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Estim ation ofSubsurfaceTem peraturesin the TattapaniG eotherm alField,C entralIndia,from Lim ited Volum e ofM agnetotelluric Data and BoreholeTherm ogram sUsing a C onstructiveBack-Propagation NeuralNetw ork

Anthony E.Akpan*

Applied GeophysicsProgramm e,PhysicsDepartment,University ofCalabar,Calabar,CrossRiverState,Nigeria

M ahesh Narayanan and T.H arinarayana1

NationalGeophysicalResearch Institute,Hyderabad,India

Received 28 M ay 2013;accepted 26 October2013

ABSTRACT: A constructiveback-propagationcodethatwasdesignedtorunas a single-hidden-layer,feed-forward neuralnetwork (SLFFNN) has been

Earth Interactions d Volum e18 (2014) d PaperNo.6 d Page1

CopyrightÓ 2014,Paper18-06;68951 words,7 Figures,0 Animations,1 Tables.http://EarthInteractions.org

initialization conditionsthatcan optim ally solvenonlinearproblem susing therandom weightinitialization approach.Available one-dim ensional(1D)M T-derived resistivity data and borehole tem perature recordsfrom the Tattapanigeotherm alfield,centralIndia,were collated and digitized at10-m intervals.The two datasetswere paired to form a setofinput–outputpairs.The paireddata were randomized,standardized,and partitioned into three mutually ex-clusivesubsets.Thevarioussubsetshad52% (laterincreasedto61% ),30% ,and18% (laterreduced to 9% )fortraining,validation,and testing,respectively,inthe firstand second training phases.The second training phasewasm eanttoassess the influence of the training data volume on network perform ance.Standard statisticaltechniquesincluding adjusted coefficientofdeterm ination(R2a),relativeerror(e),absolute average deviation (AAD),root-m ean-squareerror(RM SE),andregressionanalysiswereusedtoquantitativelyratenetworkperformance.A m anually designed two-hidden-layer,feed-forward networkwith20and15neuronsinthefirstandsecondlayerswasalsoadoptedinsolvingthesam e problem.Perform anceratingswere observed to be 0.97,3.75,4.09,1.41,1.18,and1.08forR2a,AAD,e,RM SE,slope,andintercept,respectively,com pared to an eof20.33 observed with them anually designed network.TheSLFFNN isthusastructurally flexiblenetworkthatperform sbetterin spiteofthesm allvolumeofdatausedintestingthenetwork.Thenetworkneedstobetested further.

K EYW O RDS: SLFFNN;M agnetotellurics;Resistivity;Boreholethermograms;CBP;Performance;Tattapani,India

1. IntroductionThediscovery oftheartificialneuralnetwork (ANN)technology from aconflu-

enceofdifferentscientificandnonscientificdisciplineshasgivenbirthtoapowerfulresearch toolwith applicationscutting acrossdisciplines.In thegeoscienceworld,the ANN technique hasenjoyed enhanced patronage in generating precise infor-m ationonmanygeoscientificproblemsthatborderoncontrol,prediction,inversion,classification,patternrecognition,anddatacompression,whichwerehithertosolvedwith mathematicaland statisticaltechniques with limited accuracy.Informationgenerated by theANN procedureisfreefrom theknown limitationsofthepreex-isting,em piricallybasedmodelingtoolsbecausetheANN techniquedoesnotneedm athem aticalequationstoguideitsoperation.Instead,theANN extractsinformationdirectlyfrom aseriesofinput–outputdatapairswithoutanypriorassumptionsabouttheirnatureandinterrelations(M andaletal.2009;M oghaddam etal.2010;Spichaketal.2011).To m ake reliable predictions,the requisite condition thatthe ANNm ethod needswhile solving com plicated problems is to have a large volume of


propertyestim ationandcharacterization(Calderon-M acıasetal.2000;AminianandAmeri2005),lithologic boundary discrim ination (M aitiand Tiwari2009,2010),disasterforecasting (M arzban and Stumpf1996;M arzban 2000;Boseetal.2008),weatherprediction (Hayatiand M ohebi2007;Hayatiand Shirvany 2007;Jin etal.2008),and many others.Theawarenesscreated by thecapabilitiesoftheANNsinsolving problemswith diversecom plexity hasattracted im menseinterestfrom re-searchersinboth applicationsand modeldesigns.Neuralnetworkshavethecapacity to solveawiderangeofproblem sthatwas

hitherto notsatisfactorily solved using the traditionalm athematicalorstatisticaltoolsirrespective ofthe inherentlevelofcom plexity in the problem.The feed-forward,multilayerneuralnetworks(FFM NNs)thatutilize function approxim a-tion technique in solving problem s are very outstanding in solving complexproblemsbecauseofitsstructuralflexibility,good capabilities,and presenceofalargepoolofalgorithm s(Negnevitsky2005;Stathakis2009;SharmaandChandra2010).Neuralnetwork architectureand topology arethem ajorfactorsthatinflu-encetheaccuracyofresults,generalizationcapacity,andtrainingspeed(Stathakis2009;SharmaandChandra2010).Stathakis(Stathakis2009)hasdocum entedtheproceduresforselecting opticalnetwork topology needed fortackling any prob-lem .Theyincludethetraditionaltrialanderror,heuristicsearch,exhaustivesearch,andconstructivem ethod,aswellasusingasynergybetweenneuralnetworksandgenetic algorithm s,which autom atically searches foroptimally perform ing to-pology based on anovelfitnessfunction.The conventionaltrialand errorm ethod isthe mostpopulartraining method

com monlyusedbym anyANN users.Theprocedureinvolvesmanualselectionofanetwork topology thatisusually im plemented using a trialand errorapproach.Such networkssometimeslack theneeded capacity to yield optimalresultsespe-cially ifdesigned by inexperienced users(Stathakis2009)and com pulsorily re-quires a large volum e of data for use in training the network.Perform anceoptimizationisaveryimportantconsiderationthatm ustbem adewhileplanningtodesignamultilayerperceptron(M LP)networkthatcanoptimallysolveproblem s.Networkswith minimalcomplexity and optim alperformance capability are them ostdesirable forthe ANN usercomm unity.Automated design network archi-tecturehasrecentlybeenintroducedandisbecom ingverypopularconsideringthelevelofdesirability (M aitietal.2011).A m ajorproblem thathascontinued toplague the m anualdesign approach isthatallfactorsthatcan influence networkperform ance(networktopology;thenatureoftheapplicableactivationfunctionforeachnode;typeofoptim izationmethod;andthevaluesofthetrainingparam eterssuchasepoch,learningrate,mom entum,initialweightadjustmentprocedure,andso on)mustbeproperly setbefore such networkscan perform optim ally Other-


objectiveisnottodevelopanew m ethodbuttoadaptanexistingconstructiveback-propagation (CBP)neuralnetwork algorithm to estim atesubsurfacetem peraturesfrom a smallvolum e ofM T data and borehole therm ogram s.A single,hiddenm ultilayer perception feed-forward neuralnetwork (SLM LPNN),which worksautomatically using the principle of random weightinitialization process,wasim plemented in theneutralnetwork toolbox availablein M ATLAB (Dem uth andBeale 2002).The constructive neuralnetworks(CNNs)belong to a group ofal-gorithm swith adaptivestructure.Such algorithm snorm ally solveproblemsfromthe prim itive perceptron m odeland dynam ically increasethe node population instepsofone untilthe m axim um numbercomputed from the data structure isat-tained (Ash 1989;Parekh etal.2000;Sharmaand Chandra2010).Thisarchitec-turaladaptation can only stop ifany ofthepresetstopping conditions(theupperbound num berofnodescalculated from the size ofthe training dataset,the ini-tialization conditions,and the m inimum threshold perform ance)have been at-tained.In any case,thebestoptimized solution willbereturned.

2. A review ofsom e m ethodsofestim ating subsurfacetem peratureResearchershavealwayswanted to develop adependablerelationship thatcan

be used to estimate temperature ofEarth’sinteriorfrom itsphysicalproperties.Consequently,m any em piricaland sem iempiricalrelationshipshave been devel-oped and areusually used in estimating subsurfacetem peraturesalthough some-tim eswithlimitedapplicability.Archie’slaw (Archie1942)isapopularm emberofsuchem piricallybasedrelationships.Them odifiedform ofArchie’slaw connectsthe bulk electricalconductivity ofa formation (sb)with the conductivity ofthesaturating fluid (sw)and thesolid rock m atrix according to Equation (1)as

sb5 aswumSn1 ss, (1)

wherea,m,andnareformation-dependentconstants;Sisthesaturationindex;andssistheconductivityoftherockm atrix,which wasassumed tobenonexistentintheoriginalArchie’slaw.Theconductivityofthesolidphaseofarockmatrix(ss)hasbeen found to bedependenton itsabsolutetemperatureespecially underlab-oratory conditionsaccording to thesem iconductorequation as

ss5 soexp(2 Ei/kT), (2)

where so isa constantthatrepresentsthe conductivity ofa theoretically infinitetem perature,Ei is the activation energy,and k is Boltzm ann’s constant(M eju


property relationship.Such relationships include a temperature–resistivity rela-tionship ofaltered basaltthatFl venz etal.(Fl venz etal.1985)introduced intheirwork.Therelationship isgiven by

rT 5ro

[11 a(T 2 To)][11 b(T 2 To)], (3)

where rT and ro are the resistivities attemperature T and some reference tem -peratureTo.Theempiricalconstantsa and b arealso defined attemperatureTo.Harinarayana etal.(Harinarayana etal.2006)used the em piricalrelationship

showninEquation(4)toestim atesubsurfacetem peratureinPugaregionofIndia,

TZ 5 TS1QS

k2

ASZ2

2k, (4)

whereTZisthetem perature(8C)atdepthZ(km ),TSisthesurfacetem perature(8C),QSisthesurfaceheatflux,kisthetherm alconductivityoftheEarthm aterials,andAS is the rate ofradioactive heatproduction.Insufficientknowledge aboutthenatureofthem aterialsinsideEarth’sinteriorandtheassociatedprevailingextentofheterogeneity have notpermitted reliable estim atesto be m ade.In some cases,absolute lack ofcorrelation between resultsm ade underlaboratory and field con-ditionsdegraded researcher’sconfidence on the workability ofthese assum ption-laden relationships.Spichak etal.(Spichak etal.2007;Spichak etal.2011)and Spichak and

Zakharova (Spichak and Zakharova 2009)reported an entirely new approach ofestimating subsurface temperaturesfrom magnetotelluric (M T)data and adjacentborehole(BH)thermograms.TheirprocedurewasbasedonusinganANN thathasbeen trained on the basisofcorrespondence between the M T data and BH ther-m ogram s.AnANN isafunctionalim itationofthefascinatingandhighlysuccessfulprocessingpowersofthenaturalbiologicalneuron.Justlikethebiologicalneurons,ANNshavethecapabilitytoacceptm ultipleinputsfrom severalsources,integrate,and simultaneously processalltheinputsusing itsparallelprocessing capabilities(BasandBoyaci2007;M aitiandTiwari2010).TheANNsarenonlinearandentailacompletely data-driven,m odern datamodelingprocedurethatdoesnotrequireanyinitialmodelto stand on.Itis now an efficienttoolformodeling complex andnonlineargeophysicalproblemsthatcannotbesuccessfullyhandledbyconventionalempiricalequations(M aitietal.2007;M aitietal.2011).Estim atesm ade by theANNsarereliablebecausethey arefreefrom problemsassociated with using em-piricalrelationssincetheANNsdonotrequireany m athem aticalequationtoworkwith butratherextractinform ation directly from setsofthe input–outputpairsof


areaarein theform ofhot-waterspringsdischarging hotwaterattem peraturesof 508–98.58C, with siliceous sinter deposited at the base of the springs(Shankeretal.1987;Jainetal.1995).Allthesem anifestationsoccurinm arshygrounds and hydrotherm ally altered clay zones,covering an area of about0.1km 2 (Shankeretal.1987).Thestudyareafallswithinthesouthernm arginofTattapani–Ram kolacoalfield,

atthe contactofArchaean rockswith the LowerGondwana rocks.The thermalactivity is controlled by the east-northeast–west-southwestTattapanifaultandothernortheast–southwesttrending crossfaults.TheTattapanifaultseparatestheArchaean rocks exposed in the southern side from the lowerGondwana groupexposedtowardthenorthernandnorthwesternpartsofthefault(seeFigure1).TheProterozoicbasementrockscomprisegneisses,diorites,biotiteschists,actinolite–tremolite schists,kyanite–sillim anite schists,granulites,amphibolite bands ofphyllites,quartzites,andgraphitesinsom eplaceswhilethelowerGondwanagroupconsistsofsandstonesandshales.Effectsofshearingareclearlym anifestedintherocksnearthemainTattapanifault.M ostofthenum eroushotspringspresentintheareaarelocated along theTattapanifault(Sarolkarand Das2006).

4. M Tand borehole tem perature data acquisition

Figure 1.M ap ofIndia show ing the (a)location and (b)generalized geology oftheTattapaniregion,borehole locations,and M Tstations.


can be found in Harinarayana etal.(Harinarayana etal.2000)and Veeraswam yand Harinarayana(Veeraswamy and Harinarayana2006).Despitethesedesirabledata treatments,allthe datasets were visually checked forproblem s associatedwith staticshiftby directly checking forconsistency and correlating adjacentda-tasets.Thelackofstaticshiftproblem sinthecurvesthatwerefinallyusedinthisstudy (see Figure 2)is in accordance with the findings ofHarinarayana etal.(Harinarayanaetal.2000).Inspiteoftheseeminglylackofstaticshiftproblem sasdeduced from qualitativeassessmentand correlation ofthecurves,thedatawerestillsubjected to quantitative dim ensionality assessm ent.From the resultsofthequantitative dim ensionalanalysis,any data thatwere observed to be undertheinfluenceof3D structuresweresum m arilydropped.Dimensionalityassessm entofthe data was perform ed by using the rotationally invariantam plitude skewness(SkewS)param eterofSwift(Swift1967).According to Swift(Swift1967),theSkewS param etercomputed from Equation (5)should ideally be0 for1D and 2Dstructuresbut,in environm entswhere the subsurface isdom inated by 3D struc-tures,theSkewS parameterwillideally begreaterthan zero (SkewS . 0)(Rybinetal.2008),

SkewS5Zxx1 ZyyZxy2 Zyx

. (5)

Resultsobserved are shown in Figure3 where itcan be seen thatforfrequenciesabove1Hz,theSkewSvaluesarewellbelow 0.1(, 0.1)(thedashedlineinFigure3),which issuggestive ofa subsurface thatisdominated by 1D/2D structures.SkewSvaluesobservedforfrequenciesbelow 1Hzweregenerallygreaterthan0.1(SkewS.0.1),suggestingthat3D structuresdom inatethesubsurfaceatthesedepths.BasedontheresultsofearlierM T studiesofthearea(Harinarayanaetal.2000),regionaltectonics,and theobserved SkewS values,theshallow subsurfacewasassum ed tobepredom inantly1D.Theverticalm agneticfield(Hz)recordsthatwereobservedto benoisy werealso dropped.Som e M T data were randomly selected and som e ofthe data processing pro-

cedures already conducted by the M TS data processing unitofNGRIwere re-peated.The repetition wasdone in orderto com pare the two setsofresultsandconsequently develop confidencein theotherdatasets.Thedataprocessing stepsthatwerepeated included robustsingle-station processing to downweightordis-card outliers of impedance estim ates ateach frequency band of m easurem ent(SutarnoandVozoff1989;JuppandVozoff1997);transform ingtheremainingdatafrom tim edomaintofrequencydomainusingthefastFouriertransform algorithm;and computing crosscorrelationsbetween pairsofmutually orthogonalmagnetic


Figure 2.TypicalM Tresistivity and phase curves.


Tenboreholethermogramsrecordedwithintheshallow depthrangeof100–500minthestudyareawereacquiredfrom publishedworkofShankeretal.(Shankeretal.1987).M ostofthe temperature recording stationsselected were located in closeproxim ity to theM T stations(Figure1b).Theresistivity and temperaturevariationwith depth curves(Figure5)weredigitized in a10-m interval.M aximum tem per-atureobserved from thetemperaturerecordswas1128C at500-m depth,althoughhighertemperaturevalueshavebeen reported in thesam eareaatshallowerdepths(SarolkarandDas2006).Heatflow measurem entshavebeenconductedinthearea

Figure 3.G raph show ing variation ofSkew S valuesw ith frequency.


byresearcherssuchasShankeretal.(Shankeretal.1987)andJainetal.(Jainetal.1995),andresultsshow thatpeakvaluesofheatfluxesareashighas300m W m 2 2.Averageheatflow fluxisabout1906 50m W m 2 2,whichismuchhigherthanthe

Figure 5.G raphsofM T-derived resistivity and subsurface tem peratures.


and Yeung (Kwok and Yeung 1997),Parekh etal.(Parekh etal.2000),Islam andM urase(Islam andM urase2001),andSharm aandChandra(Sharm aandChandra2010).TheCNNsusuallystartsolvingproblemswithminimum networktopology(minim um num ber oflayers,nodes in the hidden layer,and connections) anddynam ically increasethenode,layer,and connection population by unity asm aybe required before a problem isoptim ally solved.The cascade-correlation algo-rithm (CCA),dynam ic node creation (DNC),and hybrid algorithm sare famousm em bersoftheCNN fam ily(Islam andM urase2001;SharmaandChandra2010).TheDNC algorithm (Ash 1989)isan efficientm em beroftheCNN fam ily and

wasintroducedtoovercometheproblemsassociatedwithlearningusingtheback-propagation algorithm including the problem ofslow learning rate and rigid to-pology,the step size problem,the localminima problem ,and the moving targetproblem.TheDNC algorithm originallyconstructsasingle,hidden-layerartificialneuralnetwork with a zero node in the hidden layer(the naive m odel);teststhenaivenetworkontheproblem ;and,intheeventoffailure,autom aticallyaddsonenode(orbatchofnodes)totheexistingnetworkandrestartsthetestsallover.Onceanoptim um num berofnodesinthehiddenlayerhavebeenfound,itisconnectedto theoutputsideweights.Advantagesofusing theDNC algorithm includerapidlearning rate,establishmentofaself-modified networksizeandtopology,andtheability to retain itsinbuiltstructureseven when there isa change in the trainingdataset.Furtherm ore,itdoesnotrequireback propagation offailuresignals.TheoriginalDNC algorithm hasundergone severalm odificationsbecause itwasob-served to havedifficulty in learning com plex problems(Islam and M urase2001).The CBP algorithm ,forinstance,isa modified version ofDNC algorithm in-

troduced by Lehtokangas(Lehtokangas1999).CBP isfastbecoming avery pop-ulartoolbecause,apartfrom enjoying allthebenefitsattributed to theDNC,itisknown to haveasimpleim plementation procedure.TheCBP also hastheadvan-tage ofbeing suitable foruse in fixed size networksaswellasitsdocumentedabilitytoutilizestochasticoptimizationroutines(SharmaandChandra2010).TheCBP isusuallydesigned withonlyonehiddenlayerandthetraditionalmethodofbackpropagatingtheerrorsignalisretained,therebycreatingasingle-layer,feed-forward neuralnetwork (SLFFNN).Such advantagesincrease the generalizationpotentialofSLFFNN with m inimalnetwork architectureand thusinfluenced ourchoicein ourpresentstudy.

6. Data pairing,scaling,and m odelparam eterizationTesting ofthenetwork wasperform ed using two differentdatasets:1)tem per-

aturevariationswith depth derived from boreholetherm ogramsand 2)resistivity


can becom pelled to work in interpolation m ode.M oderately wideM T–BH sep-aration (; 3km )was deliberately introduced into the pairing procedure so thatgeographic factors can be incorporated into the inputdata and,consequently,forced the network to function in extrapolation mode.The m oderately widelyspaced setofM T–BH data pairswasincluded in orderto testthe function ap-proximating powersofthegood network when an optim um architecturehasbeenfound (W hiteson and Stone2006).Theraw datainthedatabaseconsistofinputsandcorrespondingcorrectoutputs

(targets).Alltheraw datain thedatabaseXiwereconverted to standard scoresZsusingthemeanm andstandarddeviations oftheentiredataset(LarsenandM arx2000;CarrollandCarroll2002)[seeEquation(6)]beforetheywaspartitionedintothethreemutually exclusivesubsetsoftraining,validation,and testsets,

Zs5Xi2 m

s. (6)

Thestandardization processwasperform edin orderto ensurethatallinputswerescaleddowntovaluesthatvarybetween2 1and1 1.Thisprocedureisastandardprecautionary measureusually adopted in ensuring thatthewidevariationsin theoriginalvaluesoftheinputdatasetdo notforcethenetwork to go into saturationwhile processing the data (M aitiand Tiwari2010).The ‘‘prestd.m ’’code thatisavailableintheneuralnetworktoolboxofM ATLAB wasusedtoachievethis.Thetraining subsetwasfurtherpreprocessed using the ‘‘prepca.m ’’code in ordertofilteroutredundantdatainthetrainingdataset.Additionally,thevalidationandtestdatasetswerepreprocessed using the‘‘transm it.m ’’code.Partitioningthedataintothreem utuallyexclusiveblockswasnecessaryinorder

to guard againsttheproblem ofdataoverfitting associated with back-propagationalgorithm s.Thephenomenonofoverfittingcanoccurifthenetwork‘‘overlearns’’andconsequently‘‘m em orizes’’thetrainingdatasetsuchthatitcannotm akegoodgeneralization when confronted with new datasets.Demuth and Beale (DemuthandBeale2002),M aitietal.(M aitietal.2007;M aitietal.2011),M aitiandTiwari(M aitiandTiwari2010),andmanyotherresearchershavedescribedhow theearlystopping technique can im prove the generalization capability ofa network byelim inating overfitting problem s. By utilizing the early stopping method, athreshold learning condition can be set,underwhich the network can stop thelearning process.The early stopping procedure ensures thatthe network stopslearning wheneverthebest-fitting m odelbeyond which thenetwork cannotm akeanyreasonableincreaseintheerrorfunctionisattained.Inpractice,thisisdonebyrandomizingallinputdataandpartitioningitintothreemutuallyexclusivesubsets,


and 35% fortesting.Thevalidation setwasused by the network to fine tune itstopology with aview to sharpening and clearing any gray areasin itspredictivepowers (Negnevitsky 2005;Larochelle etal.2009).Itis notdirectly used fortraining butisdeployed forcontinuousmonitoring ofitserrorduring theprocessoftrainingsuchthat,iftheerrorattainsapresetm inim um condition,itstopsfurthertraining ofthenetwork.Thiscross-validation schemepreventsthenetwork fromm em orizing thetraining datasetand consequently leftin astatethatitcan m akeacceptable generalization (M aitietal.2007).The testing datasetthatconsistsofdatafrom M T7-BH10andM T11-BH30M T–BH datapairswereeitherfedintothenetworkinfullcorrespondencewithatrainingdatasetof174(or52% )orfedintothe network one by one.In the latteroption,the rem aining partsofthe testingdatasetwere usually added to the training dataset,thereby increasing the per-centageofthetraining datasetto 61% .Thresholdupperperform ancebenchm arkwasfixedat10% andtheinitialization

conditions thatconsistof35 trials forany ofthe random ly generated num bers(RGNs)weresetasvariables.Thenetworkwassetintorunningmode(training)byusingaconditionalstatementtosearchforthebestfittinginitializationconditions[num beroftrials(Ntrials)andthepresetrandom num ber]foraparticularnumberofnodes in the hidden layer.These initialization conditions were used by thenetwork to testtheadequacy ofarange ofnodes(N)in the hidden layer(in ourdata,N varies from 0 to 9)foroptim alperform ance by the network.Foreachinitialization condition and number of nodes in the hidden layer,the networkperform ancewasrated.Intheeventoftheperformanceindexsatisfyingthepresetthreshold perform ance target,the training process terminates and results weredisplayedinbothtabularandgraphicalform s.Besidesthecomputedperform ancerating,visualinspection ofthe graphically presented results was also usefulinqualitatively assessing theperformanceofthenetwork.

7. Netw ork design and adaptationThe neuralnetwork toolbox package in M ATLAB (Dem uth and Beale 2002)

wasusedindesigningastructurallyflexibleSLFFNN.Thenetworkwasoptimallydesigned to take in fourinputsand autom atically search forthe bestperform ingnetwork topology to solvetheproblem foreach given setofinitialization condi-tions(Ntrialsand random num ber).A generalized form oftheSLFFNN codethatwas modified to fitinto ourproblem can be found online (athttp://www.math-works.com /matlabcentral/newsreader/view_thread/308972).Thevariousstepsthatthe SLFFNN code used in executing the com m andshave been described in thework ofNegnevitsky (Negnevitsky 2005)and Sharm aand Chandra(Sharm aand


perform anceofeach random numberso that,attheend ofeach round oftrial,thebest(ornearbest)perform ing random numbercanbeselected.

8. M ethodology ofsubsurface tem perature estim ationThe problem ofdesigning a structurally flexible network thatcan be used to

estim ate subsurface tem peraturesfrom a sm allvolum e ofM T–BH data pairsofdata,which the conventionalm anually designed M LP networks cannotsolvesatisfactorily,iswhatthisstudy isoutto address.Consequently,weattempted tosolve the problem using both the conventionalm anually designed M LP and theSLFFNN.A staticstructuredM LP networkwasdesignedandtestedwiththedata.Theinputlayerofthem anuallydesignednetworkhadfiveinputsfollowedbytwohiddenlayerswhiletheoutputlayerhadonefixedneuron.Thenum berofneuronsinthefirsthiddenlayerwasfixedat20whilethenumberofneuronsinthesecondhiddenlayerwas15.Thelearningrate(a)andmomentum (b)werekeptat0.01and0.9,respectively.TheM LP networkwastaughtrepeatedlywiththegoalofattainingathreshold perform ancebenchmark of5% using easperform anceindicator.A structurallyflexible,single-layer,feed-forwardneuralnetworkwasdesigned.

Theactualarchitectureoftheintended feed-forward,back-propagation M LP wasnotpredefinedbeforetrainingbegan.However,thestructureoftheinputandoutputlayerswere uniquely known since they depend on the structure ofthe inputandoutputdata pairs.Thus,the inputlayerwas structurally designed to have fiveneurons.TheneuronsweredesignedtotakeinthreepositioncoordinatesofnearbyboreholesandM T locations,resistivity,andtemperaturevaluesatdifferentdepths.Theposition coordinatesconsistofaveragevaluesoflongitudesand latitudesoftheM T and boreholepositionsand depth and oneneuron in theoutputlayer.Thesingleneuronintheoutputlayerwasresponsibleforthetransmissionofallresults(temperature)from thenetwork to theoutsideworld.The neuronsin the hidden layeractsindividually asa summ ing junction that

integratesand modifiesalltheinputsthatithasreceived to produceanetoutputthatcan beexpressed (Demuth and Beale 2002;Hagan etal.2002;Negnevitsky2005;M aitietal.2007)as

net(l)j 5Xn

j51

w(l)jix(l2 1)i 1 b(l)j , (7)

wherenet(l)j isthenetoutputfrom thejth neuron in ahidden layer(l),x(l2 1)i isaninputvariablefortheithnodeinhiddenlayer(l)asreceivedfrom anothernodeinthe (l2 1)th layer,w(l)ji isthe strength ofthe connection weightbetween the ith


transform edfrom theiroriginalplusandm inusinfinityvariationstotheirbinaryequivalents.Forinstance,ifthenetinputsinto node(j)in thehidden layer(l)isnet(l)j ,thenthelogsigm oidtransferfunctionensuresthattheoutputfrom thisnodejistransform ed according to Equation (8)as

a(l)j 5 fj(net(l)j ), (8)

wherefjisthelog–sigm oid transferfunction and ism athematically defined as

fj51

11 e2 x, (9)

wherexisaninputparameterandedenotesthenaturallogarithm (Benaoudaetal.1999;VanderBaanandJutten2000;Negnevitsky2005;Spichaketal.2011).Thelineartransferfunction doesnottransform theoutputsfrom theoutputneuronsinany way butratherensuresthatthey are transmitted directly with theiroriginalvariationspreserved.In theoutputlayer,thecomputed network outputsatnodej(Oj)arecom pared

with theknown outputs(Dj)and theoverallnetwork errorE sum med overalltheoutputnodes,and thesam pleswereinternally computed using

E 51

2

XP

p5 1

XNo

j5 1

(Dj2 Oj)2. (10)

Thelocalerrorefrom onlythejthnodeinhiddenlayer(l)canbecalculatedaccordingto Benaoudaetal.(Benaoudaetal.1999)and M aitietal.(M aitietal.2007)as

e(l)j 5 2›E

›(net(l)j )5 fj9(net

(l)j )

X

k

e(l1 1)k w(l1 1)kj . (11)

In the network training process,the network m inim izes E by systematicallychanging thestrength oftheconnection weightsDwjiusing thefailurehistory asguide.Thisoperationim plementedbythegradientdescentlearningruleaccordingto therelation

Dw(l)ji 5 2 h(l)ji

›E

›w(l)ji

!

5 hjie(l)j x

(l2 1)i , (12)

whereh(l)ji isthelearning rate


thesame(DemuthandBeale2002;Sinetal.2006;M aitietal.2007;M oghaddametal.2010).TheR2avaluesarecom puted using theexpression

R2a5 12

XN

i5 1

(Tknown,i2 Testim ated,i)2

XN

i5 1

(Tknown,i2 Tknown,i)2

, (13)

whereTknownistheknowntemperatureatagivendepthobservationpointi,Tpredictedistheestim atedtemperatureatthesameobservationpointi,Tknown isthem eanoftheTknown,and N isthetotalnumberofobservationdatapointsinthedataset(Gem itzietal.2009;M oghaddam etal.2010).Fora given random num berR2a,otherperform ancetestswerecalculated forallNtrialsm adeby thenetwork to find asuitable structure forthe network architecture.The standard deviation,m ean,m edian,and varianceofallR2avalueswerealso com puted and plotted againstnum berofnodesin the hidden layer(see Figure 6,top and m iddle).Thiswasdone to check the robustness and sensitivity of the predictions m ade by thenetwork to thedifferentdatasets.TheRM SE iscalculated as

RM SE 5

2

664

XN

i5 1


N

3

775

vuuuut

, (14)

whileeiscom puted as

e5

XN

i5 1


XN

i5 1

T2known,i

2

66664

3

77775

vuuuuuuut

3 100% . (15)

Finally,theAAD iscomputed as


Posttraining regression analysis was also perform ed by testing the datasettoassess how the known (actual) outputand the predicted network outputs arequantitatively related,form ally establishing their reliability limit Demuth and

Figure 6.(top)G raphs show ing variation in netw ork perform ance assessed usingR2a asa function ofnum berofnodesin the hidden layerfor(a)TPM T7-BH10and TPM T30-BH11 pairscom bined,(b)TPM T30-BH11,and (c)TPM T7-BH11;(m iddle)statisticalanalyses ofnetw ork perform ance for(a) TPM T7-BH10and TPM T30-BH11 pairs com bined, (b) TPM T30-BH11, and (c) TPM T7-BH11;and (bottom ) graphs ofestim ated and know n tem peratures for(a)TPM T7-BH10 and TPM T30-BH11 pairscom bined,(b)TPM T30-BH11,and(c)TPM T7-BH11.


10. Discussion ofresultsThe perform ance ofan SLFFNN M LP has been optim ized and adapted for

solvinganonlinearproblem ofsubsurfacetem peratureestim ationfrom alim itedvolum e ofM T–BH data pairs using the CBP algorithm .The CBP algorithmbequeathstothenetworkarchitectureanadaptablestructure,whichenablesittoautom aticallyadjustitstopologytom eetwiththechallengesoftheproblem tobesolved.The static-structured m anually designed network perform swellduringtrainingbutfailstom akegoodgeneralizationwhenconfrontedwiththetestdata(see Figure 7).Thisobservation suggeststhateitherthevolum e ofthe trainingdatasetm ightnothave been largeenough forproperlearning orthenum berofhidden neuronsthatwasm anually fixed in the network forthe purpose ofac-quiringtheneededintelligenceforuseinsolvingtheproblem m ighthavecausedthenetworktooverlearnandconsequentlyfailtom akegoodgeneralizationwhenconfronted with the testing dataset.The possibility ofthe latteroption beingresponsible forthe poorperform ance ofthe m anually designed network wasruled outbecause the perform ance ofthe network wassatisfactory during thevalidationphase.Althoughithasbeenpointedoutbym anyresearchersincludingLawrence(Lawrence1994),Helleetal.(Helleetal.2001),Stathakis(Stathakis2009),andothersthatthereisnoactualruletoguideusersinselectinganoptim alnum berofhiddenlayersandneurons,itthusappearsthatallowingthenetworktoadjustitstopologytocopewiththechallengesofthedatatobesolvedisabetterpracticein network design.Solving problem s from any dataset containing lim ited volum e of data has

continuedtoposeseriouschallengestotheneuralnetworkusers’com m unitysinceneuralnetworks require a large volum e ofdata pairs from which to learn (seeM oghaddam etal.2010).Therefore,a structurally flexible network can be de-ployedtosolveawiderangeofproblem sincludingthoseinvolvinglim itedvolum eofdata,although m oretestsneed to beperform ed to ascertain thelevelofrelia-bility.Theautom atedsearchfortheoptimalnum berofneuronsinthehiddenlayerfrom zero (the naive m odel)with corresponding increase in R2a values towardunity suggestthatthenetwork isstructurally flexible.Figure 6 (bottom )shows graphicalplots ofestimated and known subsurface

tem peraturesfrom theSLFFNN.TheSLFFNN network seem sto haveperformedwellevenwiththelimitedvolum eofdatausedintrainingthenetwork.Thegradualbuild up in the numberofneuronsin the hidden layerand theircorrespondingperform ancecan beseen in thecom puted valuesofR2aplotted in Figure6 (top).The R2a values approach unity when the m aximum num berofneurons in thehiddenlayercomputedfrom thesizeofthetrainingdatasethasbeenloadedineach

Th h h iti i th di i i i d


tuationsinnetwork

perform

ancefordifferentvolumesofinputdata.

Datavolumes(%)

Initialization

conditions

Regressionresults

Statisticalresults

Validation

Testing

Randomnumber

Ntrials

Slope

Intercept

R2

AAD

eRMSE

3018

90151941

330.96

0.91

0.97

2.29

4.09

1.14

309

50151942

120.94

1.03

0.99

2.45

3.05

1.27

309

801151940

141.08

21.18

0.99

3.75

3.19

1.47


readings,even with thelow volum eofdataused fortraining thenetwork,issug-gestiveofahigh approxim ating capacity network thatistypicalofreinforcementlearning in neuralnetworks (Sutton and Barto 1998;Ferrariand Stengel2005;Zainuddin and Pauline2008).Therefore,abetterperform ance levelthatisasso-ciatedwiththerandom weightinitializationanddegreesoffreedom procedurethatthisprocedureused in tackling theproblem can beachieved when thevolumeofdataislargeenough to properly train thenetwork.Changesin subsurfacestructuralgeology (e.g.,faults,fractures,etc.)and hydro-

geologicalconditions are known to adversely affectthe norm alsubsurface con-ductivity distribution pattern (seeSpichak etal.2011).Consequently,estim atesofsubsurfaceconditionsm adeinsuchheterogeneousenvironm entsarelikelytobefraughtwith errorsand thereforeunreliable(Spichak etal.2011).Function ap-proxim ating procedures thatare usually im plem ented by the SLFFNN couldserveasareliabletoolinpredictingsubsurfaceconditionsinsuchheterogeneousenvironm ents.Neuralnetworkswithadaptivestructureshavebeenreportedtobecharacterizedbyhighfunctionapproxim atingcapabilityandconsequentlyhigh-perform anceratings(Sharm aand Chandra2010;Qing-Laietal.2010;Nicolettietal.2009).Theability oftheSLFFNNsto optim ally estim atesubsurfacetem -peraturesinspiteoftheconstraintsim posedbyperform ancelim itingfactorssuchasthesm allvolum eofthetrainingdataset,thegeologicalcom plexityprevalentatthesiteandthesom etim eswideseparationoftheM T–BH datapairsisindicativeofa high-perform ance network.Such levelofnonlinearproblem solving capa-bility,even with a sm allsize ofdata,is characteristic ofnetworks with good

Figure 7.G raphsofknow n tem peraturesagainstm anually estim ated tem peraturesat(a)TPM T30-BH11,(b)TPM T7-BH10,and (c)com bination ofTPM T30-BH11and TPM T7-BH11 pairs.


quantityofthetrainingdatasetto62% .Theresultsoftherepeatedtrainingsforeachhalfand fulldatasetare shown in Table 1.In allthe cases,betterperformance isobtainedwhentheM T–BH datapairsused fortraining thenetwork wasmorethan60% .Inthecaseofourlimitedvolum eofdataset,thetabulatedresults(seeTable1)try to capture thelittle im provementin network performancewith increase in thevolum eofthetraining dataset.Thisindicatesthatbetterpredictionscan alwaysbeobtained from the network ifa greaternumberofM T–BH pairs is used in thenetwork training phase.Consequently,theanticipated influenceofchangesin sub-surface structural,hydrogeological,and geologicalconditions can be drasticallyreducedsincethenetworkisexpectedtogenerateenoughneuronsthatcantackletheproblem headon.Thus,thetrainingmethodologyhasasignificantinfluenceontheaccuracy ofthepredictionsm adeby thenetwork.In the process ofsearching foroptim alinitialization conditions,the network

testsallavailablenodescomputedfrom thesizeofthetrainingdatastructureusingthe random weightinitialization and degree offreedom schem e.The networkusually startsthetesting processwithazeronode(thenaivemodel)inthehiddenlayerand runsthe testuntilallthe nodesare exhausted.Thisprocessmay lastfor som e time,although itis considerably less than the tim e usually spentintrainingmanuallydesignednetworksbythetrialanderrormethod(seeTable1inM oghaddam etal.2010).Forinstance,acontinuously running setof35 trialsforeach random num ber and node population in the hidden layer usually returnssatisfactorypredictionswithin45min.Oncesuchconditionsarefound,theprocessoftrainingthenetworkisusuallyfastwithrepeatableresults.Thus,theapproach,unlikewhathasbeen reported in theliterature(seeM oghaddam etal.2010),isvery econom icalwith tim e,although theoveralltim etendsto bedependentonthedesired levelofaccuracy.Thesearch foroptim alinitialization conditionstosolve the problem norm ally starts with the network attem pting to solve theproblem withoutany node in the hidden layercorresponding to Rosenblatt’sprim itive perceptron (constantornaive)m odel(Rosenblatt1962;Negnevitsky2005).Ifthe residualprediction errorisgreaterthan the presettolerance lim itafter35 attem pts(Ntrials)fora particularnum berofnodesin the hidden layerandrandom num ber,thenetworkautom aticallyincreasesthenodepopulationbyoneand restartsthesearch alloveragain.Thenetwork willkeep on trying untiltheupperbound num berofnodesin thehidden layercom puted from thesizeofthetrainingdatastructurehasbeenexhausted.Theupperboundnum berofnodeswas6 and 9 when thequantity oftraining datawas52% and 61% ,respectively.Foreach Ntrialsm adewith aspecified num berofnodesin thehidden layer,thenetwork returnsa localm inim um thatcorrespondsto aconvergence pointwiththebestvalueofadjusted coefficientofdeterm ination (R2a) Thus,by prefixing


11. C onclusionsW ehavedemonstrated how aSLFFNN can beused to estimatesubsurfacetem-

peraturesusing asm allvolum eofresistivity dataand boreholetherm ogram sfromtheTattapanigeotherm alfieldincentralIndia.Thenetworkusestherandom weightinitialization and adjusted degreesoffreedom approach in searching foroptimalperforming initialization condition forthenetwork.Thenetwork hasthecapabilityofdynamicallychangingthenetworktopologyduringtraininginordertocopewiththechallengesoftheproblem.Thenetworkperform edsatisfactorilywellinspiteoftheperform ancelim itingconstraintsimposedbythelimitedvolum eofM T–BH datapairsusedintrainingthenetwork,thenonlinearityoftheproblemstobesolved,thewide separation ofthe M T–BH pairs,and the prevailing geologicaland structuralcomplexitiesattheTattapanigeothermalfield in centralIndia.The structuraladaptability ofthe SLFFNN isevidentin the network’scontin-

uousanddesperatesearchforoptimum networkarchitecturetosolvetheproblem.The network attem pted solving the problem with no node in the hidden layercorresponding to Rosenblatt’sprim itiveperceptron (thenaiveorconstant)m odel;ifthe predicted resultdoes notsatisfy som e presetperform ance conditions,itautomatically increasesthenodepopulation by oneand repeatsthetrialsallover.Thetrialcanonlystopifeitheranacceptablesolutionisfoundortheupperboundnumberofnodesgeneratedfrom thedim ensionofthetrainingdatasetisexhausted,in which caseitwillreturn thenear-bestsolution.

Acknowledgm ents.The firstauthorisgratefulto the CouncilforScientific and In-dustrialResearch(CSIR)andThirdW orldAcadem yofSciences(TW AS)forcosponsoringand funding hisresearch attheNationalGeophysicalResearch Institute(NGRI),Hyder-abad,India,underthe CSIR-TW AS postdoctoralfellowship.Thanksare also due to thedirectorofNGRIforhisencouragements,understanding,andinterestandforkindlygrantingus permission to publish this work.W e benefited immensely from Prof.Greg Heath ofM ATLAB Central,who generously shared his single-hidden-layermultilayerperceptronneuralnetworkcodewith us.W earealso gratefulto ourcolleaguesin theM agnetotelluricDivision ofNGRIfortheirsupport,concern,and encouragement.Contributions,sugges-tions,andcriticalreviewsmadebythereviewersthatshapedtheoriginalmanuscripttothepresentform arealso appreciated.

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