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1、INVERSEPROBLEMSINGEOPHYSICSROELSNIEDERANDJEANNOTTRAMPERTDept.ofGeophysicsUtrechtUniversityP.O.Box80.0213508TAUtrechtTheNetherlandsemailrsnieder@mines.edu1.IntroductionAnimportantaspectofthephysicalsciencesistomakeinferencesaboutphysicalparametersfromdata.Ingeneral,thelawsofphysicsprovid
2、ethemeansforcomputingthedatavaluesgivenamodel.Thisiscalledthe“forwardproblem”,seefigure1.Intheinverseproblem,theaimistoreconstructthemodelfromasetofmeasurements.Intheidealcase,anexacttheoryexiststhatprescribeshowthedatashouldbetransformedinordertoreproducethemodel.Forsomeselectedexamples
3、suchatheoryexistsassumingthattherequiredinfiniteandnoise-freedatasetswouldbeavailable.Aquantummechanicalpotentialinonespatialdimensioncanbereconstructedwhenthereflectioncoefficientisknownforallenergies[Marchenko,1955;Burridge,1980].Thistechniquecanbegeneralizedforthereconstructionofaquantum
4、mechanicalpotentialinthreedimensions[Newton,1989],butinthatcasearedundantdatasetisrequiredforrea-sonsthatarenotwellunderstood.Themass-densityinaone-dimensionalstringcanbeconstructedfromthemeasurementsofalleigenfrequenciesofthatstring[Borg,1946],butduetothesymmetryofthisproblemonlytheeve
5、npartofthemass-densitycanbedetermined.Iftheseismicvelocityintheearthdependsonlyondepth,thevelocitycanbeconstructedex-actlyfromthemeasurementofthearrivaltimeasafunctionofdistance0PresentaddressofR.Snieder:Dept.ofGeophysics,ColoradoSchoolofMines,GoldenCO80401,USA0Reprintedfrom“WavefieldInv
6、ersion”,Ed.A.Wirgin,SpringerVerlag,NewYork,p.119-190,1999.120ForwardproblemModelmDatadInverseproblemFigure1.Thetraditionaldefinitionoftheforwardandinverseproblems.ofseismicwavesusinganAbeltransform[Herglotz,1907;Wiechert,1907].Mathematicallythisproblemisidenticaltotheconstructionofaspher
7、i-callysymmetricquantummechanicalpotentialinthreedimensions[Kelleretal.,1956].However,theconstructionmethodofHerglotz-Wiechertonlygivesanuniqueresultwhenthevelocityincreasesmonotonicallywithdepth[GerverandMarkushevitch,1966].Thissituationissimilarinquantummechanicswherearadiall
