复杂有耗色散地层中的FDTD方法以及在冲击探地雷达中的应用
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摘要
脉冲探地雷达是时域瞬态电磁场理论发展过程中重要的阶段性成果。它结合了瞬态电
磁场理论、时域测量技术、纳秒脉冲源技术、超宽带天线技术和信号处理技术等多门学科
的研究成果，具有无损伤、高效率、低成本和高分辨成像等优点，已成为地下浅层勘探的
重要工具之一。同时这项技术为瞬态电磁场在超宽带雷达、超宽带通讯等更广泛领域的应
用提供了一种成功的实物范例。然而，当我们检查这项技术的状况，便会发现尚有许多不
足之处需要克服。比如，目前还不能仅仅依靠雷达的测试数据给出诸如目标位置、目标性
质、大小尺寸等应用部门等最关心的问题以较精确的、较准确的回答。为此，深入理解在
不均匀土壤背景中瞬态激励场和目标间相互作用的物理机制对解决这些问题起着关键的作
用，并可望在此基础上最大限度的提取包含在反射、散射回波中的有用信息。在这种背景
下，本项研究致力于用电磁理论的数值计算方法揭示目标与背景的瞬态电磁相互作用规律
和散射特性，为建立雷达仿真系统作必要的准备。
本文选用时域有限差分方法（FDTD方法）作为研究有耗、色散、不均匀的背景土壤
中目标、环境和雷达天馈系统的瞬态电磁脉冲之间相互作用的出发点，特别对这种方法的
适应性、技术实现细节以及性能作了系统和详细的分析并取得了一些研究成果，主要包括：
1．将经典平面波在水平分层不均匀媒质中透射与反射关系和快速傅立叶交换（FFT）结
合起来，实现了脉冲平面波入射场的解析设置方法。分别用多种角度入射的瞬态场和
时谐场对计算公式和程序作了验证，得到在连接边界处数值计算误差小于3×10~(-3)（相
对于入射场幅度）的结果；
2．研究了色散媒质模型参数提取问题。提出基于最小二乘法和Debye模型的拟合方法。
对一组典型的在50MHz-1250MHz频率范围所测量的土壤复参数色散曲线进行拟合并
取得良好效果，并将其用在色散媒质中脉冲波传播问题的模拟；
3．比较了Mur吸收边界条件、有耗媒质中的GPML完全匹配层、色散媒质的PML完全
匹配层在地下目标散射场计算中的技术性能。特别关注水平分层不均匀媒质中各种吸
收边界条件的设置方法，分析了这些吸收边界条件在角点和分层媒质交界面的误差分
布特性。理论推导和数值计算表明：（1）在有耗媒质中，Mur二阶吸收边界条件和完
全匹配层（PML）都具有一定的吸收外向行波特性。在均匀空间中，PML媒质的性能
比Mur吸收边界条件好的多。但对于分层不均匀媒质，两者的数值反射误差基本相同，
其主要原因是GPML媒质在介质交界面处不能消除固有的反射，而Mur吸收边界条件
则因为采用了行波近似在这些边界处反射要小的多。（2）对色散媒质而言，Mur二阶
吸收边界条件完全不能应用，而根据PML原理得到的广义PML方法如Gadney的单
轴完全匹配媒质（本文中简称DPML）可以取得较好的效果。（3）由于这些方法所产
生的相对反射误差和晚时瞬态散射的幅度差别较小，因此FDTD方法不适合计算晚时
响应。在一般条件下，FDTD方法完全适合于计算埋地目标的早时响应。（4）从所需
的计算资源来讲，Mur二阶吸收边界条件所需的计算时间和占用内存较小，编程要易
于PML方法。（5）数值计算表明时域有限差分方法在研究分层不均匀、有耗媒质中的
计算误差一般在10~(-3)（按入射场幅度归一化）量级，因此完全适合本文研究需要。
4．通过精心设计的一组典型分层不均匀媒质情况，运用FDTD方法数值模拟了瞬态脉冲
波在地卜传播反射、透射并返回地面的整个随时间空间波动的物理过村，从而得到平
面波在水平分层不均匀、有耗士壤中的传捎机理并分析了地面上反射脉冲的形成规律，
得到关丁反射脉冲幅度、波速和其电参数的近似关系。应用FDTD方法讨论了不均匀
介质薄层的反射特性，得到关于估算垂直分辨率的近似公式以及分辨薄层的波形特征。
5．详细计算了细长目标在瞬态线源激励情况下，在多种目标、背景、埋地深度以及测量
方法的瞬态散射波形特征，讨论了其内在的物理意义。推导了波形衰减与相角增量的
计算公式，导出了计算目标深度与背景波速的近似估算方法。
6．研究了近地面平面蝶型偶极大线的模拟计算问题，讨论了近地面脉冲大线性能参数表
示问题。详细比较了FDTD计算中各种激励源的设置方法，结合一种简单的电压型馏
滤设置方法，给出了近地面平面蝶型偶极天线的近场辐射性能分析。同时计算了圆杜、
金属管和含倾斜界面的仿真雷达回波和剖面图，这些结果具有较大的实川价值。
7．提出分层随机不均匀媒质中Monte－Carlo模拟方法，运用高斯谱模型生成粗糙面，运
用随机分布方法生成分层随机分布介质模型，结合FDTD计算其界面的散射特征，）1。
fly用到高速公路的探地雷达检测方法评估中。
8．研究了一些非平稳信号处理方法在地下目标回波信号处理中的应川，提出基丁时频分
布的信号检测与估计统计量以及优化实现方案，并以简化的尺度相关模型为例研究最
优参数估计方法。从实际测量数据出发
Impulse ground penetrating radar is an important and phasic achievement of the
development of transient electromagnetic (EM) theories and a result of a combination of time-
domain measurement technique, nanosecond pulse generation technique, ultra-wide band antenna
technique and signal processing methodology. It has many advantages such as nondestructive
detection, high efficiency, low cost and high resolution imaging ability and has already become
one of the most important tools in subsurface exploration. Also, it is a successful example for the
applications of transient electromagnetic theory in many other areas such as ultra-wide band
radar, ultra-wide band communications and so on. Yet, if one has an experience about the method,
he will see many problems left for further investigations. For example, one can not tell the
location, the constituent material and the size about an unknown target with an acceptable
accuracy and exactness just by use of field test data. This arises from lack of knowledge about
the transient EM interactions between radar antenna, inhomogeneous soil, and buried targets.
Thus, a thorough and detail understanding of the physical rules about transient EM interactions in
inhomogeneous soil is of great importance for solving these tough problems. It is prospective to
extract as much valuable information as possible from the reflected radar returns based on the
knowledge. The main purpose of the study presented in the paper is to reveal the rules of the
transient EM scattering in this situation and to make a preparation for the establishment of
simulation tools for impulse ground penetrating radar systems by using the methods of
computational electromagnetics.
Compared with other numerical tools, the finite-difference finite-time method (the FDTD
method) is the most appropriate choice for the analysis of the transient EM interactions between
the antennas, the lossy, dispersive, inhomogeneous soil, and the target(s). We begin with a careful
check about the suitability, the method of implementations, and the performances of the
algorithms. Some important results are obtained in the research work. The main contributions of
the paper can be summarized as follows:
1. An analytical method of transient incident wave source conditions for total-scattered field
formulation in stratified media is introduced using classical reflection and transmission
relations under a time-harmonic incident plane wave and the fast Fourier transform (FFT)
technique. The codes are confirmed at a variety of incident angles, both in transient field and
in time harmonic field. The results show that the errors at the contiguous boundary due to this
method are less than 3x101 relative to the magnitude of the incident field.
2. A method of the complex dielectric curve fitting from measured data based on minimum least
square method and the first order Debye model is presented, which is validated by a group of
typical soil conditions covering a frequency range of 5OMHz-l250 MHz. The result shows that
the approximations are very good. The resulting parameters are then used in the investigation
for the pulse propagating in dispersive media.
3. The performances of the Mur抯 second order absorption conditions, the GPML layer for lossy
media, and the modified PML layer for dispersive media are investigated with special concern
about the implementations for horizontal stratified media. The error distributions at the corners
and at different media interfaces are explored. From theoretical derivations and numerical
computations, one
磁场理论、时域测量技术、纳秒脉冲源技术、超宽带天线技术和信号处理技术等多门学科
的研究成果，具有无损伤、高效率、低成本和高分辨成像等优点，已成为地下浅层勘探的
重要工具之一。同时这项技术为瞬态电磁场在超宽带雷达、超宽带通讯等更广泛领域的应
用提供了一种成功的实物范例。然而，当我们检查这项技术的状况，便会发现尚有许多不
足之处需要克服。比如，目前还不能仅仅依靠雷达的测试数据给出诸如目标位置、目标性
质、大小尺寸等应用部门等最关心的问题以较精确的、较准确的回答。为此，深入理解在
不均匀土壤背景中瞬态激励场和目标间相互作用的物理机制对解决这些问题起着关键的作
用，并可望在此基础上最大限度的提取包含在反射、散射回波中的有用信息。在这种背景
下，本项研究致力于用电磁理论的数值计算方法揭示目标与背景的瞬态电磁相互作用规律
和散射特性，为建立雷达仿真系统作必要的准备。
本文选用时域有限差分方法（FDTD方法）作为研究有耗、色散、不均匀的背景土壤
中目标、环境和雷达天馈系统的瞬态电磁脉冲之间相互作用的出发点，特别对这种方法的
适应性、技术实现细节以及性能作了系统和详细的分析并取得了一些研究成果，主要包括：
1．将经典平面波在水平分层不均匀媒质中透射与反射关系和快速傅立叶交换（FFT）结
合起来，实现了脉冲平面波入射场的解析设置方法。分别用多种角度入射的瞬态场和
时谐场对计算公式和程序作了验证，得到在连接边界处数值计算误差小于3×10~(-3)（相
对于入射场幅度）的结果；
2．研究了色散媒质模型参数提取问题。提出基于最小二乘法和Debye模型的拟合方法。
对一组典型的在50MHz-1250MHz频率范围所测量的土壤复参数色散曲线进行拟合并
取得良好效果，并将其用在色散媒质中脉冲波传播问题的模拟；
3．比较了Mur吸收边界条件、有耗媒质中的GPML完全匹配层、色散媒质的PML完全
匹配层在地下目标散射场计算中的技术性能。特别关注水平分层不均匀媒质中各种吸
收边界条件的设置方法，分析了这些吸收边界条件在角点和分层媒质交界面的误差分
布特性。理论推导和数值计算表明：（1）在有耗媒质中，Mur二阶吸收边界条件和完
全匹配层（PML）都具有一定的吸收外向行波特性。在均匀空间中，PML媒质的性能
比Mur吸收边界条件好的多。但对于分层不均匀媒质，两者的数值反射误差基本相同，
其主要原因是GPML媒质在介质交界面处不能消除固有的反射，而Mur吸收边界条件
则因为采用了行波近似在这些边界处反射要小的多。（2）对色散媒质而言，Mur二阶
吸收边界条件完全不能应用，而根据PML原理得到的广义PML方法如Gadney的单
轴完全匹配媒质（本文中简称DPML）可以取得较好的效果。（3）由于这些方法所产
生的相对反射误差和晚时瞬态散射的幅度差别较小，因此FDTD方法不适合计算晚时
响应。在一般条件下，FDTD方法完全适合于计算埋地目标的早时响应。（4）从所需
的计算资源来讲，Mur二阶吸收边界条件所需的计算时间和占用内存较小，编程要易
于PML方法。（5）数值计算表明时域有限差分方法在研究分层不均匀、有耗媒质中的
计算误差一般在10~(-3)（按入射场幅度归一化）量级，因此完全适合本文研究需要。
4．通过精心设计的一组典型分层不均匀媒质情况，运用FDTD方法数值模拟了瞬态脉冲
波在地卜传播反射、透射并返回地面的整个随时间空间波动的物理过村，从而得到平
面波在水平分层不均匀、有耗士壤中的传捎机理并分析了地面上反射脉冲的形成规律，
得到关丁反射脉冲幅度、波速和其电参数的近似关系。应用FDTD方法讨论了不均匀
介质薄层的反射特性，得到关于估算垂直分辨率的近似公式以及分辨薄层的波形特征。
5．详细计算了细长目标在瞬态线源激励情况下，在多种目标、背景、埋地深度以及测量
方法的瞬态散射波形特征，讨论了其内在的物理意义。推导了波形衰减与相角增量的
计算公式，导出了计算目标深度与背景波速的近似估算方法。
6．研究了近地面平面蝶型偶极大线的模拟计算问题，讨论了近地面脉冲大线性能参数表
示问题。详细比较了FDTD计算中各种激励源的设置方法，结合一种简单的电压型馏
滤设置方法，给出了近地面平面蝶型偶极天线的近场辐射性能分析。同时计算了圆杜、
金属管和含倾斜界面的仿真雷达回波和剖面图，这些结果具有较大的实川价值。
7．提出分层随机不均匀媒质中Monte－Carlo模拟方法，运用高斯谱模型生成粗糙面，运
用随机分布方法生成分层随机分布介质模型，结合FDTD计算其界面的散射特征，）1。
fly用到高速公路的探地雷达检测方法评估中。
8．研究了一些非平稳信号处理方法在地下目标回波信号处理中的应川，提出基丁时频分
布的信号检测与估计统计量以及优化实现方案，并以简化的尺度相关模型为例研究最
优参数估计方法。从实际测量数据出发
Impulse ground penetrating radar is an important and phasic achievement of the
development of transient electromagnetic (EM) theories and a result of a combination of time-
domain measurement technique, nanosecond pulse generation technique, ultra-wide band antenna
technique and signal processing methodology. It has many advantages such as nondestructive
detection, high efficiency, low cost and high resolution imaging ability and has already become
one of the most important tools in subsurface exploration. Also, it is a successful example for the
applications of transient electromagnetic theory in many other areas such as ultra-wide band
radar, ultra-wide band communications and so on. Yet, if one has an experience about the method,
he will see many problems left for further investigations. For example, one can not tell the
location, the constituent material and the size about an unknown target with an acceptable
accuracy and exactness just by use of field test data. This arises from lack of knowledge about
the transient EM interactions between radar antenna, inhomogeneous soil, and buried targets.
Thus, a thorough and detail understanding of the physical rules about transient EM interactions in
inhomogeneous soil is of great importance for solving these tough problems. It is prospective to
extract as much valuable information as possible from the reflected radar returns based on the
knowledge. The main purpose of the study presented in the paper is to reveal the rules of the
transient EM scattering in this situation and to make a preparation for the establishment of
simulation tools for impulse ground penetrating radar systems by using the methods of
computational electromagnetics.
Compared with other numerical tools, the finite-difference finite-time method (the FDTD
method) is the most appropriate choice for the analysis of the transient EM interactions between
the antennas, the lossy, dispersive, inhomogeneous soil, and the target(s). We begin with a careful
check about the suitability, the method of implementations, and the performances of the
algorithms. Some important results are obtained in the research work. The main contributions of
the paper can be summarized as follows:
1. An analytical method of transient incident wave source conditions for total-scattered field
formulation in stratified media is introduced using classical reflection and transmission
relations under a time-harmonic incident plane wave and the fast Fourier transform (FFT)
technique. The codes are confirmed at a variety of incident angles, both in transient field and
in time harmonic field. The results show that the errors at the contiguous boundary due to this
method are less than 3x101 relative to the magnitude of the incident field.
2. A method of the complex dielectric curve fitting from measured data based on minimum least
square method and the first order Debye model is presented, which is validated by a group of
typical soil conditions covering a frequency range of 5OMHz-l250 MHz. The result shows that
the approximations are very good. The resulting parameters are then used in the investigation
for the pulse propagating in dispersive media.
3. The performances of the Mur抯 second order absorption conditions, the GPML layer for lossy
media, and the modified PML layer for dispersive media are investigated with special concern
about the implementations for horizontal stratified media. The error distributions at the corners
and at different media interfaces are explored. From theoretical derivations and numerical
computations, one
引文
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