtoann=360resolution.Weemphasizethatthesegeoidgridsarenotsubjecttotheheightdependenterror   illustratedinFigure1a.  Tiltedplanemodelsarefitto2497residualsaboutthelowresolutiongeoidgrids,andaresummarizedinTable4.A$) xdE'xA& }  (  X\ XXX -! X X\ Model 󀀀 Offset (cm.) Tilt (ppm) RMSaboutplane (cm) Azimuth (deg)# X\ X! -5##XX X X\# X\ XXX -! X X\ 7 ЀX012.160.4026.52338# X\ X! -##XX X X\# X\ XXX -! X X\  ЀX021.020.3229.77336# X\ X! -##XX X X\# X\ XXX -! X X\ e ЀX030.070.3526.22334#OV#! -##XX#OV# X\ XXX -! X X\ - ЀX040.430.3525.99335# X\ X! -##XX X X\# X\ XXX -! X X\ ] ЀX050.760.3526.10335#OV#! -p##XX#OV#OV#XX -!#OV % A$) xdE]xA߀#OV#! -k##XX#OVL#   Table4.Tiltedplanefitsto2497lowresolutiongeoidmodelresiduals.'}(  M   Theseresultsareremarkable,inthattheyshowasignificanttilttotheNorthwest.ThisconfirmsthesuspicionthatthelargeblueandmagentafeatureseenintheRockiesinFigure6isatleastpartlyduetopreliminarygeopotentialmodelcommissionerror.ItshouldbeberecalledthatthistiltwasnotevidentwhentheGPSbenchmarkresidualstothehighresolutiongeoidmodelswerecomputedinTable2.Itappearsthatthehighresolutionmodelsremovenotonlyomissionerror,butalsocommissionerror.  Toproduceanimageforevaluation,smoothedresidualsurfaceswerecomputedbymeansofcollocationforall5lowresolutionmodels.Empiricalcovariancefunctionswerefittothecovariancestatisticsofthedetrendedgeoidresiduals.Thefunctionfitswerevirtuallyidentical,withonlyminorvariationsofL=200kmandC0=(0.21)2m2.ThetrendsreportedinTable4 )#( wererestoredtothedetrendedpredictedsignalgrids,resultinginthefinalsmoothedresidualsurfaces.Thesurfacesforthe5modelswerevirtuallyidentical,soonlytheresultformodelX01isdisplayedinFigure7. -M', #r      ThecomplexstructureseeninFigure7isarepresentationofthepreliminarymodelomissionerrorassampledbytheGPSbenchmarks.Thesesmallfeaturescanbereducedwithmorestringentsmoothing,andshouldbeignored.Similarly,featuresintheoceans(e.g.nearFlorida)orbeyondtheU.S.bordersshouldalsobeignored,sincethoserepresentextrapolations.Further,thepresenceorabsenceofstructureinthecentralU.S.hasnomeaning,duetotheabsenceofGPSbenchmarks.  However,anoffsetinthenorthernRockies,relativetotheplains,isevidentinFigure7.Unfortunately,shortagesofGPSbenchmarksinthecentralU.S.,intheCarolinas,andsouthofHouston,Texas,makeitdifficulttocommentonotherfeaturesdiscussedinFigure6.ItisnoteworthythatthetiltinFloridaisstillevident;aresultconsistentwithahypothesizedprobleminthelevelnetwork.  TofurtherstudythebehaviorofthepreliminarymodelsandthepossibilityofcommissionerrorintheRockies,505GPSbenchmarksintheregionof38$49$Nand230$   255$Ewereextractedfromthefulldataset.Thesepointswerethengroupedinto500m x elevationcohorts,andstatisticswerecomputed.Thestatisticsaregroupedbymodel,orderedbyascendingelevation,anddisplayedinTable5.Inaddition,thedifferencesinaverageoffset,and,secondandthirddifferencesaredisplayed.Only15GPSbenchmarksarepresentinthe over2500mcohort,andthosestatisticsarenotfelttobereliable.A$) xdExA X\ XXX -! X X\   X01ModelCohortn 󀀀 RMS (cm)# X\ X! -@# -! X X\#OV#! -!# -!#OVԀ Offset (cm.) 򀀀Diff2ndDiff# X\ X! -# -! 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Table5supportsmanyoftheconclusionsdevelopedfromanalysisofTable3.ModelsX03throughX05seemtobemembersofafamily,andseemorderedrelativetosometunableparameter(s).ModelsX01andX02aredistinctfromthatfamily,andfromoneanother.UnlikethesituationinTable3,someinterpretationcanbeappliedtotheaverageoffsetcolumn.WhilewedonotknowthetruevalueofthedatumoffsetassociatedwithNAVD88(inGRS80),thehighresolutiongeoidmodelresultsinTable3indicatethatitisavaluecloserto10cmthanto40cm.Thisleadsustoprefermodelswithasmalleraverageoffset,sincewefeeltheoffsetisϜpartlyduetocommissionerror.ModelsX02andX05lookbest.Ingeneral,modelX05hassmalleroffsetspercohort,butshowsgreaterheightdependenceofthoseoffsetswhencomparedtoX02.ThesmallerRMSabouttheaverageoffsetsofmodelX02isalsoattractive.ua'XX ConclusionsandRecommendations#XX'ua #  *#+   Differencingthegeoidgridscomputedfromthepreliminarygeopotentialmodelcoefficientsshowthetestmodelstobeverysimilar.Soitisclearthatanyofthetestmodelsrepresentsamajoradvanceovercurrentlyavailablegeopotentials.Withthissaid,wehavea -'/ @r    +?@r    +?preferenceformodelsX05andX02,withaveryslightedgetowardsX05.Thispreferenceisbasedoncriteriaoflowheightdependenceofgeoidresiduals,lowaverageoffsetsoftheresiduals(forthecaseofthelowresolutionmodels),andlowRMSvalues.Itisinterestingthatthesecriterialeadtothesamemodelpreferencesirrespectiveofthehighresolutionorthelowresolutionapproach.  Sinceislikelythatthefinalmodelwillnotbeanyoneofthebetatestmodels,itisworthwhiletostatethatanidealmodelwouldcombinecharacteristicsfromX02andX05.However,allthepreliminarymodelssharethecommissionerrorseenintheregion38$49$N,   230$255$E.Ifthiseffectcouldbelessened,thenthefinalmodelwouldbeevenbetter. p  ua'XX Acknowledgments#XX'ua#  P    Workofthisscoperequiredthecontributionsofdozensofindividuals.VirtuallyeveryemployeeoftheNationalGeodeticSurveyhasassistedinthiseffort.Thegravitydataset,theNAVD88project,andtheGPSstateupgradeprogramwerevitaltothisstudy.TheDefenseMappingAgency(DMA)providedamajorportionoftheNGSlandgravitydataset,andwasinstrumentalincreationofthevarious3"and30"elevationgridsinexistence.Dr.WalterSmith,NOAA,providedthealtimeterderivedgravityanomalies.ua'XX \  #XX'ua#ua'XXReferences#XX'ua#  P Gleason,D.M.,1990:ObtainingEarthsurfaceandspatialdeflectionsoftheverticalfromfreeairgravityanomalyandelevationdatawithoutdensityassumptions.J.Geophys.Res.,95(B5), 4 67796786.Heiskanen,W.A.andH.Moritz,1967:PhysicalGeodesy.W.H.Freeman,SanFrancisco,364  pp.A46-55.Milbert,D.G.,1995:ImprovementofahighresolutiongeoidheightmodelintheUnitedStatesbyGPSheightonNAVD88benchmarks.In:BalminoG.,F.Sano(ed),NewGeoidsinthe "\ World,InternationalAssociationofGeodesy,BulletindInformationN.77,IGeSBulletinN.4, #L 1336.Milbert,D.G.,andD.Schultz,1993:GEOID(TheNationalGeodeticSurveyGeoidComputationProgram).GeodeticServicesDivision,NationalGeodeticSurvey,NOAA,SilverSpring,MD.Milbert,D.G.,1991:GEOID90:AhighresolutiongeoidfortheUnitedStates.EOS,72(49), D*#& 545554.Moritz,H.,1992:GeodeticReferenceSystem1980.Bull.Geod.,66(2),187192. -&) ЇMortiz,H.,1980:AdvancedPhysicalGeodesy.HerbertWichmannVerlag,Karlsruhe,500pp. h Rapp,R.H.,1996:Useofpotentialcoefficientmodelsforgeoidundulationdeterminationsusingasphericalharmonicrepresentationoftheheightanomaly/geoidundulationdifference.SubmittedtoJournalofGeodesy. 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((!$ Zilkoski,D.B.,J.H.Richards,andG.M.Young,1992:ResultsofthegeneraladjustmentoftheNorthAmericanVerticalDatumof1988.Surv.andLandInfo.Sys.,52(3),133149. *$'   ,p&)  ua'XX&   FigureCaptions#XX'uas #  h Figure1 ` GeoidModelResidualstoGPSBenchmarkGeoidHeights'h  \    ` a)ModelX01EvaluatedattheGeoidSurface.   ` b)ModelX01Evaluatedbyan Upward/DownwardProcedure.Figure2 ` 2497LeveledBenchmarkswithNAVD88HelmertOrthometricHeightsandGPS      ` EllipsoidalHeightsintheITRF93,?,?(1995.0)ReferenceSystem.Figure3 ` TrendSurfacetoRelateHighResolutionModel9622toGPSBenchmark??.Figure4 ` DifferencesbetweenTrendSurfaces7?6?6?7?6?6?;?;?6?(6?8?Contour8?9?8?e8?9?:?Interval=0.01m):? T  Ѐ   ` a)throughj)DifferencesinallCombinations. D  Figure5 ` HighResolutionModel9621HeightResidualstoGPSBenchmarkGeoidHeights $ Figure6 ` HighFrequencyComponentofGeoidHeightforHighResolutionModel9621 l    ` RelativetoPreliminaryGeopotentialCoefficientModelX02.Figure7 ` TrendSurfacetoRelateLowResolutionGeoidModelX01toGPSBenchmarks <    ` (ContourInterval=0.05m).