MODELLING LOCAL GPS/LEVELLING GEOID WITH THE ASSESSTMENT OF INVERSE DISTANCE WEIGHTING AND GEOSTATISTICAL KRIGING METHODS B. Erol a, *, R. N. elik a a ITU, Civil Engineering Faculty, Geodesy Division, 34469 Maslak Istanbul, Turkey - (bihter, celikn)@itu.edu.tr Commission IV, WG IV/1 KEY WORDS:Geodesy, Modelling, DEM/DTM, Method, Observations, Accuracy, Spatial Infrastructures, GPS ABSTRACT: Accordingtowidespreaduseofsatellitebasedpositioningtechniques,especiallyGPS(GlobalPositioningSystem),agreater attentionhasstartedtobepaidtoprecisedeterminationofgeoidmodelswithanaimtoreplacethegeometriclevelling measurementswithGPSmeasurementsduringgeodeticandsurveyingworks.Inthispaper,geoidmodellingisevaluatedasa surfacefittingproblemaccordingtoGPSandLevellingdataandthemainfocusismodellingthegeoidofalocalareaasan analyticalsurfacetoservepracticalgeodeticapplicationssuchaslargescalemapproduction,GIS(GeographicalInformation Systems)basedstudies,engineeringapplicationsetc.Duringimplementationofthesubject,twodifferentinterpolationmethods, InverseDistanceWeighting(IDW)andKrigingmethodsareevaluatedaccordingtocreatedprogramroutines.Inthispaper,in additiontocomparethesetwopopularinterpolationtechniques,mainlyitisgoingtoemphasizetheapplicabilityofinterpolation techniquesasatoolformodellingthegeoidinalocalareapreciselyusingGPS/Levellingdatatoservepracticalapplicationsof geodesy.Also, the importance of precise local geoid models as a part of geodetic infrastructure is underlined with Turkey example.

* Corresponding author.This is useful to know for communication with the appropriate person in cases with more than one author. 1.INTRODUCTION Toaccomplishthetransformationbetweenellipsoidalheights andorthometricheightshasbecomemoreimportantsincethe satellitebasedpositioningtechniques,especiallyGPS(Global PositioningSystem),werebeingusedinawiderangeof geodetic and surveying applications.Using GPS technique, the positionsaredeterminedasrelatedtogeocentricWGS84 (WorldGeodeticSystem1984)referenceellipsoidofwhich surfaceisassumedasthedatumofpointsheightswhichare derivedfromGPSmeasurements.However,inmostofthe geodetic applications, it is necessary to use orthometric heights referenced to geoid. Thegeoidheight(orgeoidundulation)canbedefinedasthe separationofthereferenceellipsoidwiththegeoidsurface measuredalongtheellipsoidalnormal(seeFigure1).The classicalGauss-Listingdefinitionofthegeoidisgivenasan equipotentialsurfaceoftheEarth'sgravityfieldthatcoincides withthemeansealevel(Fotopoulos,2003).Today,itiswell knownthatthisisnotastrictlycorrectdefinitionasmeansea level departs from the equipotential surface by up to two meters duetovariousoceanographicphenomena,suchasvariable temperature,salinity,instantaneoussea surface topography etc. (Fotopoulos, 2003). The fundamental expression of relationship between ellipsoidal heightsobtainedfromGPSmeasurementsandheightswith respecttoaverticalgeodeticdatumestablishedfromspirit levellingdatawithgravimetriccorrectionsisasgiveninthe equation 1 (Heiskanen and Moritz, 1967). h H N = 0(1) where h = ellipsoidal height H = orthometric height N = geoid undulation Figure 1.Relations between ellipsoidal height, orthometric height and geoid undulation (geoid height) (http://kartoweb.itc.nl/geometrics/, accessed August 2000) So,asparalleltothewiderangeofapplicationsofGPS technique,ademandingforahighprecisiongeoidmodel preferably referring to a global geocentric datum appeared.As theresultofthis,configuringageodeticinfrastructureconcept hasbecomeincludedtostructureaprecisegeoidmodelasan important component of these studies. In this chronology, the method of GPS /Levelling for obtaining orthometricheightscannotbeassumedasanewconcept.In fact,asaresultofcasestudiesthathavebeenconductedin different regions of Turkey, it is proved that the GPS/Levelling canprovideaviablealternativetotraditionaltechniquesasif levellingmeasurementtechniques,whichareonerous applicationsaswellknown.However,theanswerofthis questionWhataccuracylevelcanbereachedusingthis approach?isstillanimprovableissueasrelatedtomany conditions and one of them is used method of geoid modelling. Inthispaper,geoidmodellingisevaluatedasasurfacefitting problemaccordingtoGPSandLevellingdataandthefocusis modellingthegeoidofalocalareaasananalyticalsurfaceto serve practical geodetic applications.During implementation of the subject, two different interpolation methods are considered. There are several interpolation methods to transform point data, however,itisimportanttodetermine,whichoneismore appropriate and give better and clear parameter solution for the implementeddataandtheproblem.Modernspatial interpolation methods are based on geometric and geostatistical aspects. Here in, two interpolation methods, Inverse Distance Weighting and Geostatistical Kriging were applied for generating a digital geoidheightsmodelinalocalarea.Thestudyareais45x50 km2andcoveredbyheterogeneousandappropriately distributed301referencepointsofwhichpositionsareknown in ITRF datum and orthometric heights are in Turkish National Vertical Datum. Densityofthereferencepointsisappropriateforthegeoid modelling in the region (1 point\ 3 km).There is not a specific reasonforappreciatingespeciallythesetwointerpolation methodsinthestudy.Themainpurposeofthestudyisto emphasizethenecessityofaprecisegeoidmodeltoserve practicalpurposesofgeodesy.Thepresentedinterpolation methodswereprogrammedastobecomputationmodulesofa geoidmodellingsoftwareandtheresultsofthecomputation algorithmsaregoingtobementionedmorethanthe mathematicalbasesofthe methods because of that they can be reached in the literature easily (see Isaaks and Srivastava, 1989; Watson, 1992; Yanalak 1997). 2.LOCAL PRECISE GEOID MODELS TO SERVE PRACTICAL GEODETIC APPLICATIONS Aprecisegeoidmodelisanimportantpartofthegeodetic infrastructure.Geodeticinfrastructureincludeswholeofthe geodeticnetworks,whichrealizethecoordinatesystemsand datumdefinitionsasthebases of geodetic works (Aksoy et al., 1999). Restructuringandrevisingeffortsofgeodeticinfrastructure continuesinTurkeyandalsostructuringthenationalgeoid modelofTurkeyisapartoftheseefforts.Sofar,regional geoidmodelofTurkeyrevisedseveraltimes.Thelatestgeoid modelisTurkeyGeoid1999A(TG99A).TG99Ageoid modelsatisfiesnecessaryaccuracyforproducinglargescale mapsandroutinesurveyingapplications.However,insome parts, this gravimetric based refined geoid model stays weak in accuracy for practical geodetic applications. In this case, determining local precise geoid models using more intensivedatainareaswherenationalgeoidmodeldoesnot have sufficient absolute accuracy is purposed as a solution.By thisway,weakerpartsofthenationalgeoidmodelwillbe supportedbylocaltheselocalsolutionsandthecomponentof geodetic infrastructure as well. Fromthisviewpoint,toservethepracticalgeodeticand surveyingapplications,determiningthegeoidmodelofa limitedareaasananalyticalsurfaceusinghomogeneously distributed and GPS/Levelling reference points with appropriate density constitutes the subject of this study. 3.SURFACE FITTING METHODS IN MODELLING GPS/ LEVELLING GEOID IN A LOCAL AREA Among the geoid modelling techniques as based on geometrical approach,fittingasurface,whichdependsonthereference points that are chosen in the critical and characteristic locations ofthefieldtorepresenttrendofgeoidsurface,isacommon methodinsmallareasforlocalstudies.Representinggeoid heightsasmathematicallyformulatedsurfaceandcalculating thegeoidheightsinnewmeasuredpointsaccordingtoGPS technique constitutes the idea in these kinds of studies (Erol and elik, 2004). However,ina local area, determined geoid model with surface fittingtechniqueworkswithintheareathatsurroundedand covered by reference points properly and these kinds of models doesnotprovidereliableresultsfortheextrapolationpoints.One another important handicap in local geoid determination is datuminconsistencyproblembutinthisstudy,thisproblemis notgoingtobeconsideredbecauseofthefocusisontesting surfacefittingalgorithmsasageometricalapproachfor modelling a local geoid according to GPS and Levelling data. Thereareseveralimportantfactorsthataffecttheaccuracyof GPS/ Levelling geoid model (Erol and elik, 2004).These are; Distributionandnumberofreferencestations(GPS/ Levellingstations)(thesepointsmustbedistributed homogeneously to the coverage area of the model and have to be chosen to figure out the changes of geoid surface). The accuracy of GPS derived ellipsoidal heights (h) and the heights derived from levelling measurements (H). Characteristic of the geoid surface in the area. Used method while modelling the geoid (researches showed that there is not a unique model works properly for realizing the geoid surface of different areas) (Erol and elik, 2004). Interpolationmethodsaremostcommonapproachesthatare usedwhilemodellingthegeoidheights(N)inalocalarea.Therearedifferentinterpolationalgorithmsandeachofthem canhavedifferentresultswhen interpreting your data.Inverse distanceweightingandgeostatisticalKrigingaretwoofthem andverypopularalsoinmodellingspatialdataaswellasin evaluationofsomeotherdatasetsofdifferentdisciplines (Anonym, 1999). In this study these two methods have been chosen to model the geoidundulationsinalocalareaandtoprovethattheygive satisfying results to serve practical applications in geodesy.By applyingthem,theprovidedresultsandtheperformancesof both are compared each other.These two interpolation methods constitutessolutionalgorithmsofageoidmodellingsoftware thatisstillindevelopingprocessastheproductofathesis study, and because of that in this study, these two methods were evaluatedtoinvestigateandcomparetheperformanceof interpolation algorithms in local geoid modelling. InverseDistanceWeighting(IDW)isaweightedaverage interpolator.Thepowerparametercontrolshowtheweighting factorsdropoffasdistancefromthereferencepointincreases (Anonym,1999).Thisisfastandhasrelativelysimple mathematical algorithm. Inequation2and3,therelationsaregiventocalculatethe geoid undulation point k as related to the surrounding reference pointsi=1,2,3..n.Thepointkandneighborreferencepoints contributedtothecomputationofgeoidundulationvalueof pointkaccordingtocertainweightvaluesareillustratedin Figure 2 (in the figure i=7). ===n1 iin1 ii iPP NN'(2) where N = geoid undulation value of point k Ni = geoid undulation value of ith reference point Pi = weight of ith reference point niid1P =(3) where di=distancebetweeninterpolationpointkandith reference point in kilometer n= power of the distance, it can be 1, 2, 3 or 4. In the case study, n has chosen 3 empirically. Figure 2.Interpolation point k of which geoid undulation value is going to be computed and neighbor reference points that contributes the computation of geoid undulation at point k. In general, it is not usual to contribute all the reference points to thecomputationofgeoidundulationvalueaccordingtoIDW interpolationmethod.Becauseofthataboundaryisdescribed tosurroundanareatoincludethereferencepointswhichare goingtobecontributetothecomputation.Thereareseveral waystodescribetheboundary.Oneofthemistodeterminea circleasbeingtheinterpolationpointinthecenterofit.The radius of the circle is determined according to conditions of the coverareaofthestudyandinthisexamplea3kmradiuswas describedbyconsideringthetopographicalpropertiesofthe local area. Krigingisageostatisticalapproachtointerpolatedatabased uponspatialvarianceandhasprovenusefulandpopularin many fields in geodesy as well.This is a considerably flexible methodandsimilartoIDWwherebyproximityandinfluence areassumedtoberelated,Krigingrecognizesthatspatial varianceisafunctionofdistance(Wilson,1996).Itcanbe custom fit to a data set by specifying the appropriate variogram model.Thevariogrammodelmathematicallyspecifiesthe spatialvariabilityofthedataset.Theweightsofreference points, which are applied to data points during calculations, are direct functions of the variogram model (Anonym, 1999). Computinganexperimentalvariogramfromyourdataisthe onlycertainwaytodeterminewhichvariogrammodelyou shoulduse.Adetailedvariogramanalysiscanofferinsights intothedatathatwouldnototherwisebeavailable,andit allowsforanobjectiveassessmentofthevariogramscaleand anisotropy (Anonym, 1999).In an other word structural effects havebeenaccountedforthesurfacemaybemodeledonthe basisofvariationsasafunctionofdistance.Amathematical modelisfittothisdatain ordertointerpolateunknownvalues atotherlocations.Spatialvariationmayalsobemore predominateincertaindirections.Anisotropyimpliesthe preferreddirectionofhigherorlowercontinuitybetweendata points.Anisotropy is applied by specifying an anisotropy ratio which states: Give more weighting to points located along one axisversuspointslocatedalonganotheraxis.Therelative weighting is defined by an anisotropy ratio (Wilson, 1996). Themethodbasedontherecognitionthatthespatialvariation ofanyproperty,knownasaregionalizedvariable,istoo irregular to be modeled by a smooth mathematical function but canbedescribedbetterbyastochasticsurface.The interpolation proceeds by first exploring and then modelling the stochasticaspectsoftheregionalizedvariable.Theresulting informationisthenusedtoestimateweightsforinterpolation points and in the Kriging method, the logic is similar. Geostatistics,similartoanyformofstatistics,hastwomain criteriawhichmustbemet.Allstatisticalmodelshave assumptions which need to be recognized and expectantly meet beforethemodelisused.Thesecondcriterion,astatistical analysis,toperformonthedataset,shouldbeexploredfor normalityandspatialvariancewithdatatransformations applied as necessarily.Geostatistical assumptions are reviewed byIssaaksandSrivastava,1989.Regionalizedvariabletheory assumesthatspatialvariationisthesumofthreecomponents (Wilson, 1996); A first order effect, in another word, known as a structural component, which is defined by a constant trend.This part gives trend and is depicted graphically in a variogram as the sill.Itimpliesthatatthese values of the lag (distance) thereisnospatialdependencebetweenthedatapoints based on variance. Asecondordereffectisdefinedasarandomspatially correlatedcomponent.Referredtoasvarianceof differences and this is a function of distance between sites (thedistanceiscalledaslag).Varianceincreasesfrom randomnoise(thenugget)tothesillasdistanceincreases.Thedistanceisimportantasitspecifiesthedistancesite differences are spatially dependent. Random noise or residual error, known as the nugget. ForadetailedexplanationaboutKrigingMethodIsaaksand Srivastava, 1989; Watson, 1992 and Deutsch and Journel, 1992 can be seen. Inthecasestudy,universalKrigingmethodwasappliedwith linearvariogrammodel.Theresultsofthecasestudywillbe given under following title. 1 3 4 5 6 7 k 2 4.NUMERICAL EXAMPLE Thestudyhasbeencarriedoutwithin45x50km2localregion usingGPS/Levellingdata.TheareaisinthewestofTurkey near Agean Sea, Izmir Metropolitan Region.For modeling the geoidintheareaasananalyticalsurface,twointerpolation methodswereapplied.Mathematicalalgorithmsofthesetwo interpolationmethodshavebeenprogrammedastocomprise thecomputationmodulesofa LocalGeoid Modeling software, however,thementionedsoftwarehasnotbeencompletedyet, ontheotherhanditisplanedtogivethefinalversiontothe software by the year of 2005. Inthearea,181oftotally301GPS/Levellingpointswere decidedtouseasreferencepoints(modellingbenchmarks), whichcontributesthecomputationofnewpointsaccordingto appropriateinterpolationmethod,andrest120GPS/Levelling points (testing benchmarks) were used for testing the developed interpolationalgorithmsaccordingtoIDWandKriging Methods.Duringtheselectionoftestpointsset topography of theregionandalsokeepingthehomogeneousdistributionof referencepointssetwereconsidered.Thedistributionofboth point sets can be seen in Figure 3. 26.8 26.85 26.9 26.95 27 27.05 27.1 27.15 27.2 27.25 27.3LONGITUDE38.2538.338.3538.438.4538.5LATITUDEModeling BenchmarksTesting BenchmarksREFERENCE POINTS OF IZMIR LOCAL GEOID Figure 3.The distributions of modelling benchmarks and test benchmarks in the area of study(Erol and elik, 2004). IDWandKriginginterpolationalgorithmswereapplied accordingtodevelopedprogramcodes.IDWinterpolation methodhasasimplecomputationalgorithmthanKriging method.However,themostcomplicatedpartismodellingthe variogram, after that the rest of the process is similar to IDW in thatprescribedweightscontrolthe interpolation.Variogram is animportanttooltoexplorespatialdatasets.Accordingto handled results: AccordingtoIDWmodelandusingthepreviouslymentioned modelingparameters,themodelwastestedfor121testpoints.Thetestwascarriedoutbyanalyzingthedifferencebetween computed geoid heights from the model and geoid heights from measurements.As the result of these analysis, it was seen that the model fitted the data with 3.42 cm root means square error.Thisisalsoanindicatorforusabilityofthemodelfor computing the new points. WhileUniversalKrigingmethodwasappliedwiththesame data set, it was seen that the model fitted the data with 3.07 cm rootmeanssquareerror.Iftheseresultsarecomparedeach other,itcanbesaidthatKrigingmethodcanhandlethedata better than IDW. Aboutthecomputationtime,theperformanceofboth techniques are similar each other.However, as it is mentioned before, the algorithm of Kriging method is much more complex than IDW for programming. At the end of the analysis, it was decided that both interpolation methodgavereasonableresultsforthislocalareaandtheyare applicableformodellingthegeoidofthisareaasananalytical surfaceandtoservepracticalgeodeticapplications.Onthe other hand, according to researches it mustnt be neglected that theeligibilityofamathematicalmodeltoanareawhile modelingtheGPS/Levellinggeoidisstrictlydependsonthe properties of study area.So, it is necessary to test a developed modelthatifitisappropriateformodellingthegeoidofan area, before applying the new data. 5.CONCULISON AND FUTURE WORKS ItisobviousthatafterGPStechniqueshavebecomewide spreadingeodeticpurposes,geoidmodeldeterminationfor especiallyuseinpracticalapplicationsofgeodesyhasan increasedimportance.So,thisstudysmainwastoproduce some solutions to practical applications, such as large scale map production,GISapplications,engineeringsurveying applications etc. especially at a national level. Therefore,to test the performances and applicability of surface fittingmethodsingeoidmodellingusingGPS/Levelling measurements in a local area, this study was carried out. Inthestudy,twointerpolationtechniques,InverseDistance WeightingandKrigingwereappliedseparately.Inboth applications,181GPS/Levellingpointswereusedasa referencepointssetforgeneratingthemodelandanother121 GPS/Levellingpointsastestpointssetwereusedtotestthe structured models. Intheresults,ithasbeenseenthatbothinterpolationmethods areapplicableformodellingthegeoidofstudyareaasan analytical surface; however, Kriging method fits the data better than IDW method according to test criteria. Thecomputationtimeissimilarforbothmethods,whilethe programcodesrun.Ontheotherhand,Krigingmethodhas muchmorecomplexcomputationalgorithmthanIDWmethod has. Localgeoidmodelshaveaspecialimportanceespeciallyfor geodeticinfrastructureofTurkey.Because,regionalgeoid modelofTurkeyhasntgothomogeneousaccuracyinevery part of the Turkey.Because of that in some part of the country, it is necessary to be supported with precise local geoid models.Surfacefittingmethodsareverypracticalwiththisrespect; however, it is important to assess the appropriate surface fitting approach.Soitisnecessarytotestthemodelbeforeapplying the new points. Asthefutureworksitisplanningtoprogramsomeotherdata modelingtechniquestoaddthedevelopinggeoidmodelling software.Alsotocombineregionalgeoidmodelandlocal geoidmodeltocreateawholeandimprovedgeoidmodelfor the study areas is planning. 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