探地雷达中的逆时偏移及速度估计
|
摘要
偏移技术是探地雷达数据处理中一项重要的技术环节,其目的是使绕射波收敛和倾斜反射波归位,而寻求雷达高频电磁波所遵循的上行波方程又是波动方程偏移技术的关键。本文从几种常规的雷达偏移技术入手,讨论了它们的实现过程及优缺点,再从传统15°有限差分偏移算法遵循的波动方程出发,分析了该算法难以使陡倾角反射界面偏移归位的产生根源,进而引入了不同于浮动坐标变换或延时坐标变换的线性变换,通过对原波动方程进行了精确推导而未采用任何近似,得到了与15°偏移方程形式相同的均匀介质中线性变换有限差分偏移方程,在不同的差分格式及边界条件约束下实现雷达波场的逆时外推,论文在前人研究的基础之上,研究了该偏移算法在非均匀层状变速介质中的实现情况,通过对加入了速度参数的波动方程进行推导,得到非均匀变速介质中含混合因子项的线性变换差分逆时偏移方程,解决了原偏移算法在非均匀介质中对速度参数的高敏感性问题,并编写了相应的Matlab实现程序,应用该程序对复杂的非均匀变速介质地电模型的正演剖面和实测剖面进行了偏移处理,分析得到的偏移结果体现了混合因子项的作用,即在非均匀介质中同时存在主要反射体和多次反射质点的情况下,可以根据勘探目的,选择是否考虑差分方程式中的混合项来对其进行偏移处理。
在此基础上,而在运行线性变换差分逆时偏移程序进行波场外推,需要输入速度矩阵,实际操作中常是根据偏移成像后图像的效果,对输入的速度矩阵进行不断的修正,这样来逐渐逼近地下介质真实的速度参数,这就给该算法的偏移结果带来一定误差,所以准确速度参数是该偏移算法实现的关键,文章深入研究了波速估计,提出了一种较精确的适合该偏移算法的波速估计方法,它结合输入得到的准确速度矢量矩阵,再次对前面实测资料的偏移效果进行优化,通过对比二者的偏移效果,基于波速估计的非均匀介质中线性变换差分逆时偏移算法极大地提高了原算法的偏移精度。
本文得到国家自然科学基金项目、教育部博士点新教师基金资助项目、湖南省自然基金重点项目、湖南省科技计划项目的联合支助。
Migration techniques is a crucial step in Ground penetrating radar data processing, the aim of which is to make the diffraction wave converged and let the tilt reflection homing, and the key of this techniques is the to find the upcoming wave equation solution, firstly, the paper discussed the migration of several conventional method as well as the merit and disadvantages of each method, then the paper analyzes the traditional 15 degree equation migration algorithm, of which is unable to correctly migrate steep-dip reflections(reflections propagating at large angle),then the linearly transformed ideology is derivated, of which is essentially different from the time-retarded coordinates transformation, through more accurate and stable computation, a none-approximate equation is obtained with special difference scheme to achieve steep-dip reflection migration in constant-velocity media, then the paper extends the application it into laterally inhomogeneous media to improve the former case, a new equation is derivated with a mixed factor items added, and the preparation of the corresponding Matlab program. Then apply it to dealing with a more complicated variable-velocity model and measured data, the imaging results shows the advantage of the improved linear transformation of the migration algorithm is more flexible in dealing with the geological model of main reflector or multiple reflection particles.
On this basis, the paper discusses the velocity estimates, for the velocity matrix is needed to input when running the former migration program, while, the actual operation is through repeated input attempt to approximate the true velocity matrix value, which is based on the migration effect, thus a more accurate velocity estimation method is used to reduce the errors in the process of velocity matrix importing, through which to optimize the accuracy of final migration effect with the migration program in the former case.
The paper was funded by a joint support of the National Natural Science Fund, the Ministry of Education Doctoral Foundation of new teachers, the key project of Hunan Natural Science Foundation, the Science and technology projects of Hunan Province.
在此基础上,而在运行线性变换差分逆时偏移程序进行波场外推,需要输入速度矩阵,实际操作中常是根据偏移成像后图像的效果,对输入的速度矩阵进行不断的修正,这样来逐渐逼近地下介质真实的速度参数,这就给该算法的偏移结果带来一定误差,所以准确速度参数是该偏移算法实现的关键,文章深入研究了波速估计,提出了一种较精确的适合该偏移算法的波速估计方法,它结合输入得到的准确速度矢量矩阵,再次对前面实测资料的偏移效果进行优化,通过对比二者的偏移效果,基于波速估计的非均匀介质中线性变换差分逆时偏移算法极大地提高了原算法的偏移精度。
本文得到国家自然科学基金项目、教育部博士点新教师基金资助项目、湖南省自然基金重点项目、湖南省科技计划项目的联合支助。
Migration techniques is a crucial step in Ground penetrating radar data processing, the aim of which is to make the diffraction wave converged and let the tilt reflection homing, and the key of this techniques is the to find the upcoming wave equation solution, firstly, the paper discussed the migration of several conventional method as well as the merit and disadvantages of each method, then the paper analyzes the traditional 15 degree equation migration algorithm, of which is unable to correctly migrate steep-dip reflections(reflections propagating at large angle),then the linearly transformed ideology is derivated, of which is essentially different from the time-retarded coordinates transformation, through more accurate and stable computation, a none-approximate equation is obtained with special difference scheme to achieve steep-dip reflection migration in constant-velocity media, then the paper extends the application it into laterally inhomogeneous media to improve the former case, a new equation is derivated with a mixed factor items added, and the preparation of the corresponding Matlab program. Then apply it to dealing with a more complicated variable-velocity model and measured data, the imaging results shows the advantage of the improved linear transformation of the migration algorithm is more flexible in dealing with the geological model of main reflector or multiple reflection particles.
On this basis, the paper discusses the velocity estimates, for the velocity matrix is needed to input when running the former migration program, while, the actual operation is through repeated input attempt to approximate the true velocity matrix value, which is based on the migration effect, thus a more accurate velocity estimation method is used to reduce the errors in the process of velocity matrix importing, through which to optimize the accuracy of final migration effect with the migration program in the former case.
The paper was funded by a joint support of the National Natural Science Fund, the Ministry of Education Doctoral Foundation of new teachers, the key project of Hunan Natural Science Foundation, the Science and technology projects of Hunan Province.
引文
[1]粟毅,黄春琳,雷文太.探地雷达理论与应用[M].北京：科学出版社,2006.
[2]李大心.探地雷达方法与应用[M].地质出版社,1994.
[3]王惠濂.探地雷达专辑[J].中国地质大学学报(地球科学),1993,18(3)：249-367.
[4]毕德显.电磁场理论[M].电子工业出版社,1985.
[5]谢处方,饶克谨.电磁场与电磁波[M].北京：高等教育出版社,1987.
[6]傅良魁.应用地球物理教程[M].地质出版社,1991.
[7]赵凯华,陈熙谋.电磁学[M].北京：高等教育出版社,2005.
[8]谢辉.电磁场有限元法在GPR正演模拟中的应用：[硕士学位论文].郑州：郑州大学,2003.
[9]冯德山.基于小波多分辨率探地雷达正演及偏移处理研究：[硕士学位论文].中南大学,2006.
[10]裴建新.衰减介质中地质雷达数据正演模拟和叠前偏移方法研究[硕士学位论文].青岛：中国海洋大学,2004(1)：1-66.
[11]刘喜武,刘洪.地震盲反褶积综述[J].地球物理学进展,2003,18(2)：203-209.
[12]马在田.论反射地震偏移成像[J].勘探地球物理进展,2002,25(3)：1-5.
[13]李世华,杨有发.物探数据处理[M].北京：地质出版社,1995.
[14]李振春,张军华.地震数据处理方法[M].北京：石油大学出版社,2004.
[15]郑海山.地震偏移技术研究进展[J].世界地质,2000,19(4)：393-401.
[16]陈玉译,Claerbout J. F.地球物理数据处理基础[M].石油化学工业出版社,1979.
[17]马在田.高阶方程偏移的分裂算法[J].地球物理学报,1983,26(4)：377-388.
[18]贺振华,王才经,李建朝.反射地震资料偏移处理与反演方法[M].重庆大学出版社,1989.
[19]周彬忠,包吉山.复杂地区地震波场的正演模拟与反时偏移域有限差分法[J].石油地球物理勘探,1988,23(3)：257-262.
[20]张关泉.利用低阶偏微分方程组的大倾角差分偏移[J].地球物理学报,1986(3)：273-282.
[21]赵永辉,吴健生,万明浩,等.有限差分法探地雷达波动方程偏移成像[J].物探化探计算技术,2001(1)：47-51.
[22]冯德山,戴前伟,左德勤.地质雷达二维时域有限差分正演[J].勘察科学 技术,2004(6)：35-37.
[23]张钋,李幼铭.几类叠前深度偏移方法的研究现状[J].地球物理学进展,2000,15(2)：30-39.
[24]陈树文,刘洪.三维叠前深度偏移的准三维算法研究[J].地球物理学进展,2001,16(4)：23-28.
[25]邓世坤.探地雷达图像的正演合成[J].地球科学：中国地质大学学报,1993,18(3)：285-293.
[26]邓世坤.克希霍夫积分偏移法在探地雷达图象处理中的应用[J].地球科学-中国地质大学学报,1993(03),18(3)：303-309.
[27]邓世坤,王惠濂.探地雷达图像的正演合成与偏移处理[J].地球物理学报,1993,36(4)：528-536.
[28]戴前伟, 冯德山, 何继善.Kirchhoff偏移法在探地雷达正演图像处理中的应用[J].地球物理学进展,2005,20(3)：849-853.
[29]李文忠.F—K偏移技术在地质雷达资料处理方面的应用[J].物探化探计算技术,1998,20(3)：280-283.
[30]李成方.偏移技术在探地雷达数据处理中的研究[硕士学位论文].成都：成都理工大学,2002.
[31]李成方,王绪本,薛克勤,等.偏移技术在GPR资料处理中的研究[J].物探与化探,2004,28(5)：451-452.
[32]王尚旭,牟永光.有限元素法全倾角波动方程偏移[J].地球物理学报,1989,32(3)：308-318.
[33]底青云,许琨,王妙月.衰减雷达波有限元偏移[J].地球物理学报,2000,43(2)：257-263.
[34]Teng Y, Dai T. Finite-element prestack reverse-time migration for elastic wave [J]. Geophysics,1989,54(9):1204-1208.
[35]张美根,王妙月.各向异性弹性波有限元叠前逆时偏移[J].地球物理学报,2001,44(5)：711-719.
[36]薛东川,王尚旭.波动方程有限元叠前逆时偏移[J].石油地球物理勘探,2008,4(1)：17-21.
[37]Sayedelahl Aya, Bording R. Phillip. A Study of Seismic Inverse Methods for Radar Signal Processing[J]. Computer society,2005,58-62.
[38]Bay sal E, Kosloff DD, Sherwood JWC. Reverse-time migration[J]. Geophysics,1983,48(11):1514-1524.
[39]C Mosher C., S Wu R., J Foster D. Phase shift migration using orthogonal beamlet transforms[J]. SEG 72nd Annual Meeting,2002.
[40]Young. A collection formulation of wave equation[J]. Computer Aided Seismic Analysis and Discrimination,1981,11(19):47-53.
[41]Zhiming Li. Imaging steep-dip reflection by the linearly transformed wave equation method[J]. SEG 56th Annual Meeting,1986.
[42]Grav Samuel H.地震偏移问题及其解决方案[J].勘探地球物理进展,2002,25(2)：44-60.
[43]Gray S. H. Seismic Imaging[J]. Geophysics,2001(66):15-17.
[44]Gray S. H., Etgen J., Dellinger J., et al. Seismic problems and solutions[J]. Geophysics,2001(66):1622-1640.
[45]Rubsion E. A.,汪贤驯,吴晖译.地球物理数据偏移[M].北京：石油工业出版社,1985.
[46]Tana MT, Neider NS,马在田译. 波动方程偏移引论[M].北京：石油工业出版社,1984.
[47]牟永光.地震勘探资料数字处理方法[M].北京：石油工业出版社,1981.
[48]石油化学工业部石油地球物理勘探局计算中心站编译.常用地震资料处理程序[M].北京：石油化学工业出版社,1978.
[49]黄德济,贺振华,包吉山.地震勘探资料数据处理[M].北京：地质出版社,1990.
[50]薛桂霞,邓世坤,刘秀娟.逆时偏移在探地雷达信号处理中的应用[J].煤田地质与勘探,2004,32(1)：55-57.
[51]Zhu Jinming, Lines L. R. Comparison of Kirchhoff and reverse-time migration methods with applications to prestack depth imaging of complex structures[J]. Geophysics,1998,63(4):1166-1176.
[52]刘喜武,刘洪.波动方程地震偏移成像方法的现状与进展[J].地球物理学进展,2002,17(4)：582-591.
[53]李广场.有限差分法探地雷达波动方程偏移[硕士学位论文].南京：河海大学,2004(01).
[54]马在田.地震成像技术有限差分法偏移[M].石油工业出版社,1989.
[55]Gazdag. Wave equation migration with the phase-shift method[J]. Geophysics.1978,43:1342-1351.
[56]Gazdag. Migration of seismic Data by phase-shift plus interpolation[J]. Geophysics.1984,49:124-131.
[57]贺振华,G. H. F. Gardner. F-K波动方程偏移的频率域插值方法[J].石油物 探,1985,24(1)：1-22.
[58]石油化学工业部石油地球物理勘探局编.地震勘探数字技术[M].北京：科学出版社,1977.
[59]Schneider. Integral formulation for migration in two and three dimensions[J]. Geophysics,1978,43(1):49-76.
[60]李鸣祉.地震勘探资料数字处理[M].江苏：中国矿业大学出版社,1989.
[61]吴国明,龚小金.三维波动方程克希霍夫积分偏移在150机上的实现[J].石油地球物理勘探,1983(2)：127-133.
[62]冯德山,戴前伟.探地雷达小波域三维波动方程偏移[J].地球物理学报,2008,51(2)：566-574.
[63]董渊,杨慧珠,杜启振.有限元有限差分法二维波动逆时偏移初探[J].石油大学学报(自然科学版),2003,27(6)：25-29.
[64]修志杰.偏移及速度估计在探地雷达信号处理中的研究[硕士学位论文].北京：中国科学院研究生院,2006.
[65]Stolt R H. Migration by Fourier transforms [J]. Geophysics,1978,43(1): 23-48.
[66]陈明逵.Toeplitz三对角线性方程组的快速并行算法[J].工程数学学报,1987,4(3)：59-63.
[67]陈明逵.解特殊三对角线性方程组的修正追赶法[J].西安交通大学学报,1982,16(5)：58-92.
[68]高德荫,徐峰,潘乃德.一类特殊三对角线性方程组的快速近似追赶法[J].计算数学,1981(1)：10-17.
[69]Zhiming Li. Wave field extrapolation by the linearly transformed wave equation[J]. Geophysics,1986,51(8):1538-1551.
[70]James Irving, Rosemary Knight. Numerical modeling of ground-penetrat-ing radar in 2-D using MATLAB[J]. Computers Geosciences,2006,32(9): 1246-1258.
[71]Lampe Bernhard, Holliger Klaus, Green Alan. Finite-difference time-domain simulation tool for ground-penetrating radar antennas [J]. Geophysics,2003,68(3):971-985.
[72]Leuschen CJ, Plumb RG. A matched-filter-based reverse-time migration algorithm for ground-penetrating radar data[J]. IEEE Transactions on Geoscience and Remote Sensing,2001,39(5):929-936.
[73]蒲俊,吉家锋,伊良忠.MATLAB 6.0数学手册[M].上海：浦东电子出版社, 2005.
[74]偏微分方程的MATLAB解法[M].武汉：武汉大学出版社,2001.
[75]赵嘉.基于MATLAB的FDTD算法编程[J].广西工学院学报,2006,17(4)：43-46.
[76]熊彬,阮百尧.MATLAB在有限差分法中的应用[J].桂林工学院学报,2001,21(2)：104-109.
[77]工程电磁场数值计算[M].北京：机械工业出版社.2004.
[78]王飞,裴永祥.有限差分方法的MATLAB编程[J].新疆师范大学学报(自然科学版),2003,22(4)：22-27.
[79]张志涌,杨祖樱.掌握和精通MATLAB[M].北京：北京航空航天大学出版社,2000.
[80]Stephen J Chapman. MATLAB Programming for Engineers(Second Edition)[M].北京：科学出版社,2002.
[81]Partial Differential Equation Toolbox For Use with MATLAB[M]. The MathWorks,1999.
[82]罗延钟, 张桂青.电子计算机在电法勘探中的应用[M].武汉：武汉地址学院出版社,1987.
[83]沈飚,孙忠良,曲孝良.探地雷达波波动方程研究及其正演模拟[J].东南大学学报,1994,24(2)：114-117.
[84]王长清.FD-TD法中的吸收边界条件[J].无线电电子学汇刊,1998,1：49-57.
[85]王长清,祝西里.电磁场计算中的时域有限差分法[M].北京：北京大学出版社,1994.
[86]Roberts RL, Daniels J. Finite-difference time domain (FDTD) forward modeling of GPR data[J].5th Int. Conf. Ground Penetrating Radar, 1994:185-204.
[87]Yee KS. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[J]. IEEE Transations on antennas and propagation,1966(14):302-307.
[88]李文超,王绪本,邓薇.探地雷达FDTD正演模拟及F-K偏移处理[J].2007(03)：33-36.
[89]Michael Jervis, Mrinal K Sen, Paul L Stoffa. Prestack migration velocity estimation using nonlinear methods[J]. Geophysics,1996, 60(6):138-150.
[90]于景兰,王春和.探地雷达探测地下目标时的波速估计[J].地球物理学进展,2003,18(3)：477-480.
[91]袁明德.探地雷达检测中如何计算速度[J].物探与化探,2003,27(03)：220-222.
[92]陆基孟.地震勘探原理[M].北京：石油大学出版社,1996.
[93]何樵登.地震勘探原理和方法[M].地质出版社,1986.
[94]张安学,蒋延生,汪文秉.探地雷达频率波数域速度估计和成像方法的实验研究[J].电子学报,2001,29(3)：315-317.
[95]Toru Kaneko. Rader image processing for locating underground linear objects[J]. IEICE Trans,1991,74(10):3451-3458.
[96]孔令讲,周正欧.浅地层探地雷达波速测量方法的研究[J].电子学报,2002,30(9)：1330-1332.
[2]李大心.探地雷达方法与应用[M].地质出版社,1994.
[3]王惠濂.探地雷达专辑[J].中国地质大学学报(地球科学),1993,18(3)：249-367.
[4]毕德显.电磁场理论[M].电子工业出版社,1985.
[5]谢处方,饶克谨.电磁场与电磁波[M].北京：高等教育出版社,1987.
[6]傅良魁.应用地球物理教程[M].地质出版社,1991.
[7]赵凯华,陈熙谋.电磁学[M].北京：高等教育出版社,2005.
[8]谢辉.电磁场有限元法在GPR正演模拟中的应用：[硕士学位论文].郑州：郑州大学,2003.
[9]冯德山.基于小波多分辨率探地雷达正演及偏移处理研究：[硕士学位论文].中南大学,2006.
[10]裴建新.衰减介质中地质雷达数据正演模拟和叠前偏移方法研究[硕士学位论文].青岛：中国海洋大学,2004(1)：1-66.
[11]刘喜武,刘洪.地震盲反褶积综述[J].地球物理学进展,2003,18(2)：203-209.
[12]马在田.论反射地震偏移成像[J].勘探地球物理进展,2002,25(3)：1-5.
[13]李世华,杨有发.物探数据处理[M].北京：地质出版社,1995.
[14]李振春,张军华.地震数据处理方法[M].北京：石油大学出版社,2004.
[15]郑海山.地震偏移技术研究进展[J].世界地质,2000,19(4)：393-401.
[16]陈玉译,Claerbout J. F.地球物理数据处理基础[M].石油化学工业出版社,1979.
[17]马在田.高阶方程偏移的分裂算法[J].地球物理学报,1983,26(4)：377-388.
[18]贺振华,王才经,李建朝.反射地震资料偏移处理与反演方法[M].重庆大学出版社,1989.
[19]周彬忠,包吉山.复杂地区地震波场的正演模拟与反时偏移域有限差分法[J].石油地球物理勘探,1988,23(3)：257-262.
[20]张关泉.利用低阶偏微分方程组的大倾角差分偏移[J].地球物理学报,1986(3)：273-282.
[21]赵永辉,吴健生,万明浩,等.有限差分法探地雷达波动方程偏移成像[J].物探化探计算技术,2001(1)：47-51.
[22]冯德山,戴前伟,左德勤.地质雷达二维时域有限差分正演[J].勘察科学 技术,2004(6)：35-37.
[23]张钋,李幼铭.几类叠前深度偏移方法的研究现状[J].地球物理学进展,2000,15(2)：30-39.
[24]陈树文,刘洪.三维叠前深度偏移的准三维算法研究[J].地球物理学进展,2001,16(4)：23-28.
[25]邓世坤.探地雷达图像的正演合成[J].地球科学：中国地质大学学报,1993,18(3)：285-293.
[26]邓世坤.克希霍夫积分偏移法在探地雷达图象处理中的应用[J].地球科学-中国地质大学学报,1993(03),18(3)：303-309.
[27]邓世坤,王惠濂.探地雷达图像的正演合成与偏移处理[J].地球物理学报,1993,36(4)：528-536.
[28]戴前伟, 冯德山, 何继善.Kirchhoff偏移法在探地雷达正演图像处理中的应用[J].地球物理学进展,2005,20(3)：849-853.
[29]李文忠.F—K偏移技术在地质雷达资料处理方面的应用[J].物探化探计算技术,1998,20(3)：280-283.
[30]李成方.偏移技术在探地雷达数据处理中的研究[硕士学位论文].成都：成都理工大学,2002.
[31]李成方,王绪本,薛克勤,等.偏移技术在GPR资料处理中的研究[J].物探与化探,2004,28(5)：451-452.
[32]王尚旭,牟永光.有限元素法全倾角波动方程偏移[J].地球物理学报,1989,32(3)：308-318.
[33]底青云,许琨,王妙月.衰减雷达波有限元偏移[J].地球物理学报,2000,43(2)：257-263.
[34]Teng Y, Dai T. Finite-element prestack reverse-time migration for elastic wave [J]. Geophysics,1989,54(9):1204-1208.
[35]张美根,王妙月.各向异性弹性波有限元叠前逆时偏移[J].地球物理学报,2001,44(5)：711-719.
[36]薛东川,王尚旭.波动方程有限元叠前逆时偏移[J].石油地球物理勘探,2008,4(1)：17-21.
[37]Sayedelahl Aya, Bording R. Phillip. A Study of Seismic Inverse Methods for Radar Signal Processing[J]. Computer society,2005,58-62.
[38]Bay sal E, Kosloff DD, Sherwood JWC. Reverse-time migration[J]. Geophysics,1983,48(11):1514-1524.
[39]C Mosher C., S Wu R., J Foster D. Phase shift migration using orthogonal beamlet transforms[J]. SEG 72nd Annual Meeting,2002.
[40]Young. A collection formulation of wave equation[J]. Computer Aided Seismic Analysis and Discrimination,1981,11(19):47-53.
[41]Zhiming Li. Imaging steep-dip reflection by the linearly transformed wave equation method[J]. SEG 56th Annual Meeting,1986.
[42]Grav Samuel H.地震偏移问题及其解决方案[J].勘探地球物理进展,2002,25(2)：44-60.
[43]Gray S. H. Seismic Imaging[J]. Geophysics,2001(66):15-17.
[44]Gray S. H., Etgen J., Dellinger J., et al. Seismic problems and solutions[J]. Geophysics,2001(66):1622-1640.
[45]Rubsion E. A.,汪贤驯,吴晖译.地球物理数据偏移[M].北京：石油工业出版社,1985.
[46]Tana MT, Neider NS,马在田译. 波动方程偏移引论[M].北京：石油工业出版社,1984.
[47]牟永光.地震勘探资料数字处理方法[M].北京：石油工业出版社,1981.
[48]石油化学工业部石油地球物理勘探局计算中心站编译.常用地震资料处理程序[M].北京：石油化学工业出版社,1978.
[49]黄德济,贺振华,包吉山.地震勘探资料数据处理[M].北京：地质出版社,1990.
[50]薛桂霞,邓世坤,刘秀娟.逆时偏移在探地雷达信号处理中的应用[J].煤田地质与勘探,2004,32(1)：55-57.
[51]Zhu Jinming, Lines L. R. Comparison of Kirchhoff and reverse-time migration methods with applications to prestack depth imaging of complex structures[J]. Geophysics,1998,63(4):1166-1176.
[52]刘喜武,刘洪.波动方程地震偏移成像方法的现状与进展[J].地球物理学进展,2002,17(4)：582-591.
[53]李广场.有限差分法探地雷达波动方程偏移[硕士学位论文].南京：河海大学,2004(01).
[54]马在田.地震成像技术有限差分法偏移[M].石油工业出版社,1989.
[55]Gazdag. Wave equation migration with the phase-shift method[J]. Geophysics.1978,43:1342-1351.
[56]Gazdag. Migration of seismic Data by phase-shift plus interpolation[J]. Geophysics.1984,49:124-131.
[57]贺振华,G. H. F. Gardner. F-K波动方程偏移的频率域插值方法[J].石油物 探,1985,24(1)：1-22.
[58]石油化学工业部石油地球物理勘探局编.地震勘探数字技术[M].北京：科学出版社,1977.
[59]Schneider. Integral formulation for migration in two and three dimensions[J]. Geophysics,1978,43(1):49-76.
[60]李鸣祉.地震勘探资料数字处理[M].江苏：中国矿业大学出版社,1989.
[61]吴国明,龚小金.三维波动方程克希霍夫积分偏移在150机上的实现[J].石油地球物理勘探,1983(2)：127-133.
[62]冯德山,戴前伟.探地雷达小波域三维波动方程偏移[J].地球物理学报,2008,51(2)：566-574.
[63]董渊,杨慧珠,杜启振.有限元有限差分法二维波动逆时偏移初探[J].石油大学学报(自然科学版),2003,27(6)：25-29.
[64]修志杰.偏移及速度估计在探地雷达信号处理中的研究[硕士学位论文].北京：中国科学院研究生院,2006.
[65]Stolt R H. Migration by Fourier transforms [J]. Geophysics,1978,43(1): 23-48.
[66]陈明逵.Toeplitz三对角线性方程组的快速并行算法[J].工程数学学报,1987,4(3)：59-63.
[67]陈明逵.解特殊三对角线性方程组的修正追赶法[J].西安交通大学学报,1982,16(5)：58-92.
[68]高德荫,徐峰,潘乃德.一类特殊三对角线性方程组的快速近似追赶法[J].计算数学,1981(1)：10-17.
[69]Zhiming Li. Wave field extrapolation by the linearly transformed wave equation[J]. Geophysics,1986,51(8):1538-1551.
[70]James Irving, Rosemary Knight. Numerical modeling of ground-penetrat-ing radar in 2-D using MATLAB[J]. Computers Geosciences,2006,32(9): 1246-1258.
[71]Lampe Bernhard, Holliger Klaus, Green Alan. Finite-difference time-domain simulation tool for ground-penetrating radar antennas [J]. Geophysics,2003,68(3):971-985.
[72]Leuschen CJ, Plumb RG. A matched-filter-based reverse-time migration algorithm for ground-penetrating radar data[J]. IEEE Transactions on Geoscience and Remote Sensing,2001,39(5):929-936.
[73]蒲俊,吉家锋,伊良忠.MATLAB 6.0数学手册[M].上海：浦东电子出版社, 2005.
[74]偏微分方程的MATLAB解法[M].武汉：武汉大学出版社,2001.
[75]赵嘉.基于MATLAB的FDTD算法编程[J].广西工学院学报,2006,17(4)：43-46.
[76]熊彬,阮百尧.MATLAB在有限差分法中的应用[J].桂林工学院学报,2001,21(2)：104-109.
[77]工程电磁场数值计算[M].北京：机械工业出版社.2004.
[78]王飞,裴永祥.有限差分方法的MATLAB编程[J].新疆师范大学学报(自然科学版),2003,22(4)：22-27.
[79]张志涌,杨祖樱.掌握和精通MATLAB[M].北京：北京航空航天大学出版社,2000.
[80]Stephen J Chapman. MATLAB Programming for Engineers(Second Edition)[M].北京：科学出版社,2002.
[81]Partial Differential Equation Toolbox For Use with MATLAB[M]. The MathWorks,1999.
[82]罗延钟, 张桂青.电子计算机在电法勘探中的应用[M].武汉：武汉地址学院出版社,1987.
[83]沈飚,孙忠良,曲孝良.探地雷达波波动方程研究及其正演模拟[J].东南大学学报,1994,24(2)：114-117.
[84]王长清.FD-TD法中的吸收边界条件[J].无线电电子学汇刊,1998,1：49-57.
[85]王长清,祝西里.电磁场计算中的时域有限差分法[M].北京：北京大学出版社,1994.
[86]Roberts RL, Daniels J. Finite-difference time domain (FDTD) forward modeling of GPR data[J].5th Int. Conf. Ground Penetrating Radar, 1994:185-204.
[87]Yee KS. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[J]. IEEE Transations on antennas and propagation,1966(14):302-307.
[88]李文超,王绪本,邓薇.探地雷达FDTD正演模拟及F-K偏移处理[J].2007(03)：33-36.
[89]Michael Jervis, Mrinal K Sen, Paul L Stoffa. Prestack migration velocity estimation using nonlinear methods[J]. Geophysics,1996, 60(6):138-150.
[90]于景兰,王春和.探地雷达探测地下目标时的波速估计[J].地球物理学进展,2003,18(3)：477-480.
[91]袁明德.探地雷达检测中如何计算速度[J].物探与化探,2003,27(03)：220-222.
[92]陆基孟.地震勘探原理[M].北京：石油大学出版社,1996.
[93]何樵登.地震勘探原理和方法[M].地质出版社,1986.
[94]张安学,蒋延生,汪文秉.探地雷达频率波数域速度估计和成像方法的实验研究[J].电子学报,2001,29(3)：315-317.
[95]Toru Kaneko. Rader image processing for locating underground linear objects[J]. IEICE Trans,1991,74(10):3451-3458.
[96]孔令讲,周正欧.浅地层探地雷达波速测量方法的研究[J].电子学报,2002,30(9)：1330-1332.