Generalized priors in Bayesian inversion problems

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1、AdvancesinWaterResources36(2012)3–10ContentslistsavailableatScienceDirectAdvancesinWaterResourcesjournalhomepage:www.elsevier.com/locate/advwatresGeneralizedpriorsinBayesianinversionproblemsPeterK.KitanidisStanfordUniversity,CivilandEnvironmentalEngineering,473ViaOrtega,Stanford,

2、CA94305,UnitedStatesarticleinfoabstractArticlehistory:Inverseproblems,whenonlyharddataareconsidered,aremathematicallyill-posedbecausetheyhaveAvailableonline14June2011multiplesolutionsandaresensitivetosmallchangesinthedata.Anexampleisthedeterminationofhydraulicconductivityfromhead

3、data.BayesianmethodsprovideageneralstochasticframeworkwithinKeywords:whichonecansolvesuchproblems.Theapproachallowsonetocombineharddatawithotherinforma-Inversemodelingtioninordertoexploretherangeofpossiblesolutions.Weconsidertheroleofgeneralized(alsoknownInterpolationasimproper)p

4、robabilitydistributionswithinthisframework.ThesedistributionsareveryusefulbecauseProbabilisticanalysistheyrepresentinformationparsimoniously(withsimpleequationsandfewparameters)andareidealinUncertaintyquantificationtermsofrepresentingsituationswithlimitedinformation.Theyareparticu

5、larlyusefulinintroducingpriorinformationaboutthedependentvariablesoftheinverseproblemandstructuralparametersthatdescribethedegreeofcontinuityorsmoothnessofdependentvariables.Additionally,theyareveryusefulfromacomputationalperspectivebecausetheycanleadtoformulationswithspecialstru

6、cture,suchaswithhighdegreeofsparsityorstructurethatcanbeexploitedthroughhighperformancecomputingalgorithms.Weexamineinthecontextofasimpleexamplesomeoftheconsequencesofusingdifferentgeneralizedpriors.Ó2011ElsevierLtd.Allrightsreserved.1.Inverseproblems2pty1¼ecosð3ÞTTheterminvers

7、eproblemissometimesusedassynonymforparameterestimation.AlthoughthismaybeavaliddescriptionThen,thesolutionisinsomecases,adistinguishingfeatureofaninverseproblemisdy02pe2ptthatitismathematicallyill-posedinthesenseofHadamard.SuchsðtÞ¼sinð4ÞdtTTproblemsarestudiedin,forexample,Laur

8、ientiev[17].Aproblemissaidtobewellposedi