探地雷达二维有限元正演模拟
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摘要
探地雷达(Ground Penetrating Radar, GPR)是探测地下结构及其分布规律的一种重要浅层地球物理勘探方法,探地雷达正演模拟则是探地雷达研究的一个重要课题,其模拟结果对野外实际工作、室内数据处理与解释具有重要的指导意义。目前探地雷达正演模拟的方法主要有射线追踪法、时域有限差分法和有限单元法,有限单元法的优越性体现在它能模拟较复杂的模型,且有更好的稳定性。故本文选择有限单元法进行探地雷达正演模拟。
论文研究得到了国家自然科学基金项目“基于小波有限元探地雷达正演及偏移处理研究”(40804027)、国家自然科学基金项目“复杂GPR模型无网格法正演及多参数混合智能优化反演(41074085)”、湖南省自然基金重点项目(09JJ3084)等科研项目的支持。其主要内容如下：
1.从Maxwell方程组出发,推导了探地雷达有限元波动方程,给出了伽辽金法求解该微分问题的推导过程,并详细阐述了各主要步骤的细节处理方法,如：质量矩阵、衰减系数矩阵、刚度矩阵的具体求解过程,激励源的加载办法、差分代替微商将探地雷达有限元方程转化成对线性方程组的措施等。
2.引入了大型稀疏矩阵的存储方法、不完全LU分解技术的大型线性方程组求解的Biconjugate Gradient Stabilized (BICGSTAB)算法,并结合差分代替微商,讨论了时间步长和网格步长须满足的数值稳定性条件,讨论了集中质量矩阵法对求解有限元方程的影响及其优缺点。
3.阐述了基于透射机理的透射边界条件和基于衰减机理的Sarma边界条件的原理,详细推导这两种边界条件的探地雷达有限元理论公式；通过在Sarma边界条件衰减层内加入过渡带,压制了介质区和衰减层交界面处的人为反射。以中心电磁脉冲源模型为例,对比了两种边界条件对人工截断边界的处理效果。
4.考虑到透射边界条件与Sarma边界条件不同的理论机制,提出了一种结合透射边界条件和Sarma边界条件的混合边界条件,主要思想是使探地雷达波经过Sarma边界条件的衰减吸收后再通过透射边界条件将剩余能量透射出去,集成了二者的优势。以二维均匀模型的全波场快照为例,证明了该混合边界条件对到达截断边界处的探地雷达波的处理优于单一边界条件。
5.编制了带衰减项与不带衰减项的探地雷达二维有限元正演模拟程序,并自定义了衰减比的概念及计算公式,通过对衰减媒质和非衰减媒质的模拟计算,说明了GPR波在衰减媒质中传播时会产生频散和被媒质吸收。因此,对于常见的衰减媒质中进行正演模拟时,运用带衰减项的雷达波方程进行正演模拟,显然更符合雷达波在地下介质中传播的实际情况。最后,通过对一系列的典型地电模型进行探地雷达正演模拟,加深了对探地雷达波在地下介质中的传播规律的理解和掌握,有助于更好地指导探地雷达资料的地质解释。
Ground penetrating radar (GPR) is an important shallow layer geophysical exploration method which detects the underground structure and the distribution law. The GPR numerical simulation is one of the most significant subjects in the GPR research. The GPR numerical simulation result can help us to work outside and data processing and geologic interpretation of radar data. Currently, the methods of GPR forward simulation are ray tracing method, finite difference time domain method (FDTD) and finite element method (FEM). The finite element method superiority manifests in the simulation of the complex model. So the FEM is used to GPR numerical simulation in this paper.
The paper was funded by a joint support of the National Natural Science Fund "The research of GPR' s forward modeling and migration based on finite element of wavelet" (40804027),"The complex GPR element-free method forward simulation and multi-parameter hybrid intelligent optimization inversion"(41074085) and the Hunan province natural fund important project(09JJ3084). The main content of this paper is as follows:
1. The FEM wave equation method was derived from the Maxwell system of equations. The process of the derivation about the differential coefficient equation which was solved by the Galerkin method was elaborated, such as how to calculate the mass matrix, the attenuation coefficient matrix, the stiffness matrix, the method of disposing the source, in order to transform the GPR finite element equation to system of linear equations uesing difference to instead of the differential coefficient, etc.
2. The method to storage the large-scale sparse matrix was introduced. The Biconjugate Gradient Stabilized (BICGSTAB) algorithm based on the LU incompletely decomposition technology was introduced to solve the large linear systems. The rules of the time step and gridding step which are influence the numerical value stability were given out, discussed the influence of the lumped mass matrix to solve the FEM equation and its advantages and disadvantages.
3. The principle of the transmitting boundary condition which is based on transmission mechanism and the Sarma boundary condition which is based on attenuation mechanism were described in this paper. The formula of the two kinds of boundary condition which are used in GPR FEM forward simulation were derivated in detail. Through joined the transitional layer in the Sarma boundary condition to suppress the articicial reflection which is engendered from the interface between the medium area and the damping region. Taking the center electromagnetic pulse source model as example, compared the results of the two kinds of boundary condition.
4. Considered the different theoretical mechanism of the transmitting boundary condition and the Sarma boundary condition, a mixed boundary condition which combined with the two kinds of boundary condition was given out in this paper. The main idea is the power of the GPR waves were attenuated by the Sarma boundary condition and then transmitted the residual energy by the transmitting boundary condition. And take the two-dimensional equal model entire wave field snapshot as the example, proved the effect of the mixed boundary condition better than just one kind of boundary condition.
5. Compiled the 2D GPR forward simulation program which including the attenuation term or not. Defined the concept and the formula of the attenuation ratio. Through to simulate the attenuation medium and non attenuation medium models, explained the GPR waves transmit in the attenuation medium were absorbed by the medium and the frequency dispersion. Therefore, when the medium is attenuation medium, forward simulation must use the equation which including the attenuation term, obviously conforms to the actual situation of the GPR waves which disseminates in the underground medium. Finally, the results of the typical geoelectric structures can help us to understand the propagation law of GPR waves, so as to guide geologic interpretation of GPR data.
论文研究得到了国家自然科学基金项目“基于小波有限元探地雷达正演及偏移处理研究”(40804027)、国家自然科学基金项目“复杂GPR模型无网格法正演及多参数混合智能优化反演(41074085)”、湖南省自然基金重点项目(09JJ3084)等科研项目的支持。其主要内容如下：
1.从Maxwell方程组出发,推导了探地雷达有限元波动方程,给出了伽辽金法求解该微分问题的推导过程,并详细阐述了各主要步骤的细节处理方法,如：质量矩阵、衰减系数矩阵、刚度矩阵的具体求解过程,激励源的加载办法、差分代替微商将探地雷达有限元方程转化成对线性方程组的措施等。
2.引入了大型稀疏矩阵的存储方法、不完全LU分解技术的大型线性方程组求解的Biconjugate Gradient Stabilized (BICGSTAB)算法,并结合差分代替微商,讨论了时间步长和网格步长须满足的数值稳定性条件,讨论了集中质量矩阵法对求解有限元方程的影响及其优缺点。
3.阐述了基于透射机理的透射边界条件和基于衰减机理的Sarma边界条件的原理,详细推导这两种边界条件的探地雷达有限元理论公式；通过在Sarma边界条件衰减层内加入过渡带,压制了介质区和衰减层交界面处的人为反射。以中心电磁脉冲源模型为例,对比了两种边界条件对人工截断边界的处理效果。
4.考虑到透射边界条件与Sarma边界条件不同的理论机制,提出了一种结合透射边界条件和Sarma边界条件的混合边界条件,主要思想是使探地雷达波经过Sarma边界条件的衰减吸收后再通过透射边界条件将剩余能量透射出去,集成了二者的优势。以二维均匀模型的全波场快照为例,证明了该混合边界条件对到达截断边界处的探地雷达波的处理优于单一边界条件。
5.编制了带衰减项与不带衰减项的探地雷达二维有限元正演模拟程序,并自定义了衰减比的概念及计算公式,通过对衰减媒质和非衰减媒质的模拟计算,说明了GPR波在衰减媒质中传播时会产生频散和被媒质吸收。因此,对于常见的衰减媒质中进行正演模拟时,运用带衰减项的雷达波方程进行正演模拟,显然更符合雷达波在地下介质中传播的实际情况。最后,通过对一系列的典型地电模型进行探地雷达正演模拟,加深了对探地雷达波在地下介质中的传播规律的理解和掌握,有助于更好地指导探地雷达资料的地质解释。
Ground penetrating radar (GPR) is an important shallow layer geophysical exploration method which detects the underground structure and the distribution law. The GPR numerical simulation is one of the most significant subjects in the GPR research. The GPR numerical simulation result can help us to work outside and data processing and geologic interpretation of radar data. Currently, the methods of GPR forward simulation are ray tracing method, finite difference time domain method (FDTD) and finite element method (FEM). The finite element method superiority manifests in the simulation of the complex model. So the FEM is used to GPR numerical simulation in this paper.
The paper was funded by a joint support of the National Natural Science Fund "The research of GPR' s forward modeling and migration based on finite element of wavelet" (40804027),"The complex GPR element-free method forward simulation and multi-parameter hybrid intelligent optimization inversion"(41074085) and the Hunan province natural fund important project(09JJ3084). The main content of this paper is as follows:
1. The FEM wave equation method was derived from the Maxwell system of equations. The process of the derivation about the differential coefficient equation which was solved by the Galerkin method was elaborated, such as how to calculate the mass matrix, the attenuation coefficient matrix, the stiffness matrix, the method of disposing the source, in order to transform the GPR finite element equation to system of linear equations uesing difference to instead of the differential coefficient, etc.
2. The method to storage the large-scale sparse matrix was introduced. The Biconjugate Gradient Stabilized (BICGSTAB) algorithm based on the LU incompletely decomposition technology was introduced to solve the large linear systems. The rules of the time step and gridding step which are influence the numerical value stability were given out, discussed the influence of the lumped mass matrix to solve the FEM equation and its advantages and disadvantages.
3. The principle of the transmitting boundary condition which is based on transmission mechanism and the Sarma boundary condition which is based on attenuation mechanism were described in this paper. The formula of the two kinds of boundary condition which are used in GPR FEM forward simulation were derivated in detail. Through joined the transitional layer in the Sarma boundary condition to suppress the articicial reflection which is engendered from the interface between the medium area and the damping region. Taking the center electromagnetic pulse source model as example, compared the results of the two kinds of boundary condition.
4. Considered the different theoretical mechanism of the transmitting boundary condition and the Sarma boundary condition, a mixed boundary condition which combined with the two kinds of boundary condition was given out in this paper. The main idea is the power of the GPR waves were attenuated by the Sarma boundary condition and then transmitted the residual energy by the transmitting boundary condition. And take the two-dimensional equal model entire wave field snapshot as the example, proved the effect of the mixed boundary condition better than just one kind of boundary condition.
5. Compiled the 2D GPR forward simulation program which including the attenuation term or not. Defined the concept and the formula of the attenuation ratio. Through to simulate the attenuation medium and non attenuation medium models, explained the GPR waves transmit in the attenuation medium were absorbed by the medium and the frequency dispersion. Therefore, when the medium is attenuation medium, forward simulation must use the equation which including the attenuation term, obviously conforms to the actual situation of the GPR waves which disseminates in the underground medium. Finally, the results of the typical geoelectric structures can help us to understand the propagation law of GPR waves, so as to guide geologic interpretation of GPR data.
引文
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