摘要
混凝土是建筑工程中应用最广的材料之一,其结构的安全性一直是人们关注的焦点.鉴于混凝土结构的特殊性,在对其进行维修,加固或改建前,通常需要对其进行无损检测,从而获得损伤的位置、大小和属性等信息.因此,混凝土结构的无损检测具有非常重要的意义.
探地雷达无损检测具有分辨率高,检测速度快,使用方便等优点,很受工程地球物理界的青睐.目前,这种无损检测技术已经成为混凝土结构检测的一项重要手段.然而,探地雷达的数据解释工作,与对空雷达理论、地震勘探方法、以及日新月异的硬件技术相比较其还远未达到系统、成熟的阶段.目前这项工作主要是先对数据进行预处理,然后再通过人工分析进行识别,还没有形成机械化的处理方式.高频雷达波的衰减特性使某些缺陷的目标信号比较弱从而难以识别,并且由于人的经验不同常会发生误判和错判等问题,导致这种方法的准确性及工作效率均较低.这种状况严重制约了探地雷达技术在混凝土无损探测方面的发展,因此急需开发更加直观的、自动识别系统.混凝土无损检测属于小尺度检测问题.其特点是检测对象的尺度很小、检测精度和分辨率要求高、介质的均匀性较差等.对于复杂的检测问题,用基于射线理论的雷达波层析成像,和基于Born或Rytov弱散射近似假定的波形反演都已无能为力,必须从参数识别(介电常数或电导率)的角度出发,发展全波形数值反演方法.
本文以电磁波传播原理为理论依据,以探地雷达为探测手段,建立符合混凝土结构和观测系统的数学模型,从正演、反演两方面对混凝土无损检测进行研究,并针对具体数据进行处理,取得了满意的效果.本文部分研究内容与成果来源于与“长江工程地球物理勘测武汉有限公司(隶属于长江水利委员会长江勘测规划设计研究院)”的合作项目.
目前,人们对电磁波在混凝土介质及其缺陷中的传播过程还不十分清楚,尤其是缺乏混凝土中不同缺陷反射特征的分析,这导致混凝土探地雷达剖面图的解释有很大差异.本文采用时域有限差分方法模拟混凝土中电磁波传播规律,这不但能够分析电磁波在混凝土缺陷处的反射特征,还能为反演迭代算法提供正演结果.为了使模拟结果更加准确,网格剖分满足数值稳定性条件,同时避免数值色散现象的发生,在边界处用Mur吸收边界条件代替理论分析中经常用到的Dirichlet或Newman边界条件,削弱电磁波在边界处的反射强度.在数值算例中,针对混凝土构件中经常出现的空洞、积水及钢筋等异物,着重分析电磁波在不同介质分界面处的反射特征,并对这三种介质在相同或不同深度的反射强度进行比较,减小分析探地雷达剖面图时产生的误差.
混凝土探地雷达剖面图从形式上只能给出缺陷的位置信息,而缺陷的尺度及属性信息需要专业人士的进一步分析.为了使探地雷达数据的解释工作更加准确、直观,本文从参数识别的角度,通过反演混凝土结构介电常数的方式实现对混凝土内部缺陷的识别.这种思想在地震勘探领域有一定的研究,但是在混凝土无损检测方面还未见到类似的应用.究其主要原因是,这种参数识别方法抽象成数学问题即Maxwell方程反问题,它是一个高度不适定问题,接收数据的微小扰动会造成解的巨大变化.另外,这个反问题具有高度的非线性性,所以会存在多个局部极值,当选取的迭代初值与真实的混凝土结构相差很大时,经典的局部收敛算法通常只会收敛到局部极值,无法收敛到全局最优解,所以不能准确刻画混凝土内部结构.针对这两个难点,本文根据正则化思想,设计了同伦优化方法,它不但能够克服问题的不适定性,还可以根据不同的同伦参数改变局部区域内问题的凸性,从而实现大范围收敛.为了提高算法的收敛性,避免迭代过程中出现分叉、回旋等不收敛现象,对每一步的迭代解扰动后,加大解的容许集.在数值算例中,将真实模型、雷达剖面图和反演结果进行比较,结果显示,仅仅通过雷达剖面图还无法准确刻画混凝土结构,而反演结果更加准确、直观地反映了缺陷的位置、大小及属性.
针对混凝土参数识别方法中出现的局部极值问题,引入小波技术结合Tikhonov正则化,设计了小波多尺度方法.通过小波分解将反问题分解到不同尺度,大尺度下问题呈现凸性,从大尺度到小尺度逐级求解每个尺度上的子反问题,从而实现大范围收敛.在正则化参数选取方面,由于在实际应用中无法获得数据误差信息??,采用GCV方法选择正则化参数.数值算例从三个空洞共存的模型入手,验证了算法的收敛性及稳定性,讨论不同小波对数值结果的影响.然后以从简单到复杂的方式,针对混凝土中缺陷及钢筋的各种存在情况,实现介电常数反演,并结合实际数据说明了该方法能够准确地识别混凝土的内部结构.
总之,本文以混凝土无损检测为出发点,研究电磁波在混凝土不同缺陷处的反射特征.设计大范围收敛的同伦优化方法和小波多尺度方法,解决参数识别方法中出现的不适定性及局部极值问题,这两种方法的解释结果更具定量化和可视性,克服了传统解释方法的缺点,不仅能够探测混凝土缺陷的位置,而且能够比较准确地确定缺陷的尺度及属性.
Concrete is one of most widely used material which is applied in architectural en-gineering. Its structure safety has been the focus of attention. In view of particularityof concrete structure, it is necessary to carry out nondestructive testing before repair-ing, strengthening or reconstructing the concrete. Then we can obtain the informationof defect about location, size and property. So concrete nondestructive testing has veryimportant significance.
GPR nondestructive testing has the advantages of high resolution, fast detect speedand convenient operation, so it wins the good graces of engineering geophysics. Thistechnique has become an important means for concrete nondestructive testing at present.However, when the work of GPR data explanation is compared to radar theory, seismicexploration and fast developed hardware technology, it is found that this work is far awayfrom systematic and mature stage. Now this work mainly is artificial analysis and estima-tion after data precondition, it has not become mechanized processing mode. It is difficultto estimate some of defect signals because of attenuation of electromagnetic wave withhigh frequency. So it often occurs erroneous estimation due to difference of person’sexperience, it results in low accuracy and efficiency of this method. This circumstanceseriously restrict development of GPR technique in aspects of concrete nondestructivetesting, it is badly in need of more visual and automatic estimated system. Concretenondestructive testing belongs to small size testing problem compared with resource ex-ploration. Its Characteristics are small size of object, high accuracy and resolution, badhomogeneity. For complicated testing problem, radar CT tomography based on ray the-ory and waveform inversion based on Born and Rytov weak scattering approximation areall inability. So it must start from coefficient estimation(permittivity or conductivity),develops full waveform numerical inversion method.
In this paper, the theoretical foundation and testing means are based on electromag-netic wave propagation theory and GPR respectively. We build TM model which is accordwith concrete structure and observation system and investigate concrete nondestructivetesting from forward modeling and inversion aspects.
At present, people don’t clearly understand propagation law of electromagnetic wave in concrete and its defect, especially lack quantitative analysis for different re?ection in-tensity of defect. This results in enormous difference for GPR profile explanation. Thispaper uses the finite difference in time domain for simulating electromagnetic wave prop-agation law. It not only analyze re?ection characterize of electromagnetic wave on con-crete’s defect but also provide forword results for inversion iteration algorithm. In orderto simulate propagation law of electromagnetic wave in concrete more accurately andavoid the change of frequency and accumulated error, numerical dispersion is consideredat the same time. At the boundary, we employ Mur absorbing boundary condition insteadof Dirichlet or Newman boundary condition that are usually used for theory analysis, itcan weaken electromagnetic wave re?ection intensity. In Numerical examples, accordingto cavity, water and reinforcement that always appears in concrete, we analyze re?ectioncharacteristics of different medium. Through the comparison of three mediums in sameor different depth, we can decrease error of analysis for GPR profile.
Concrete GPR profile only present location of defect formally, the size and propertyneed further analysis by expert. In order to make the explanation of GPR data more ac-curate and visual, this paper implements the estimation to concrete defect by inversionof permittivity in view of coefficient estimation. There is certain investigation about thisthought in the field of seismic exploration, but the similar application is not appeared inconcrete nondestructive testing. The main reason lies in highly ill-posed and tremendouschange of solution with little disturbance of observe data when this coefficient estimationmethod is abstracted to inverse problems of Maxwell’s equations. In addition this inverseproblem is highly nonlinear, existence of local minimum makes classical method invalidthat couldn’t accurately estimate the structure of concrete. According to this two diffi-culties, this paper designed homotopy optimization method which has the regularizaitoneffect. It can not only overcome the ill-posedness but also implement wide convergenceby selecting different homotopy coefficient for changing convexity. In order to improvealgorithm convergency and avoid phenomenon of non-convergence such as bifurcation,cyclotron and so on, we disturb solution and increase ensemble. In Numerical examples,comparing the true model, radar profile and inversion, results showed that it couldn’taccurately estimate the structure of concrete only by GPR profile. However, inversionresults could estimate the location, size and property of defect more accurately.
According to multi-local extremum problem in concrete inversion imaging, we in- troduce wavelet multiscale method into inverse problem of Maxwell’s equations. We de-signed wavelet multiscale method combined with Tikhonov regularizaiotn. Inverse prob-lem is decomposed onto different scale by wavelet decomposition. The inverse problempresents convexity on large scale, we solve sub-inverse problem on every scale from largeto small, thus large range convergence is implemented. Considering how to select reg-ularization parameter, because it is impossible to obtain data error ??, we employ GCVmethod to select regularization parameter. The simulated example started from singlecavity model, verified the convergence and stability of wavelet multiscale algorithm. Ac-cording to all kinds of structure of concrete, we complemented inversion of permittivityfrom simplicity to complexity, and verified that this method could accurately estimate thestructure of concrete combined with practical data.
In a word, this paper starts from concrete nondestructive testing, investigates electro-magnetic wave re?ection characterize in different defect. We sign homotopy method andwavelet multiscale method which is wide convergence, resolve ill-posedness and localminimum problem appeared in coefficient estimation method. The explanation of the twomethods are more quantitative and visual, overcame the shortcoming of traditional expla-nation method. The two method not only test the location but also accurately determinethe size and property of defect.
引文
[1]罗骐先.水工混凝上缺陷检测和处理[M].中国水利水电出版社, 1997: 20–90.
[2]王继成,许锡宾,赵明阶.水工硷结构损伤诊断技术研究综述[J].重庆交通学院学报,2004,23(4): 19–29.
[3]顾淦臣.国内外土石坝重大事故剖析–对若干土石坝重大事故的再认识[J].水利水电科技进展,1997,17(1): 13–70.
[4]汝乃华,姜忠胜.大坝事故与安全·拱坝[M].中国水利水电出版社, 1995:1–80.
[5]吴丰收.混凝土探测中探地雷达方法技术应用研究[M].吉林大学博士论文,2009.
[6]国家建筑工程质量监督检验中心.混凝土无损检测技术[M].中国建材工业出版社,1996.
[7]邱平.新编混凝土无损检测技术[M].中国环境科学出版社, 2002.
[8]庞超明.试验设计与混凝土无损检测技术[M].中国建材工业出版社, 2006.
[9]罗兴盛.混凝土无损检测技术开发及应用研究[M].重庆大学硕士论文,2008.
[10]吴威,王德元,王奎.超声脉冲波法检测混凝土缺陷试验分析[J].西南公路,2006, 4: 63–65.
[11]熊永红,陈义群,余才盛,等.脉冲回波法在三峡工程混凝土检测中的应用[J].工程地球物理学报,2005,2: 97–100.
[12]蒋济同,范晓义.红外热像技术在混凝土检测中的应用现状和发展趋势[J].无损检测,2011,33: 52–55.
[13]陈兵,姚武,吴科如.声发射技术在混凝土研究中的应用[J].无损检测, 2000,22: 387–396.
[14]陈长征,罗跃纲,白秉三,等.结构损伤检测与智能诊断[M].科学出版社,2001.
[15]李大心.探地雷达方法与应用[M].地质出版社, 1994.
[16]田钢,刘菁华,曾绍发.环境地球物理教程[M].地质出版社, 2005.
[17]曾昭发,刘四新,王者江,等.探地雷达方法原理及应用[M].科学出版社,2006.
[18]粟毅,黄春琳,雷文太.探地雷达理论与应用[M].科学出版社, 2006.
[19]陈理庆.雷达探测技术在结构无损检测中的应用研究[M].湖南大学硕士论文,2008.
[20]徐茂辉.混凝土中钢筋检测的探地雷达方法[M].汕头大学硕士学位论文,2004.
[21]夏才初,潘国荣.土木工程监测技术[M].中国建筑工业出版社, 2001.
[22] Scullion T, Lau C L, Chen Y. Performance Specifications of Ground PenetratingRadar[J]. Proceedings of the Fifth International Conference on Ground PenetratingRadar, Kitchener, Ontario, Canada, 1994.
[23] Maser K R. Highway Speed Radar for Pavement Thickness Evaluation[J]. Proceed-ings of the Fifth International Conference on Ground penetrating radar, Kitchener,Ontario, Canada, 1994.
[24] Attoh N O, Roddis W K. Pavement Thickness Variability and its Effect on De-termination of Moduli and Remaining Life[J]. Transport Res. Rec., 1994, 1449:39–45.
[25] Hugenschmidt J. Gpr Inspecting of a Mountain Motorway in Switzerland[J]. J.Appl. Geophys., 1998, 40: 95–104.
[26] Benedetto A, Benedetto F. Gpr Experimental Evaluaton of Subgrade Soil Charac-teristics for Rehabilitation of Roads[J]. International conference on ground pene-trating radar, Etats-Unis, 2002, 4758: 708–714.
[27] Benedetto A, Blasiis M R. Road Pavement Diagnosis[J]. Quarry Construct., 2001,6: 93–111.
[28] Bungey J H, Millard S G. Use of Impulse Radar as a Method for the Non-destructive Testing of Concrete Slabs[J]. Build. Res. Inf., 1992, 20(3): 152–156.
[29] Pitt J B. Use of Impulse Radar as a Method for the Non-destructive Testing ofConcrete Slabs[J]. Build Res. Inf., 1992, 20: 152–156.
[30] Bungey J H, Millard S G, Shaw M R. In?uence of Reinforcing Stecl on RadarSurveys of Concretestructure[J]. Constr. Build Mater., 1994, 8: 119–126.
[31] Shaw P, Bergstrom J. In-situ Testing of Reinforced Concrete Structures UsingStress Waves and High Frequency Ground Penetrating Radar[J]. NDT and Condi-tion Monitoring, 1998, 42: 454–461.
[32] Ivashov S I, Makarenkov V I, Masterkov A V. Concerete Floor Inspection withHelp of Subsurface Radar[J]. Proc. Spie, 2000, 4084: 552–555.
[33]黄玲,曾昭发,王者江,等.钢筋混凝土缺陷的探地雷达检测模拟与成像效果[J].物探与化探,2007,31: 181–185.
[34]黄玲,曾昭发,刘四新,等.钢筋混凝土缺陷的探地雷达(gpr)检测研究[J].中国地球物理第二十一届年会论文集,2005.
[35]徐美庚,殷宁骏,杨梦蛟.一种新的混凝土桥梁无损测试技术[J].中国铁道科学,1999,20(3): 61–69.
[36]谢雄耀,李永盛,黄新材.地质雷达检测在保护性建筑结构加固中的应用[J].同济大学学报,2000,28(1): 19–23.
[37]倪光正,杨仕友,钱秀英,等.工程电磁场数值计算[M].机械工业出版社,2004.
[38]谢德馨,杨仕友.工程电磁场数值分析与综合[M].机械工业出版社, 2009.
[39] Yee K Z. Numerical Solution of Initial Boundary Value Problems InvolvingMaxwell’s Equations[J]. IEEE T. Antenn. Propag., 1966, 14(3): 302–307.
[40] Jones F W, Pascoe L J. The Perturbation of Alternating Geomagnetic Fields byThree-dimensional Conductivity Inhomogeneities[J]. Geophys. J. Roy. Astr. Soc.,1972, 27: 479–484.
[41] Kahnert M. Boundary Symmetries in Linear Differential and Integral EquationProblems Applied to the Self-consistent Green’s Function Formalism of Acousticand Electromagnetic Scattering[J]. Opt. Commun., 2006, 265: 383–393.
[42] Weaver J T. Mathematical Methods for Geo-electromagnetic Induction[M]. JohnWiley and Sons, 1994.
[43] Bruno O, Elling T, Paffenroth R, et al. Electromagnetic Integral Equations Requir-ing Small Numbers of Krylov-subspace Iterations[J]. J. Comput. Phyisc., 2009,228: 6469–6183.
[44] Liu S B, Zhou T, Liu M L, et al. Wentzel-kramer-brillouin and Finite-differenceTime-domain Analysis of Terahertz Band Electromagnetic Characteristics of Tar-get Coated with Unmagnetized Plasma[J]. J. Sys. Engi. and Electro., 2008, 19:15–20.
[45] Wei S, Huang Z X, Wu X L, et al. Application of the Symplectic Finite-differenceTime-domain Scheme to Electromagnetic Simulation[J]. J. Comput. Phys., 2007,225: 33–50.
[46] Zhao Y, Hao Y. Full-wave Parallel Dispersive Finite-difference Time-domain Mod-eling of Three-dimensional Electromagnetic Cloaking Structures[J]. J. Comput.Phys., 2009, 228: 7300–7312.
[47] Sijoy C, Chaturvedi S. Finite Difference Time Domain Algorithm for Electromag-netic Problems Involving Material Movement[J]. J. Comput. Phys., 2009, 228:2282–2295.
[48] Mimouni A, Rachidi F, Azzouz Z. A Finite-difference Time-domain Approachfor the Evaluation of Electromagnetic Fields Radiated by Lightning Strikes to TallStructures[J]. J. Electro., 2008, 66: 504–513.
[49] Maa F A. Fast Finite-difference Time-domain Modeling for Marine-subsurfaceElectromagnetic Problems[J]. Geophysics, 2007, 72: 19–23.
[50] Zhao Y, Argyropoulos C, Hao Y. Full-wave Finite-difference Time-domain Simu-lation of Electromagnetic Cloaking Structures[J]. Opt. Express, 2008, 16: 6717–6730.
[51] Chen J Q, Zhou B H, Zhao F, et al. Finite-difference Time-domain Analysis of theElectromagnetic Environment in a Reinforced Concrete Structure When Struck byLightning[J]. IEEE Tran. on Electro. Compat., 2010, 52: 914–920.
[52] Wang T, Hohmann G W. A Finite-difference Time-domain Solution for Three-dimensional Electromagnetic Modeling[J]. Geophysics, 1993, 58: 797–809.
[53] Wang T, Tripp A. Fdtd Simulation of Em Wave Propagation in 3-d Media[J].Geophysics, 1996, 61: 110–120.
[54] Weiss C J, Newman G A. Electromagnetic Induction in a Fully 3-d AnisotropicEarth[J]. Geophysics, 2002, 67: 1104–1114.
[55] Commer M, Newman G. A Parallel Finite-difference Approach for 3d TransientElectromagnetic Modeling with Galvanic Sources[J]. Geophysics, 2004, 69: 1192–1202.
[56]刘四新,曾昭发.频散介质中地质雷达波传播的数值模拟[J].地球物理学报,2007, 1: 320–326.
[57] Xu K, Ding D Z, Fan Z H, et al. Fsai Preconditioned Cg Algorithm Combinedwith Gpu Technique for the Tinite Element Analysis of Electromagnetic ScatteringProblems[J]. Finite Elem. Anal. Des., 2010, 47: 387–393.
[58] Ozgun O, Mittra R, Kuzuoglu M. Parallelized Characteristic Basis Finite Ele-ment Method (cbfem-mpi)―a Non-iterative Domain Decomposition Algorithm forElectromagnetic Scattering Problems[J]. J. Comput. Physics., 2009, 228: 2225–2238.
[59] Li P J. Coupling of Finite Element and Boundary Integral Methods for Electro-magnetic Scattering in a Two-layered Medium[J]. J. Comput. Physics., 2010, 229:481–497.
[60] A. Fisher D. White G R. An Efficient Vector Finite Element Method for NonlinearElectromagnetic Modeling[J]. J. Comput. Physics., 2007, 225: 1331–1346.
[61] Pardo D, Demkowicz L, Torres-Verdin C, et al. A Self-adaptive Goal-orientedHp-finite Element Method with Electromagnetic Applications. Part Ii: Electrody-namics[J]. Comp. Meth. Appl. Mech. Engi., 2007, 196: 3585–3597.
[62] Paulsen K D, Linch D R, Strohbehn J W. Three-dimensional Finite, Boundary, andHybrid Element Solutions of the Maxwell Equations for Lossy Dielectric Media[J].IEEE Trans. Microwave Theory Tech., 2001, 36: 682–693.
[63] Livelybrooks D. Program 3dfeem, a Multidimensional Electromagnetic Finite El-ement Model[J]. Geophys. J. Int., 1993, 114: 443–458.
[64] Lager I E, Mur G. Generalized Cartesian Finite Elements[J]. IEEE Trans. Magn.,1998, 34: 2220–2227.
[65] Zunoubi M R, Jin J M, Donepudi K C, et al. A Spectral Lanczos DecompositionMethod for Solving 3-d Low-frequency Electromagnetic Diffusion by the Finite-element Method[J]. IEEE Trans. Antennas Propogat., 1999, 47: 242–248.
[66] Mitsuhata Y, Uchida T. D Magnetotelluric Modeling Using the T-?? DocumentFinite-element Method[J]. Geophysics, 2004, 69: 108–119.
[67] Godev K N, Lazarov R D, Makarov V L, et al. Homogeneous Difference Schemesfor One-dimensional Problems with Generalized Solutions[J]. Mat. Sb., 1986, 131:159–184.
[68] Mattiussi C. An Analysis of Finite Volume, Finite Element, and Finite DifferenceMethods Using some Concepts from Algebraic Topology[J]. J. Comput. Phys.,1997, 133: 289–309.
[69] Haber E, Ascher U M. Fast Finite Volume Simulation of 3d Electromagnetic Prob-lems with Highly Discontinuous Coefficients[J]. SIAM J. Sci. Comput., 2001, 22:1943–1961.
[70]樊明武,颜威利.电磁场积分方程法[M].机械工业出版社, 1988.
[71] Xiong Z. Em Modeling Three-dimensional Structures by the Method of SystemIteration Using Integral Equations[J]. Geophysics, 1992, 57: 1556–1561.
[72] Xiong Z, Tripp A C. Electromagnetic Scattering of Large Structures in LayerednEarth Using Integral Equations[J]. Radio Sci., 1995, 30: 921–929.
[73] Ting S C, Hohmann G W. Integral Equation Modeling of Three-dimensional Mag-netotelluric Response[J]. Geophysics, 1981, 46: 182–197.
[74] Levadoux D P. Stable Integral Equations for the Iterative Solution of Electromag-netic Scattering Problems[J]. Comptes Rendus Physique, 2006, 7: 518–532.
[75] Botha M M. Solving the Volume Integral Equations of Electromagnetic Scatter-ing[J]. J. Comput. Physic., 2006, 218: 141–158.
[76] Kirsch A. An Intergral Equation Approach and the Interior Transmission Problemfor Maxwell’s Equations[J]. Inverse Problems and Imaging, 2007, 1: 107–127.
[77] Kirsch A. An Integral Equation for Maxwell’s Equations If a Layered Mediumwith an Application to the Factorization Method[J]. J. Integral Equations Appl.,2007, 19: 333–358.
[78] Ben J M, Braaten D, Nelson R M, et al. Electric Field Integral Equations for Elec-tromagnetic Scattering Problems with Electrically Small and Electrically Large Re-gions[J]. IEEE T. Antenn. Propag., 2008, 56: 142–150.
[79] Salon S J, Angelo J. Applications of the Hybrid Finite Element-boundary ElementMethod in Electromagnetics[J]. IEEE T. Magnet., 1988, 24: 80–85.
[80] Tusboi H, Misaki T. An Analysis of Three Dimensional Electromagnetic Field byUsing Boundary Element Method,[J]. 8th Inr. Gnf On BEM, Tokyo, September1986, ComputationalMechanics publications, Southampton, 1986.
[81] Enokizono M, Kagawa R, Nakamura T N. Non Linear Analysis of Magnetic Fieldby Boundry Element Method Taking Into Aemunt External Power Sources[J]. 8thInr. Gnf On BEM, Tokyo, September 1986, ComputationalMechanics publications,Southampton, 1986.
[82]刘龙. Maxwell方程数值解的多重网格方法与自适应方法[M].哈尔滨工业大学硕士学位论文,2006.
[83] Mulder W A. A Multigrid Solver for 3d Electromagnetic Diffusion[J]. Geophys.Prospect, 2006, 54: 633–649.
[84] Rickard Y. An Efficient Wavelet-based Solution of Electromagnetic Field Prob-lems[J]. APPL. Numer. Math., 2008, 58: 472–485.
[85] Li S S, Xiao F. A Global Shallow Water Model Using High Order Multi-momentConstrained Finite Volume Method and Icosahedral Grid[J]. J. Comput. Physic.,2010, 229: 1774–1796.
[86] Ney M M. Method of Moments as Applied to Electromagnetic Problems[J]. IEEET. Microw. Theory., 1985, 33: 972–980.
[87] Guimaraes F G, Saldanha R R, Mesquita R C, et al. A Meshless Method for Elec-tromagnetic Field Computation Based on the Multiquadric Technique[J]. IEEE T.Magn., 2007, 43: 1281–1284.
[88]黄卡玛,赵翔.电磁场中的逆问题及应用[M].科学出版社, 2005.
[89]姚参林.蒙特卡洛非线性反演方法及应用[M].冶金工业出版社, 1997.
[90] Tikhonov A N, Arsenin V A. Solutions of Ill-posed Problems[M]. Winston, 1977.
[91]刘继军.不适定问题的正则化方法及应用[M].科学出版社, 2005.
[92]韩波,李莉.非线性不适定问题的求解方法及其应用[M].科学出版社, 2011.
[93] Engl H W, Hanke M, Neubauer A. Regularization of Inverse Problems[M]. KluwerAcademic Publishers, 1996.
[94] Haber E. Quasi-newton Methods for Large-scale Electromagnetic Inverse Prob-lems[J]. Inverse Problems, 2005, 21: 305–323.
[95] Haber E, Heldmann S. An Octree Multigrid Method for Quasi-static Maxwell’sEquations with Highly Discontinuous Coefficients[J]. J. Comput. Phys., 2007,223: 783–796.
[96] Acar R, Vogel C R. Analysis of Total Variation Penalty Methods[J]. Inverse Prob-lems, 1994, 10: 1217–1229.
[97] Haber E, Heldmann S. An Octree Multigrid Method for Quasi-static Maxwell’sEquations with Highly Discontinuous Coefficients[J]. J. Comput. Phys., 2007,223: 783–796.
[98] Daubechies I, Defrise M, Mol C D. An Iterative Thresholding Algorithm for LinearInverse Problems with a Sparsity Constraint[J]. Comm. Pure Appl. Math., 2004,57: 1413–1457.
[99] Ramlau R, Teschke G. A Projection Iteration for Nonlinear Operator Equationswith Sparsity Constraints[J]. Numer. Math., 2006, 104: 177–203.
[100] Vogel C R. Computational Methods for Inverse Problems[M]. SIAM, 2002.
[101] Kirsch A. An Introduction to the Mathematical Theory of Inverse Problems[M].Springer, 1996.
[102] Bunks C, Zaleski F M, Chavent G. Multiscale Seismic Waveform Inversion[J].Geophysics, 1995, 60: 1457–1473.
[103] Donoho D L. Nonlinear Solution of Linear Inverse Problems by Wavelet-vagueletteDecomposition[J]. Appl. Comput. Harmon A., 1995, 11: 1–46.
[104] Abramovich F, Silverman B W. Wavelet Decomposition Approaches to StatisticalInverse Problems[J]. Biometrika, 1998, 85: 115–129.
[105] Kaltenbacher B. Some Newton-type Methods for the Regularization of Non-linearIll-posed Problems[J]. Inverse Problems, 1997, 13: 729–753.
[106] Liu J. A Multiresolution Method for Distributed Parameter Estimation[J]. SIAMJ. Sci. Comput., 1993, 14: 289–405.
[107] Chavent G, Liu J. Multiscale Parameterization for the Estimation of a DiffusionCoeffient in Elliptic and Parabolic Problems[J]. In Fifth IFAC Symposium on Con-trol of Distributed Parameter Systems., 1989.
[108] Chardaire C, Chavent G, Jaffr?′? J, et al. Multiscale Representation for SimultaneousEstimation of Relative Permeabilities and Capillary Pressure[J]. In Proceedings ofthe 65th SPE Annual Technical Conference and Exhibition. New Orleans, 1990.
[109] Liu J, Guerrier B, Benard C. A Sensitivity Decomposition for the RegularizedSolution of Inverse Heat Conduction Problems by Wavelets[J]. Inverse Problems,1995, 11: 1177–1187.
[110] Fu H S, Han B. A Wavelet Multiscale Method for the Inverse Problems of a Two-dimensional Wave Equation[J]. Inverse Probl. Sci. En., 2004, 12: 643–656.
[111] Zhang X M, Liu K A, Liu J Q. The Wavelet Multiscale Method for Inversion ofPorosity in the Fluid-saturated Porous Media[J]. Appl. Math. Comput., 1995, 180:419–427.
[112] Xu Y X, Wang J Y. A Multiresolution Inversion of Onedimensional Magnetotel-luric Data[J]. Chinese J.Geophys., 1998, 41: 704–711.
[113]马坚伟,朱亚平,杨慧珠.二维地震波形小波多尺度反演[J].工程数学学报,2004, 21: 109–113.
[114]李世雄,刘家琦.小波变换和反演数学基础[M].地质出版社, 1994.
[115] Keller H B, Perozzi D J. Fast Seismic Ray Tracing[J]. SIAM J. Appl. Math., 1983,43: 981–992.
[116] Jegen M D, Everett M E, Schultz A. Using Homotopy to Invert GeophysicalData[J]. Geophysics, 2001, 66: 1749–1760.
[117] Dunlavy D M, Leary D P, Klimov D, et al. Hope: A Homotopy OptimizationMethod for Protein Structure Prediction[J]. J. Comput. Biol., 2005, 12: 1275–1288.
[118] Han B, Kuang Z, Liu J Q. A Monotonous Homotopy Method for Solving theResistivities of the Earth[J]. Chinese J.Geophys., 1991, 34: 517–522.
[119] Han B, Fu H S, Li Z. A Homotopy Method for the Inversion of a Two-dimensionalAcoustic Wave Equation[J]. Inverse Probl. Sci. En., 2005, 13: 411–431.
[120] Han B, Fu H S, Liu H. A Homotopy Method for Well Log Constraint WaveformInversion[J]. Geophysics, 2007, 72: R1–R7.
[121] Han B, Fu H S, Gai G. A Wavelet Multiscale-homotopy Method for the InverseProblem of Two-dimensional Acoustic Wave Equation[J]. Appl. Math. Comput.,2007, 190: 576–582.
[122]傅红笋.地震勘探中波动方程反问题的小波、同伦方法[M].哈尔滨工业大学博士学位论文,2006.
[123]李壮.探地雷达反问题的同伦算法研究[M].哈尔滨工业大学博士学位论文,2007.
[124] Newman G A, Alumbaugh D L. Three-dimensional Massively Parallel Electro-magnetic Inversion[J]. Geophys. J. Int., 1997, 128: 345–354.
[125] Haber E, Ascher U M, Oldenburg D W, et al. 3-d Frequency Domain Csem Inver-sion Using Unconstrained Optimization[J]. In 72st Ann. Internat. Mtg., Soc. Expl.Geophys., 2002: 653–656.
[126] Mackie R L, Madden T R. Three-dimensional Magnetotelluric Inversion UsingConjugate Gradients[J]. Geophys. J. Int., 1993, 115: 215–229.
[127] Avdeev D B. Three-dimensinal Electromagnetic Modelling and Inversion fromTheory to Application[J]. Surv. Geophys., 2005, 26: 767–799.
[128] Fletcher R, Reeves C M. Function Minimization by Conjugate Gradients[J]. Com-put. J., 1964, 7: 149–154.
[129] Newman G A, Alumbaugh D L. Three-dimensional Magnetotelluric Inversion Us-ing Non-linear Conjugate Gradients[J]. Geophys. J. Int., 2000, 140: 410–424.
[130] Rodi W, Mackie R L. Nonlinear Conjugate Gradients Algorithm for 3-d Magne-totelluric Inversion[J]. Geophysics, 2000, 66: 174–187.
[131] Mackie R L, Rodi W, Watts M D. 3-d Magnetotelluric Inversion for ResourceExploration[J]. In 71st Ann. Internat. Mtg., Soc. Expl. Geophys., 2001: 1501–1504.
[132] Siripunvaraporn W, Egbert G, Lenbury Y, et al. Three-dimensional MagnetotelluricInversion: Data Space Method[J]. Phys. Earth Planet. Inter., 2004, 150: 3–14.
[133] Sasaki Y. Full 3-d Inversion of Electromagnetic Data on Pc[J]. J. Appl. Geophys.,2001, 46: 45–54.
[134] Haber E, Ascher U M, Aruliah D A, et al. On Optimisation Techniques for SolvingNonlinear Inverse Problems[J]. Inverse Problems, 2000b, 16: 1263–1280.
[135]王长清.现代计算电磁学基础[M].北京大学出版社, 2005.
[136]王秉中.计算电磁学[M].科学出版社, 2002.
[137]何兵寿,张会星.地质雷达正演中的频散压制和吸收边界改进方法[J].地质与勘探,2000,36: 59–63.
[138] Ascher U M, Haber E. A Multigrid Method for Distributed Parameter EstimationProblem[J]. Electron. Trans. Numer. Ana., 2003, 15: 1–17.
[139] Baboolal S, Bharuthram R. Two-scale Numerical Solution of the Electromag-netic Two-?uid Plasma-maxwell Equations:shockand Soliton Simulation[J]. Math.Comput. Simulat., 2007, 76: 3–7.
[140] Haber E. Quasi-newton Methods for Large Scale Electromagnetic Inverse Prob-lems[J]. Inverse Problems, 2005, 21: 305–323.
[141] Albanese R, Monk P B. The Inverse Source Problem for Maxwell’s Equations[J].Inverse Problems, 2006, 22: 435–455.
[142] Han J M, Kim S T. Self-dual Maxwell-chern-simons Theory on a Cylinder[J].Inverse Problems, 2011, 44: 1023–1036.
[143] C. E. Kenig M S, Uhlmann G. Inverse Problems for the Anisotropic MaxwellEquations[J]. Duke Math. J., 2011, 157: 369–419.
[144]李庆扬,关治,白峰杉.数值计算原理[M].清华大学出版社, 2003.
[145]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].科学出版社, 1997.
[146]黄象鼎,曾钟钢,马亚南.非线性数值分析的理论与方法[M].武汉大学出版社,2004.
[147] Chen J Q, Xu Y F, Zou J. An Adaptive Inverse Iteration for Maxwell EigenvalueProblem Based on Edge Elements[J]. J. Comput. Physic., 2010, 229: 2649–2658.
[148] Fu C L, Zhu Y B, Qiu C Y. Wavelet Regularization for an Inverse Heat ConductionProblem[J]. J. Math. Anal. Appl., 2003, 288: 212–222.
[149] Wahba G. Splinve Models for Observational Data[M]. SIAM, 1990.
[150] Charles K C. An Introduction to Wavelets[M]. Science Press, 1995.