3D Geological Modeling: Solving as a Classification Problem with
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119 3D Geological Modeling: Solving as a Classification Problem with the Support Vector Machine By Alex Smirnoff, Eric Boisvert, and Serge J. Paradis Geological Survey of Canada GSC-Quebec 490 de la Couronne Quebec, Canada G1K 9A9 Telephone: (418) 654-3716 Fax: (418) 654-2615 e-mail: [email protected] ABSTRACT The process of creating multi-unit 3D geological models by successive unit interpolation may be tedious and time-consuming. Here, we propose to automate this procedure through presenting the problem as a classifica- tion task and solving it simultaneously with the Support Vector Machine (SVM), a method known from the field of artificial intelligence. Experiments with various input data and kernel parameters demonstrated that the SVM has great potential in 3D reconstructions from sparse geologi- cal information. An extended version of this paper has been accepted for publication in “Computers and Geosci- ences” (Smirnoff et al., 2008). INTRODUCTION Often, geologists are faced with a variety of diverse information that requires generalization and analysis. 3D modeling software packages such as Gocad of Earth Decision Sciences have proven an excellent means for data presentation and interpretation. The procedure normally requires reconstruction of individual geological units using surfaces interpolated from control points with subsequent fusion of these units into a single model. The popular interpolation techniques include nverse Distance Weighting (DW), Discrete Smooth nterpolation (DS), and various flavors of kriging preceded by semi-vario- gram analysis. The above procedure can easily become a tedious and time-consuming task when a complex geomodel is considered. n addition, the traditional interpolation techniques assume reasonable areal coverage of the input data. Therefore, there is a strong need for an algorithm that would automate the process of model creation even in cases when only a few pieces of information on regional geology, (e.g., a few sparse cross-sections) are available. Finding such an algorithm and testing its performance on available data sets was the objective of this study. Here, we propose the use of the Support Vector Ma- chine (SVM), a tool routinely applied in the field of image analysis and pattern recognition. The SVM is becoming increasingly popular and has been successfully used to solve classification and regression problems in biol- ogy (e.g., Noble et al., 2005), hydrology (e.g., Yu et al., 2004), medicine (e.g., El-Naqa et al., 2002), and environ- mental science (e.g., Gilardi et al., 1999). n this study, we demonstrate that the application of SVM in geology allows sparse data to be efficiently combined in order to reconstruct shape, area, and volume of multiple geologi- cal units. METHODOLOGY The SVM Algorithm The SVM algorithm is based on the Statistical Learn- ing Theory developed by V. Vapnik (Vapnik, 1995). t uses a set of examples with known class information to build a hyperplane that separates samples of differ- ent classes. n machine learning theory, this is known as supervised learning as opposed to unsupervised learn- ing when no a priori class information is available. This initial dataset is known as a training set, and every sample within it is characterized by features upon which classifi- cation is based. Figure 1A demonstrates this for the one- dimensional (single-feature) case. The samples closest to the hyperplane are termed support vectors (filled marks in Figure 1). n more complicated, non-linear cases, the task of discovering the separator is turned into a linear task by transferring input data into a higher-dimensional space known as the feature space. Figure 1B shows a non-sepa- rable one-dimensional data set in the input space. The problem is easily solved through re-mapping data to a higher, two-dimensional space where a linear solution is found (Figure 1C). Functions satisfying certain conditions and known as kernels are normally employed for this
Transcript of 3D Geological Modeling: Solving as a Classification Problem with
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3D Geological Modeling: Solving as a Classification Problem with the Support Vector Machine
ByAlexSmirnoff,EricBoisvert,andSergeJ.Paradis
GeologicalSurveyofCanadaGSC-Quebec
490delaCouronneQuebec,CanadaG1K9A9Telephone:(418)654-3716
Fax:(418)654-2615e-mail:[email protected]
ABSTRACT
Theprocessofcreatingmulti-unit3Dgeologicalmodelsbysuccessiveunitinterpolationmaybetediousandtime-consuming.Here,weproposetoautomatethisprocedure through presenting the problem as a classifica-tiontaskandsolvingitsimultaneouslywiththeSupportVector Machine (SVM), a method known from the field of artificial intelligence. Experiments with various input data andkernelparametersdemonstratedthattheSVMhasgreatpotentialin3Dreconstructionsfromsparsegeologi-calinformation.Anextendedversionofthispaperhasbeenacceptedforpublicationin“ComputersandGeosci-ences”(Smirnoffetal.,2008).
INTRODUCTION
Often,geologistsarefacedwithavarietyofdiverseinformationthatrequiresgeneralizationandanalysis.3DmodelingsoftwarepackagessuchasGocadofEarthDecisionScienceshaveprovenanexcellentmeansfordatapresentationandinterpretation.Theprocedurenormallyrequiresreconstructionofindividualgeologicalunitsusingsurfacesinterpolatedfromcontrolpointswithsubsequentfusionoftheseunitsintoasinglemodel.Thepopularinterpolationtechniquesinclude�nverseDistanceWeighting(�DW),DiscreteSmooth�nterpolation(DS�),and various flavors of kriging preceded by semi-vario-gramanalysis.
Theaboveprocedurecaneasilybecomeatediousandtime-consumingtaskwhenacomplexgeomodelisconsidered.�naddition,thetraditionalinterpolationtechniquesassumereasonablearealcoverageoftheinputdata.Therefore,thereisastrongneedforanalgorithmthatwouldautomatetheprocessofmodelcreationevenincaseswhenonlyafewpiecesofinformationonregionalgeology,(e.g.,afewsparsecross-sections)areavailable.Findingsuchanalgorithmandtestingitsperformanceonavailabledatasetswastheobjectiveofthisstudy.
Here,weproposetheuseoftheSupportVectorMa-chine (SVM), a tool routinely applied in the field of image analysisandpatternrecognition.TheSVMisbecomingincreasinglypopularandhasbeensuccessfullyusedtosolve classification and regression problems in biol-ogy(e.g.,Nobleetal.,2005),hydrology(e.g.,Yuetal.,2004),medicine(e.g.,El-Naqaetal.,2002),andenviron-mentalscience(e.g.,Gilardietal.,1999).�nthisstudy,wedemonstratethattheapplicationofSVMingeologyallows sparse data to be efficiently combined in order to reconstructshape,area,andvolumeofmultiplegeologi-calunits.
METHODOLOGY
The SVM Algorithm
TheSVMalgorithmisbasedontheStatisticalLearn-ingTheorydevelopedbyV.Vapnik(Vapnik,1995).�tusesasetofexampleswithknownclassinformationtobuildahyperplanethatseparatessamplesofdiffer-entclasses.�nmachinelearningtheory,thisisknownassupervisedlearningasopposedtounsupervisedlearn-ingwhennoaprioriclassinformationisavailable.Thisinitialdatasetisknownasatrainingset,andeverysamplewithin it is characterized by features upon which classifi-cationisbased.Figure1Ademonstratesthisfortheone-dimensional(single-feature)case.Thesamplesclosesttothe hyperplane are termed support vectors (filled marks in Figure1).
�nmorecomplicated,non-linearcases,thetaskofdiscoveringtheseparatoristurnedintoalineartaskbytransferringinputdataintoahigher-dimensionalspaceknownasthefeaturespace.Figure1Bshowsanon-sepa-rableone-dimensionaldatasetintheinputspace.Theproblemiseasilysolvedthroughre-mappingdatatoahigher,two-dimensionalspacewherealinearsolutionisfound(Figure1C).Functionssatisfyingcertainconditionsandknownaskernelsarenormallyemployedforthis
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Figure 1.Buildingtheseparatinghyperplane,inseparableandnon-separableone-dimensionalcase.Filledmarksrepresentsupportvectors.(A)Linearlyseparablecaseanddecisionfunctionininputspace;(B)non-separablecaseininputspace.(C)trainingdatare-mappedintotwo-dimen-sionalfeaturespaceusingφ(x) = (x, x2) and linearsolutioninthisspace;(D)solutionre-plottedbackininputspace.
transfer(e.g.,Abe,2005).Thesolutionbecomesnon-lin-earwhenshownintheoriginaldataspace(Figure1D).
Once the equation for the optimal classifier is found, newdatawithunknownclassinformation(testsamples)can be classified based on the value of this decision func-tion.Unlikemostinterpolationmethodsbasedontheprin-ciplethatvaluesatpointscloserinspacearemoresimilar,the SVM is a boundary classification method where the boundaryisbuiltbasedontheinitialtrainingsetamongwhichonlyasmallnumberofsamples(supportvectors)are involved in the final decision making.
TheclassicalSVMtaskisabinary(two-class)clas-sification. However, a number of methods have been developed to support multi-class classification through variouscombinationsofbinarymethodssuchas“one-against-all”,“one-against-one”,etc.(seeHsuandLin,2002forreferences).Therefore,theSVMapproachisalsoapplicableformodelswithmorethantwoclasses.MoredetaileddescriptionsoftheSVMalgorithmareavailablefromanumberofsources(e.g.,CristianiniandShawe-Taylor,2000;Abe,2005).
SVM Application to 3D Modeling
ToapplytheSVMalgorithmtoourgeologicalrecon-structions, we defined the 3D space-partitioning task as a pure spatial classification problem. Three coordinates uniquelydescribeeverypointinthe3Dreconstructionspace.However,onlyalimitednumberofthosepointspossessdescriptionsorclassinformationthatcanbeiden-tified through well drilling, surficial geology mapping, and seismic profiling. The class information describes the geologicalunittowhicheachparticularpointbelongs.Therefore,thepointswithknownclasslabelsbecomesamplesintheSVMtrainingset,andpointcoordinatesinthethree-dimensionalspaceareusedassamplefeatures.Once a classification model based on this training set is built,therestofthepointsinthereconstructionspacecanbe classified based on their coordinates (features).
WeemployedoneofthemanySVMimplementationsfreelyavailableovertheinternet,namelyL�BSVMdevel-opedattheNationalTaiwanUniversity(seeChangandLin,2001fordetaileddescription).AsrecommendedinHsuetal.(2004),weusedL�BSVMwiththeradialbasisfunction(RBF)kernel,themostgeneralformofkernelresultinginapredictionmodelcontrolledbyonlytwohy-perparameters,Candγ.Asinglesolutionisobtainedforeverypairofparameters,anditissensitivetothechoiceoftheirvalues.However,selectingtheappropriatevaluesisadarkartnormallydoneonatry-and-seebasis.
For multi-class classification, LIBSVM uses “one-against-one”approach,whichwasshowntobeadvanta-geoustoothermethodsforpracticaluse(HsuandLin,2002).�naddition,asetofin-houseJavautilitieshasbeendevelopedforscaling,validation,andformatconversionpurposes.
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Data
A3DgeomodelcreatedattheGeologicalSurveyofCanada,Quebec,inthecourseoftheEsker/Abitibiproject(Bolducetal.,2005)wasusedasthereferencedatasetinalloftheexperiments.Thismodelisbasedonsurfacegeology,well,andcross-sectiondata.SixgeologicalunitsweresequentiallyinterpolatedfromtheabovecontrolpointsusingtheDiscreteSmooth�nterpolation(DS�)technique(Mallet,1989)intheGocadG�S.
Experimental Work
Theexperimentalworkwasdesignedtoperformthefollowingtasks:(1)investigatewhethertheSVMcanbeefficiently used in binary geological reconstructions, (2) testtheSVMformulti-unitmodeling,and(3)examinehowtheresultingmodeldependsontheRBFkernelpa-rametersusedinthereconstruction.
General Approach
Thegeneralapproachtakeninalloftheexperimentswasasfollows:
Prediction
• �nGocad,createareconstructionspaceasasetofvolumeelements(voxet)oftheshaperepresenta-tiveofstudyareageometry.Thereconstructionspace was defined by a voxet with the following numberofvolumeelements(voxels)ineachdirec-tion:X-110,Y-240,Z-24.
• Addavailabledatatothereconstructionspace.Theunittypepropertyforeachofthesixgeologi-calunits(SVMclasses)wastransferredfromthestratigraphicgrid(SGrid)structurerepresentingthereferencemodel.
• �nGocad, using a DEM, define all voxet nodes abovethesurfaceasairorno-datapoints.
• Define a training set for the experiment. For the re-maininggroundpoints,settheunittypepropertytozero; these are the points that will be later classified bytrainedSVM.
• Scalecoordinatevaluesforthetrainingsetbetween0and1asrecommendedinHsuetal.(2004).
• UsingL�BSVM,buildapredictionmodelbasedonthetrainingset.Asinglereferencesetofkernelparameters(C=104andγ=102)previouslydeter-minedfrom2Dexperimentswasusedinallrecon-structionsexcepttheparametersensitivitytests.
• Scalecoordinatevaluesforthepointstobeclassi-fied and classify them using the prediction model createdinthepreviousstep.
• �mportthepointsetwithpredictedclassinformationbackintoGocad,forvisualizationandanalysis.
Validation
• Based on the available reference data, define the validationsetandextractitasasetofpointswithattachedclassproperty.
• Testpredictedclasslabelsagainstthevalidationsettodeterminehowmanyoriginalpointsineachclass and overall were adequately classified, a mea-surealsoknownastherecallrate.
Binary reconstruction
ThetrainingsetwascomposedoftheEsker/Abitibimodelpointslocatedon11arbitrarilychosenparallelsectionsorientedalongaxisX.PointsweregroupedintotwoclassesasshowninTable1.TheinputdatastatisticsaregiveninColumn4ofTable1.Asseenfromthetable,thetrainingsetwasdominatedbypointsrepresentativeofClass2,whichcombinedallmodelunitsexcepttheEskerUnit.Thevalidationsetwascomposedofalltheremain-ingmodelpoints(notincludedinthetrainingset).ThenumberofpointsusedforvalidationineachofthetwoclassesisshowninColumn1aofTable3.
Table 1.Geologicalunits,SVMclasses,andtrainingsetstatisticsforEsker/AbitibiBinaryModel.Totalnumberofpoints to be classified is 371783.
1. Geological Unit 2. SVM Class 3. All Points (#/%)b 4. Training Points (#/%)c
Esker 1 20300/5.22 995/0.26 Non-Eskera 2 368935/94.78 16457/4.23 AllUnits - 389235/100 17452/4.48
aNon-EskerunitincludedRock,Till,Clay,LittoralandOrganicunitsbNumberofallclasspointsandtheirpercentageofallmodelpointscNumberoftrainingpointsperclassandtheirpercentageofallmodelpoints
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Multi-Class reconstruction
Thesametrainingpoints,arrangedintosixclassescorrespondingtothesixgeologicalunitsfoundintheoriginalmodel,wereusedtotesttheSVMcapabilitiesin multi-class classification (Table 2). For training data statistics,seeColumn4ofTable2.Thistime,bedrock(Class6)entirelydominatedthetrainingset,withorgan-ics(Class1)beingtheleastrepresentative.ThevalidationsetalsocontainedinformationaboutallsixgeologicalunitsasshowninColumn2aofTable3.
Hyperparameters Sensitivity tests and Multiple Parameter-Set reconstructions
Toanalyzethesensitivityofpredictionresultstothevaluesofhyperparameters,Candγ,weusedasimplegridsearchprocedureasproposedinHsuetal.(2004).Thegrid search was run for the above training set configu-rations,andtherangeofparametersscannedbyevery
searchwasfrom2-8to215forCandfrom2-15to212forγincrementingparametervaluesbyapowerof2.Asinpreviousexperiments,thesuccessratewasdeterminedthroughdirectcomparisonwiththevalidationsetextract-edfromthereferencemodel.Wealsoexaminedthede-pendencyofsuccessrateonparametervaluesfortheclasswiththeleastnumberoftrainingpoints(Class1–Organ-ics).Finally,binarymodelswerebuiltwithcombinationsofparametersdrawnfromthemarginsofthereasonableworkingrange.TheseincludedlowC(2-2)–highγ(28),lowC(2-2)–lowγ(25),averageC(27)–averageγ(26),highC(214)–highγ(28)andhighC(214)–lowγ(25).
RESULTS AND DISCUSSION
Binary Reconstruction
Theoriginaleskerbody,trainingsections,andthere-sultsofbinaryreconstructionwiththereferenceparametersetareshowninFigure2.Column1bofTable3describes
Table 2. Geologicalunits,SVMclasses,andtrainingsetstatisticsforEsker/AbitibiMulti-ClassModel.Totalnumberofpoints to be classified is 371783.
1. Geological Unit 2. SVM Class 3. All Points (#/%)a 4. Training Points (#/%)b
Organics 1 1210/0.31 48/0.01 Littoral 2 3819/0.97 193/0.05 Clay 3 13295/3.42 628/0.16 Esker 4 20300/5.22 995/0.26 Till 5 15865/4.08 747/0.19 Bedrock 6 334746/86.00 14841/3.81 AllUnits -- 389235/100389235/100389235/100 17452/4.4817452/4.4817452/4.48
aNumberofclasspointsandtheirpercentageofallmodelpointsbNumberoftrainingpointsperclassandtheirpercentageofallmodelpoints
Table 3.ValidationdataandresultsforEsker/Abitibibinaryandmulti-classmodel.Numberofvalidationpointsinorigi-nal model and percentage of points properly classified by SVM.
SVM Class 1. Binary 2. Multi-Class
a. Validation Points (#/%)a b. Success (%) a. Validation Points (#/%)a b. Success (%)
1 19305/5.19 71.50 1162/0.31 18.76 2 352478/94.81 98.87 3626/0.98 37.20 3 - - 12667/3.41 57.10 4 - - 19305/5.19 67.65 5 - - 15118/4.07 45.72 6 - - 319905/86.05 95.45 All 371783/100 97.34 371783/100 89.87
aNumberofvalidationpointsperclassandtheirpercentageofallmodelpoints
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Figure 2.Binaryeskerreconstruction.(A)OriginalEskerUnit;(B)trainingset;(C)reconstructedEskerUnit.
thevalidationresults.AsseenfromTable3,thesuccessrateofSVMpredictionisexceedinglyhigh.Especiallyremarkableresults,98.87%,areachievedinClass2.�npart,thiscanbeattributedtothefactthatpointsofthisclassentirelydominatethetrainingset.WhentheSVMcannot classify a point in binary classification, it tends toattributeittothepredominantclass.Consideringthat
bedrockpointsconstitute94.81%ofallpointsthatneedto be classified (352478 of 371783 as shown in Column 1ofTable3),itisnosurprisethattheoverallsuccessofpredictionachieves97.34%.
With the above explanation in mind, the classifica-tionsuccessinClass1,whichrepresentsonlyabout6%ofthetrainingset,isstillashighas71.50%.This,inouropinion,provesthattheSVMcanbeeffectivelyusedforbinary(single-unit)reconstructionsevenwithtrainingsetssubstantiallyskewedtowardoneoftheclasses.
WefurtheranalyzedsuccessrateinClass1onallmodelsectionswheretheEskerUnitwaspresent(234sections).TheresultsarepresentedinFigure3.Thefigure clearly demonstrates that the success of prediction decreasesasthedistancefromatrainingsectionincreases.Astrainingsection#1didnotintersecttheeskerbody,the success rate on the first 18 sections drops to 0%. Therefore,ascouldbeexpected,theoverallreliabilityofpredictionisdirectlyproportionaltothedensityofsec-tionswithtrainingdata.
Multi-Class Reconstruction
TheresultsofthisexperimentarefoundinFigure4andColumn2bofTable3.Theoverallsuccessscoreis89.87%,whichismainlycontrolledbythepredominantbedrockclass(Class6).Twootherclasses,eskerandmarineclay,demonstratesuccessratesover50%.TheseunitsaresomewhatbetterrepresentedintheSVMtrainingsetthantheremainingclasses.Figure5showsthatthesuccessofreconstructionforaparticularclassinthisex-perimentisalmostdirectlyproportionaltothenumberofthoseclasspointsinthetrainingset,exceeding75%whenthenumberoftrainingpointsexceeds1%ofthetotal.
Wealsocomparedareaandvolumeofgeologicalunitsintheoriginalmodelanditsreconstructedcounter-part.TheresultspresentedinTable4andFigure6showthat,forbothareaandvolumecalculations,therecon-structedandoriginalvaluesforanyunitarethesameorderofmagnitude.Thissuggeststhatalongwithsingleunit modeling, the SVM can efficiently be applied in multi-unitvolumetricreconstructions.
Hyperparameter Sensitivity Tests and Multiple Parameter-Set Reconstructions
Figure7summarizestheresultsofgridsearchforthebestpairofhyperparametersinbinaryandmulti-classreconstructions.Thebestoverallsuccessrates,97.79%and92.03%,wereachievedat[C=21,γ=26]and[C=2-1,γ=26],respectively.Theanalysisofsuccessratesintheclasswiththeleastnumberoftrainingpointsyielded77.38%and22.46%at[C=22,γ=26]and[C=29,γ=25],correspondingly.Asseenfromtheresults,parametersarefairlystable,andasinglerangeforC[2-3-215]andγ[24-
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Figure 3.Validationresultsforbinaryeskerreconstructionfrom11parallelsections.SuccessinClass1(Esker)againstsectionnumber.Verticallinesindicatetrainingsections.Totallengthofhorizontalaxisis24kmandsectionsarespacedat100m.
Figure 4.Multi-classeskerreconstructionfrom11parallelsections.(A)Originalmodel;(B)trainingset;(C)reconstructedmodel.
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Figure 5.Validationresultsformulti-classeskerreconstructionfrom11parallelsections.Successperclassvs.numberoftrainingpointsperclass.
Figure 6.Surfaceareaandvolumecomparisonfororiginal(referencemodel)andreconstructedgeologicalunits.(A)Surfacearea;(B)volume.
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Table 4.ResultsofSVMreconstructionfrom11parallelsectionsforEsker/Abitibi.Unitareaandvolumecomparison(originalmodelvs.reconstructed).SeeTable2fortrainingsetstatistics.
Geological Unit 1. Area (m2) 2. Volume (m2. Volume (m3)
a. Original b. Reconstructed a. Original b. Reconstructed
Organics 2.55E+07 2.99E+07 8.06E+07 1.05E+08Littoral 7.56E+07 7.34E+07 2.54E+08 3.30E+08Clay 1.99E+08 1.97E+08 8.86E+08 1.01E+09Esker 1.48E+08 1.33E+08 1.35E+09 1.16E+09Till 2.15E+08 1.99E+08 1.06E+09 1.07E+09Bedrock 2.70E+08 3.06E+08 2.23E+10 2.23E+10
Figure 7.Summaryofbestresultsfromparametersensitivitytestsforbinaryandmulti-classreconstructionsandproposedrangeforRBFkernelparameters(C[2-3,215]andγ[24,29]).
29]canberecommended.Withinthisrange,higheroverallscoresandhigherscoresforover-representedclassesareachievedatsomewhatlowerCvalues.Ontheotherhand,proper classification of points in the least represented classesrequireshigherCvalues.VisualexaminationofbinarymodelsbuiltwithcombinationsofparametersdrawnfromdifferentcornersoftheaboverangealsoshowthatamoregeneralizedpicturecanbeachievedatlowerC’s(Figure8a,8b)whilehighervaluesofthisparameterpromotemoredetailedinterpretation(Figure8d,8e).AverageCvaluesresultinwell-balancedmodels(Figure 8c). The influence of the second parameter (γ)isnotasobvious.
CONCLUSIONS
OurexperimentsclearlyshowedthattheSVMwithRBF kernel can be efficiently used for both single- and multi-unit3Dreconstructions.Theprocedureisper-formedinasinglestep,whicheliminatestheneedforunit-by-unitinterpolation.Evenfromalimitedtrainingset(e.g.,severalcross-sectionssparselydistributedacrossthestudyarea)reasonablereconstructionresultscanbeachieved.
�tisimportant,however,thatallclassestoberecon-structedarereasonablyrepresentedinthetrainingset.�nthemulti-classcase,thereconstructionsuccesswas
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showntobedirectlyproportionaltothenumberofunitsamplesinthetrainingdata.Thereliabilityofpredic-tionisgreaterinthevicinityofthetrainingdata,andtherefore,thedensityoftrainingsectionsandspatialcontinuityoflithologicalunitsmaydirectlyaffectthereconstructionresults.
Thekernelparametersshouldbechosenfromtherange2-3-215forCand24-29forγ.Whenmoremodeldetailsarerequiredorclasseswithasmallnumberoftrainingpointsareinvolved,higherCvaluesshouldbeconsidered.LowerCvaluesresultinmoregeneralizedmodelswithfewerdetails.Thisfavorsclassesthatdomi-natethetrainingset.
Finally,ourresultsindicatethatwhenappropri-ateparametersarechosen,notonlythegeneralshapeofageologicalbody,butalsosuchcharacteristicsasitssurfaceareaandvolumecanbereconstructedwithresultsclosetothoseobtainedfromtheapplicationofclassicalG�Smethods.
Figure 8.BinaryreconstructionswithparametersdrawnfromdifferentcornersoftherangepresentedinFigure7.(A)lowC(2-2)–highγ(28);(B)lowC(2-2)–lowγ(25);(C)averageC(27)–averageγ(26);(D)highC(214)–highγ(28);(E)highC(214)–lowγ(25).
ACKNOWLEDGEMENTS
TheresearchwasconductedattheQuebecDivi-sionoftheGeologicalSurveyofCanada(GSC),andwasfundedbytheGroundwaterProgramoftheEarthSci-encesSectorofNaturalResourcesCanada(contributionnumber20060335).WeareespeciallygratefultoAndréeBolduc(GSC)forherinvolvementinthisworkandforhavingmadetheGocad files with initial data and 3D Es-ker/Abitibimodelusedinexperimentsreadilyavailable.
REFERENCES
Abe, S., 2005, Support Vector Machines for Pattern Classifica-tion:Springer-Verlag,London,343p.
BolducA.M.,ParadisS.J.,RiverinM.N.,LefebvreR.,andMi-chaudY.,2005,A3Deskergeomodelforgroundwaterre-search:ThecaseoftheSaint-Mathieu–Berryesker,Abitibi,Quebec,Canada,inRussell,H.,Berg,R.C.,andThorleif-
128 D�G�TALMAPP�NGTECHN�QUES‘06
son,L.H.eds.,Three-DimensionalGeologicalMappingforGroundwaterApplications,GeologicalSurveyofCanada,Ottawa,Ontario,OpenFile5048,p.17-20.
Chang,C-C.,andLin,C-J.,2001,L�BSVM:aLibraryforSup-portVectorMachines,accessedathttp://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.pdf.
Cristianini,N.,andShawe-Taylor,J.,2000,SupportVectorMachines:CambridgeUniversityPress,189p.
El-Naqa,�.,Yang,Y.,Wernik,M.N.,Galatsanos,N.P.,andNishikawa,R.,2002,Supportvectormachinelearningforthe detection of microcalcifications in mammograms: IEEE TransactionsonMedical�maging21,p.1552-1563.
Gilardi,N.,Kanevski,M.,Maignan,M.,andMayoraz,E.,1999,Environmental and Pollution Spatial Data Classification withSupportVectorMachinesandGeostatistics:WorkshopW07“�ntellegenttechniquesforSpatio-TemporalDataAnalysisinEnvironmentalApplications”,ACA�99,July,Greece,p.43-51.
Hsu,C-W.,Chang,C-C.,andLin,C-J.,2004,APracticalGuide
to Support Vector Classification, accessed athttp://www.csie.ntu.edu.tw/~cjlin/papers/guide/guide.pdf.
Hsu,C-W.,andLin,C-J.,2002,Acomparisonofmethodsformulti-classsupportvectormachines:�EEETransactionsonNeuralNetworks13(2),p.415-425.
Mallet,J-L.,1989,DiscreteSmooth�nterpolation:ACMTrans-actionsonGraphics8(2),p.121-144.
Noble,W.S,Kuehn,S.,Thurman,R.,Yu,M.,andStamatoyan-nopoulos,J.,2005,Predictingtheinvivosignatureofhumangeneregulatorysequences:Bioinformatics21(1),p.338-343.
Smirnoff,A.,Boisvert,E.,andParadis,S.J.,2008,SupportVectorMachinefor3Dmodellingfromsparsegeologicalinforma-tionofvariousorigins:ComputersandGeosciences34,p.127-143.
Vapnik,V.,1995,TheNatureofStatisticalLearningTheory:Springer-Verlag,NewYork,311p.
Yu,X.,Liong.,S-Y.,andBabovic,V.,2004,EC-SVMapproachforreal-timehydrologicforecasting:JournalofHydroinfor-matics06(3),p.209-223.
