一维波动方程波阻抗反演的同伦方法
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摘要
文中从地震勘探一维波动方程反问题出发 ,研究了一种反演地层参数的同伦方法 ,该方法把非线性方程组的求解转化成常微分方程初值问题的数值求解 ,从而给出一种稳定的计算速度快、抗噪能力强的全局收敛的反演方法 .理论模型和实例试算的结果表明了同伦方法是一种有效的反演算法 ,特别适用于非线性、多极值的地球物理反演问题 ,在地球物理非线性反演中具有广泛的应用前景
Based on the inversion of one-dimensional seismic wave equation,we propose a homotopy method for determining the parameters of the earth.With this method,the problem of the nonlinear system of equations is reduced to an initial value problem of ordinary differential equations.This algorithm is not only fast and stable but also globally convergent.Calculation results on both synthetic data and field data show the effectiveness of the homotopy method as an inversion algorithm.It is especially appropriate for non-linear geophysical inversion problems with multi-minimum values,and has widespread application prospects in non-linear geophysical inversion problems.
Based on the inversion of one-dimensional seismic wave equation,we propose a homotopy method for determining the parameters of the earth.With this method,the problem of the nonlinear system of equations is reduced to an initial value problem of ordinary differential equations.This algorithm is not only fast and stable but also globally convergent.Calculation results on both synthetic data and field data show the effectiveness of the homotopy method as an inversion algorithm.It is especially appropriate for non-linear geophysical inversion problems with multi-minimum values,and has widespread application prospects in non-linear geophysical inversion problems.
引文
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[7] GraySH ,HaginF .Towardprecisesolutionofone dimensionalvelocityinverseproblems.SIAMJ.Appl.,Math.,1982,42:346～355
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[12]ZhangDL ,HanB .Adifferentialcontinuation regularizationmethodforsolvinginverseproblemsofacousticwaveequation.JournalofHarbinInstituteofTechnology,1995,E 2(1):10～14
[13]HanB ,ZhangML ,LiuJQ .Awidelyconvergentgeneralizedpulse spectrumtechniqueforthecoefficientinverseproblemofdifferentialequations.Appl.Math.AndComp.,1997,81:97～112
[14]WangSD ,LiuJQ ,XieG .Q .2Dacousticwaveequationinver sioncombining plane waveseismogramswithwelllogs.Geo physics,1996,61:735～741
[15]冯国峰,韩 波,刘家琦.二维波动方程约束反演的大范围收敛广义脉冲谱方法.地球物理学报,2003,46(2):265～270FengGF ,HanB ,LiuJQ .Widelyconvergentgeneralizedpulse spectrummethodsfor2Dwaveequationinversion.ChineseJ.Geophys.(inChinese),2003,46(2):265～270
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[17] KellerHB ,PerozziDJ.Fastseismicraytracing.SIAMJ.Appl.Math.,1983,43:981～992
[18] VascoDW .Singularityandbranching:Apath followingformal ismforgeophysicalinverse problems.Geophys.J.Internat.,1994,119:809～830
[19] VascoDW .Regularizationandtrade offassociatedwithnonlin eargeophysicalinverseproblems:penaltyhomotopies.InverseProblem,1998,14:1033～1052
[20] BubeKP .LanganRT .Onacontinuationapproachtoregularizationforcrosswelltomography.69thAnn.Internat.Mtg.,Soc.Expl.Geo phys.,ExpandedAbstracts,1999.1295～1298
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[22] JegenMD ,EverettME ,SchultzA .Usinghomotopytoinvertgeophysicaldata.Geophysics,2001,66:1749～1760
[23] LiuJQ ,QianJL ,HanB .Amethodofwideconvergenceregionforsolvinginverse problemsofwaveequations.JournalofHarbinInstituteofTechnology,1995,E 2(4):1～4
[24] 李庆扬,莫孜中,祁力群.非线性方程组的数值解法.北京:科学出版社,1997LiQY ,MoZZ ,QiLQ .NumericalSolutionofNonlinearEqua tionsSystem(inChinese).Beijing:SciencePress,1997
[25] WayburnTL ,SeaderJD .Homotopycontinuationmethodsforcom puter aidedprocessdesign.Comput.Chem.Eng.,1987,11(1):7～25
[26] 周 洲,祝小平,张肇炽.同伦方法中辅助映射的影响.飞行力学,1997,15(3):84～89ZhouZ ,ZhuXP ,ZhangZC .Theeffectoftheauxiliarymapconstructiononsolvingnonlinearproblem.FlightDynamics(inChinese),1997,15(3):84～89
[27] 黄光远.地震勘探数据处理的一种算法———分布参数系统最优理论的应用.地球物理学报,1985,28(1):74～83HuangGY .Analgorithmforprocessingseismicexplorationda ta.ChineseJ.Geophys.(inChinese),1985,28(1):74～83
[2] ClaerboutJF .Towardaunifiedtheoryofreflectormapping.Geo physics,1971,36:467～481
[3] BecqueyM ,LavergneM ,WillmC .Acousticimpedancelogscom putedfromseismictraces.Geophysics,1979,44:1485～1501
[4] BerrymanJG ,GreeneRR .Discreteinversemethodsforelasticwavesinlayeredmedia.Geophysics,1989,45:213～233
[5] SantosaF ,SchwetlickH .Theinversionofacousticalimpedanceprofilebymethodsofcharacteristics.WaveMotion,1982,4:99～110
[6] BubeK ,BurridgeR .Theone dimensionalinverseproblemofre flectionseismology.SIAMReview,1983,25:497～559
[7] GraySH ,HaginF .Towardprecisesolutionofone dimensionalvelocityinverseproblems.SIAMJ.Appl.,Math.,1982,42:346～355
[8] GraySH .Therelationshipbetween“direct,discrete”and“itera tive,continuous”one dimensionalinversemethods.Geophysics,1984,49:54～59
[9] LiuJQ ,LiuKA ,GuoBQ ,etal.Acousticimpedanceprofilein versionfromdifferentialequation.ChineseJournalofGeophysics(inEnglish),1994,37(1):157～164
[10]韩 波,杨晓军,刘家琦.微分连续正则化方法与一维声波方程系数反演问题求解.高校应用数学学报,1994,9(A4):351～360HanB ,YangXJ ,LiuJQ .Differentialcontinuation regularizationmethodandthecoefficientinverseproblemofone dimensionalwaveequations.AppliedMathematicsAJournalofChineseUniversities(inChinese),1994,9(A4):351～360
[11]王德人.脉冲谱反演方法研究.应用数学学报,1985,3:1～9WangDeren.Theresearchonpulse spectrumtechnique.ActaMathematicaeApplicataeSinica(inChinese),1985,3:1～9
[12]ZhangDL ,HanB .Adifferentialcontinuation regularizationmethodforsolvinginverseproblemsofacousticwaveequation.JournalofHarbinInstituteofTechnology,1995,E 2(1):10～14
[13]HanB ,ZhangML ,LiuJQ .Awidelyconvergentgeneralizedpulse spectrumtechniqueforthecoefficientinverseproblemofdifferentialequations.Appl.Math.AndComp.,1997,81:97～112
[14]WangSD ,LiuJQ ,XieG .Q .2Dacousticwaveequationinver sioncombining plane waveseismogramswithwelllogs.Geo physics,1996,61:735～741
[15]冯国峰,韩 波,刘家琦.二维波动方程约束反演的大范围收敛广义脉冲谱方法.地球物理学报,2003,46(2):265～270FengGF ,HanB ,LiuJQ .Widelyconvergentgeneralizedpulse spectrummethodsfor2Dwaveequationinversion.ChineseJ.Geophys.(inChinese),2003,46(2):265～270
[16] 王则柯,高堂安.同伦方法引论.重庆:重庆出版社,1990WangZK ,GaoTA .TheIntroduceoftheHomotopyMethod(inChinese).Chongqing:ChongqingPublishingHouse,1990
[17] KellerHB ,PerozziDJ.Fastseismicraytracing.SIAMJ.Appl.Math.,1983,43:981～992
[18] VascoDW .Singularityandbranching:Apath followingformal ismforgeophysicalinverse problems.Geophys.J.Internat.,1994,119:809～830
[19] VascoDW .Regularizationandtrade offassociatedwithnonlin eargeophysicalinverseproblems:penaltyhomotopies.InverseProblem,1998,14:1033～1052
[20] BubeKP .LanganRT .Onacontinuationapproachtoregularizationforcrosswelltomography.69thAnn.Internat.Mtg.,Soc.Expl.Geo phys.,ExpandedAbstracts,1999.1295～1298
[21] EverettME .Homotopy,polynomialequations,grossboundarydata,andsmallHelmholtzsystems.J .Comput.Phys.,1996,124:431～441
[22] JegenMD ,EverettME ,SchultzA .Usinghomotopytoinvertgeophysicaldata.Geophysics,2001,66:1749～1760
[23] LiuJQ ,QianJL ,HanB .Amethodofwideconvergenceregionforsolvinginverse problemsofwaveequations.JournalofHarbinInstituteofTechnology,1995,E 2(4):1～4
[24] 李庆扬,莫孜中,祁力群.非线性方程组的数值解法.北京:科学出版社,1997LiQY ,MoZZ ,QiLQ .NumericalSolutionofNonlinearEqua tionsSystem(inChinese).Beijing:SciencePress,1997
[25] WayburnTL ,SeaderJD .Homotopycontinuationmethodsforcom puter aidedprocessdesign.Comput.Chem.Eng.,1987,11(1):7～25
[26] 周 洲,祝小平,张肇炽.同伦方法中辅助映射的影响.飞行力学,1997,15(3):84～89ZhouZ ,ZhuXP ,ZhangZC .Theeffectoftheauxiliarymapconstructiononsolvingnonlinearproblem.FlightDynamics(inChinese),1997,15(3):84～89
[27] 黄光远.地震勘探数据处理的一种算法———分布参数系统最优理论的应用.地球物理学报,1985,28(1):74～83HuangGY .Analgorithmforprocessingseismicexplorationda ta.ChineseJ.Geophys.(inChinese),1985,28(1):74～83
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