改進(jìn)的TRUST方法在航磁數(shù)據(jù)線性特征增強(qiáng)中的應(yīng)用

改進(jìn)的TRUST方法在航磁數(shù)據(jù)線性特征增強(qiáng)中的應(yīng)用Abstract：

Inordertoenhancethelinearfeaturesofaeromagneticdata,animprovedTRUST(TotalVariationReconstructionUsingSplittingandTranslating)methodisproposedinthispaper.InthetraditionalTRUSTalgorithm,itisdifficulttoselectanappropriateregularizationparameterλtobalancethedatafidelitytermandthetotalvariationterm.Therefore,anewregularizationparameterselectionstrategyisproposedintheimprovedTRUSTalgorithm,whichisbasedonthecharacteristicofthepowerspectrumofaeromagneticdata,aimingtoretainthelowfrequencycomponentsandsuppressthehighfrequencynoise.Theproposedmethodisappliedtotherectificationofaeromagneticdatainacomplexgeologicalenvironment,andtheresultsdemonstratethattheimprovedTRUSTmethodcaneffectivelyenhancethelinearfeatures,improvetheapparentsusceptibilitycontrast,andretaintheshapeoftheoriginalanomalies.

1.Introduction

Asanon-invasivegeophysicalexplorationmethod,aeromagneticsurveyhasbeenwidelyusedinmineralexploration,geothermalexploration,andoilandgasexploration.However,aeromagneticdataisinevitablycontaminatedbyvariousnoisesources,suchasculturalnoise,high-frequencyinstrumentalnoise,andregionalmagneticanomalies.Thenoiseinthedatawillaffecttheidentificationandinterpretationofgeologicalstructures,soitisnecessarytoremovethenoiseandenhancethelinearfeaturesofthedata.

Totalvariation(TV)regularizationhasbeeneffectivelyappliedinthefieldofimageprocessingandhasattractedattentionintheprocessingofgeophysicaldatainrecentyears.ThetraditionalTVmethodcaneffectivelysuppressthenoiseandenhancethelinearfeaturesofthedata,butitalsohassomedrawbacks,suchasstaircaseeffectandill-posedproblem.Therefore,amoreadvancedvariation,theTotalExponentialVariation(TEV)method,wasproposedtoovercomethedisadvantagesofTV.TheTEVmethodintroducesanexponentialfunctiontoconstructtheregularizationterm,whichcanpreservetheedgesofthediscontinuousstructuresandeffectivelyremovethenoiseinthedata.However,duetothecomplexcomputationprocessofTEV,itisdifficulttoapplydirectlyintheprocessingoflarge-scalegeophysicaldata.

TheTRUSTalgorithm,introducedbyM.Nikolova,isanextensionofTEVandhasbeensuccessfullyappliedintheprocessingofmagneticandgravitydata.TheTRUSTalgorithmbreaksdowntheTVregularizationtermintosplittingandtranslatingsteps,whichefficientlysolvethehigh-dimensionaloptimizationproblem.However,theselectionoftheregularizationparameterλiscrucialintheTRUSTalgorithm.λcontrolsthetrade-offbetweenthedatafidelitytermandthetotalvariationterm,anditdirectlyaffectstheenhancementeffectofthelinearfeaturesandthesuppressioneffectofthenoise.Inpracticalapplications,itisdifficulttochooseanappropriateλwhichcanoptimizetheenhancementandsuppressioneffects.

Inthispaper,weproposeanimprovedTRUSTmethodandapplyittotheenhancementofthelinearfeaturesofaeromagneticdata.Theimprovedalgorithmintroducesanewregularizationparameterselectionstrategybasedonthepowerspectrumofthedata,whichcanautomaticallybalancethefidelitytermandthetotalvariationterm.Theperformanceoftheproposedalgorithmisdemonstratedinacomplexgeologicalenvironment.TheresultsshowthattheimprovedTRUSTalgorithmcaneffectivelyenhancethelinearfeatures,improvetheapparentsusceptibilitycontrast,andretaintheshapeoftheoriginalanomalies.

2.TheImprovedTRUSTMethod

2.1TheTraditionalTRUSTMethod

TheTRUSTmethodisasplittingandtranslatingalgorithm,whichsolvesthefollowingoptimizationproblem:

argmin_x1/2||y-x||_2^2+λTV(x)

whereyistheobserveddata,TV(x)istheTotalVariationregularizationterm,andλistheregularizationparameter.

TraditionalTVregularizationcanbedefinedas:

TV(x)=∑_(i,j)(x_i,j-x_i-1,j)^2+(x_i,j-x_ij-1)^2

TEVregularizationcanbedefinedas:

TEV(x)=∑_(i,j)exp(-(x_i,j-x_i-1,j)^2/σ^2)+exp(-(x_i,j-x_ij-1)^2/σ^2)

whereσisaparameterthatcontrolsthedegreeofexponentialdecay.

TheTRUSTalgorithmperformsthefollowingsplittingandtranslatingsteps:

Step1:Initializexwiththeobserveddatay.

Step2:Updateubysolvingthefollowingsubproblem:

argmin_u1/2||u-y||_2^2+λ/2||?u-g||_2^2

where?uisthegradientofu,andgisthesolutionoftheconvexconjugateoftheTVorTEVregularization.

Step3:Updatexbysolvingthefollowingsubproblem:

argmin_x1/2||x-u||_2^2+λ/2||?x-g||_2^2

Step4:Updategbysolvingthefollowingsubproblem:

argmin_gλ/2||?x-g||_2^2+<g,?x>,g∈?TV(x)org∈?TEV(x)

where?TV(x)and?TEV(x)arethesubdifferentialsofTV(x)andTEV(x),respectively.

Step5:GobacktoStep2,andrepeattheiterationsuntilacertainstoppingcriterionismet.

2.2TheImprovedTRUSTMethod

InthetraditionalTRUSTalgorithm,theselectionofλiscriticaltotheperformanceofthealgorithm.Atoolargeλwillcauseanover-smoothedresult,whileatoosmallλwillnotbeabletoremovethenoiseeffectively.Toavoidthisproblem,weproposeanewregularizationparameterselectionstrategybasedonthepowerspectrumofthedata.

Thepowerspectrumofthedatadescribesthedistributionoftheenergyofthedatainthefrequencydomain.Inaeromagneticdata,thelowfrequencycomponentsareusuallyrelatedtotheregionalmagneticfield,whilethehighfrequencycomponentsareusuallyrelatedtonoiseorsmallsources.Therefore,intheimprovedTRUSTalgorithm,weusethepowerspectrumtoestimatethelowfrequencycomponentandthehighfrequencycomponentofthedata,andthendetermineλbasedonthesetwocomponents.

TheimprovedTRUSTalgorithmperformsthefollowingsteps:

Step1:Calculatethepowerspectrumofthedatay,anddeterminethefrequencycutofffc,whichisthefrequencyatwhichthespectrumdropstohalfofitsmaximumvalue.

Step2:SmooththedataywithaGaussianfilterofstandarddeviationσ1(f)={2πfc}/[(2π)T]inthefrequencydomain,whereTisthesamplinginterval.

Step3:PerformthetraditionalTRUSTalgorithmonthesmootheddata,andobtainaninitialestimationx1.

Step4:Calculatetheresidualr1=y-x1,andextractthehighfrequencycomponentr2byapplyinganotherGaussianfilterofstandarddeviationσ2(f)={2π(1-fc)}/[(2π)T].

Step5:Calculatetheroughnessoftheinitialestimationx1by:

R1=∑_(i,j)(x1_i,j-x1_i-1,j)^2+(x_i,j-x_ij-1)^2

Step6:Calculatetheroughnessofthehighfrequencycomponentr2by:

R2=∑_(i,j)(r2_i,j-r2_i-1,j)^2+(r2_i,j-r2_ij-1)^2

Step7:Determinetheregularizationparameterλby:

λ=(σ[R1/R2])^2

whereσisthestandarddeviationoftheresidualr1.

Step8:ApplythetraditionalTRUSTalgorithmontheoriginaldataywiththedeterminedλ,andobtainthefinalestimationx.

3.ExperimentResults

TheproposedimprovedTRUSTmethodisappliedtotheenhancementofthelinearfeaturesofaeromagneticdatainacomplexgeologicalenvironment.Thedataiscollectedinaporphyrycopperdeposit,wherethegeologicalcomplexityandhighnoiseposechallengestotheinterpretation.

Figure1showstheoriginalaeromagneticdatabeforeprocessing,wherethemagneticanomaliesareweakandpoorlydefinedduetothehighnoiseandlowsusceptibilitycontrast.Figure2showstheresultusingthetraditionalTRUSTmethodwithafixedλ,wheretheedgesareover-smoothedandsomefeaturesarelost.Figure3showstheresultusingtheimprovedTRUSTmethodwiththeproposedregularizationparameterselectionstrategy,wherethelinearfeaturesareenhanced,andthesusceptibilitycontrastisimproved.Figure4showsthedifferencebetweentheoriginaldataandtheresultusingtheimprovedTRUSTmethod,wheretheanomalyshapesarewellpreserved,andthenoiseiseffectivelyremoved.

Thequantitativeevaluationoftheresultsisconductedbyanalyzingthepowerspectrumofthedataandthelocalanomalyenhancementfactor(LAEF).ThepowerspectrumofthedataafterprocessingbytheimprovedTRUSTmethodshowsthatthelowfrequencycomponentsareretained,andthehighfrequencynoiseissuppressed.TheLAEFoftheanomaliesafterprocessingbytheimprovedTRUSTmethodishigherthanthatoftheoriginaldataorthetraditionalTRUSTmethod,indicatingthatthelinearfeaturesoftheanomaliesareenhanced.

4.Conclusion

Inthispaper,animprovedTRUSTmethodisproposedtoenhancethelinearfeaturesofaeromagneticdata.Theimprovedalgorithmintroducesanovelregularizationparameterselectionstrategybasedonthepowerspectrum,whichcanautomaticallybalancethefidelitytermandthetotalvariationterm.Theexperimentresultsdemonstratethattheproposedmethodcaneffectivelyenhancethelinearfeatures,improvetheapparentsusceptibilitycontrast,andretaintheshapeoftheoriginalanomalies.Theproposedmethodhasthepotentialtoimprovetheaccuracyofmineralresourceexplorationandgeologicalinterpretationincomplexgeologicalenvironments.Furthermore,theproposedmethodhassomeadvantagesoverthetraditionalTV-basedmethods.Firstly,theimprovedTRUSTalgorithmcaneffectivelyremovehigh-frequencynoisewhileretainingthelow-frequencycomponentsofthedata.Thisisachievedbyintroducingfrequency-dependentGaussianfiltersintheregularizationparameterselectionstrategy.Secondly,theproposedmethodcanpreservetheedgesoftheanomaliesandavoidthestaircaseeffect.ThisisbecausetheTEVregularizationtermintheTRUSTalgorithmcanpreservethediscontinuitiesofthestructures.Finally,theproposedmethodcanautomaticallyselecttheregularizationparameterλ,whichisasignificantchallengeinthetraditionalTV-basedmethods.

Insummary,theimprovedTRUSTalgorithmproposedinthispaperisaneffectivemethodforenhancingthelinearfeaturesofaeromagneticdata.Thepowerspectrum-basedregularizationparameterselectionstrategyisaninnovativeapproachthatcanbalancethefidelitytermandthetotalvariationtermandimprovetheaccuracyofinterpretation.Theexperimentresultsshowthattheproposedmethodcaneffectivelyremovethenoise,enhancethelinearfeatures,andretaintheshapeoftheoriginalanomalies.Theproposedmethodhasgreatpotentialforpracticalapplicationsinmineralexplorationandgeologicalinterpretationincomplexgeologicalenvironments.Moreover,theproposedmethodcanbeimprovedfurtherbyincorporatingotherconstraints,suchasdipconstraintsordipangleconstraints,toenhancetheaccuracyofinterpretation.Theseadditionalconstraintscanbeaddedthroughtheuseofpriorgeologicalknowledgeorexpertguidance,providingamoreaccurateandreliableinterpretationoftheunderlyinggeologicalstructures.

Anotheradvantageoftheproposedmethodisitscomputationalefficiency.Byusingaparallelcomputingapproachbasedonthealternatingdirectionmethodofmultipliers(ADMM),theproposedmethodcanhandlelarge-scaledatasetsefficientlyandeffectively.Thismakesitparticularlysuitableforprocessinghigh-resolutionairborneorground-basedmagneticsurveydata.

Furthermore,theproposedmethodcanbeextendedtoothergeophysicaldataprocessingtasks,suchasgravitydataprocessingorelectricalresistivitydataprocessing.Thebasicprincipleoftheproposedmethod,namelytheregularization-basedapproach,canbeappliedtoawiderangeofgeophysicaldataprocessingtasksthatinvolvetheenhancementorsuppressionofspecificfeaturesinthedata.

Overall,theproposedimprovedTRUSTalgorithmoffersapromisingsolutionfortheenhancementoflinearfeaturesinaeromagneticdataprocessing.Itsabilitytoeffectivelyremovenoise,preserveedgesandautomaticallyselecttheregularizationparametermakeitanattractivemethodforpracticalapplicationsinmineralexplorationandgeologicalinterpretation.Asaresult,thispapercontributessignificantlytotheongoingeffortstodevelopadvancedgeophysicaldataprocessingmethodsforimprovedgeologicalmappingandresourceexploration.Inadditiontoitspracticalapplicationsinmineralexplorationandgeologicalmapping,theimprovedTRUSTalgorithmalsohasimportantimplicationsforscientificresearch.Byprovidingamoreaccurateandreliablemethodforenhancinglinearfeaturesinaeromagneticdata,theproposedmethodcanhelpresearchersgainnewinsightsintothegeologicalstructuresandprocessesthatshapetheEarth'scrust.Forexample,theenhancedmagneticdatacanbeusedtoinferthelocationandgeometryofgeologicalfaults,whichcanprovidecluesaboutthetectonicforcesthathaveactedontheregionovergeologicaltimescales.Similarly,theenhancedmagneticdatacanbeusedtoidentifyareasofanomalousmagnetization,whichcanindicatethepresenceofmineraldepositsorhydrothermalsystems.

Inaddition,theimprovedTRUSTalgorithmcanhelpresolvesomeofthechallengesassociatedwithinterpretinggeologicaldatainregionswithcomplexgeologicalstructures.Bysuppressingnoiseandenhancinglinearfeatures,theproposedmethodcanprovideaclearerpictureoftheunderlyinggeologicalframework,whichcanfacili