基于矢量有限元的高频大地电磁法三维数值模拟
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摘要
由于中深度工程地球物理勘探的迫切需要,以美国EH-4电导率成像系统为代表的高频率大地电磁法在我国地球物理勘探行业应用越来越广泛。高频大地电磁法属于采集天然场信号的被动源电磁方法,采集的信号频率范围为10Hz～100KHz,研究深度从地下的十几米至上千米。论文研究的基于矢量有限元的高频率的被动源电磁测深法,紧密结合当前我国中深度地球物理勘探中电磁法大量应用的实际情况,不仅具有学术意义,对指导工程实践也有一定的价值。论文研究的重点是以标量有限元和矢量有限元为核心的、以高频率电磁波为对象的正演数值模拟方法。主要包括四部分:建立与高频大地电磁法边值问题相等价的变分问题,为有限元数值计算奠定数理基础;标量/矢量有限元方法的研究;加速标量/矢量有限元矩阵系统方程迭代求解技术的研究;实现高频大地电磁参数精确快速正演模拟的研究。希望给后续的正反演研究工作提供科学的依据和参考,在减少地球物理多解性方面做一些有益的探索和实践。
本文的主要内容和创新成果如下:
1.在前人工作的基础上,从麦克斯韦电磁场双旋度方程出发,分别利用广义变分原理和加权余量法推导了高频大地电磁场在有耗介质中边值问题的稳定泛函,建立了变分方程,为有限元的计算的稳定性和准确性提供了数理保证。
2.传统的标量有限元在解决矢量电磁场边值问题时,需要将未知量转化为标量场问题,然后进行求解。这种基于标量基函数处理矢量电磁场问题时,会造成非物理解或伪解问题的出现、异常体表面强加边界条件的不方便以及处理介质或导体边缘及角的困难性。本文利用了一种新型的矢量插值基函数来近似未知函数,将自由度赋给单元的棱边而不是节点,避免了传统标量有限元的困难,所做的工作证明,矢量有限元在高频大地电磁数值模拟中应用的非常成功,得到了较好的结果。
3.论文研究了一种新型的大型病态线性方程组的求解方法——改进的威尔金森方法。应用这种方法进行求解不仅加速了有限元线性方程组的迭代收敛速度,而且大大降低了矩阵方程的求解时间,特别是在求解大地电磁场矩阵方程的情况下,此时矩阵方程的性态极差,一般的数值求解方法很难收敛到比较满意的截断门限,但该方法依然能够取得较好的效果。
4.以典型的模型,计算并详细分析了三维情况下,标量有限元和矢量有限元高频大地电磁的响应特征、精度和速度,分别探讨了山谷地形、山脊地形和复杂地形条件下电磁异常的特点和变化规律。
在本文的结论部分还指出了一些不足和今后工作的建议。
With the development of geophysical prospecting in 1000 meters below surface, the High-Frequency Magnetotelluric (HMT) method is applied more and more extensively in our country, represented by EH-4 conductivity imaging system which is made in USA. HMT method belongs to the passive source electromagnetic method which collects natural electromagnetic signals. And the required frequency range is from 10Hz to 100 KHz, correspondingly, the prospecting depth varies from a few meters to one kilometer. The paper not only has academic value but also has some effect in guiding the practice of the engineering. Aiming at the target of HMT filed, this paper successfully implements the HMT forward numerical modeling with scalar/node and vector/edge finite element method respectively. This paper focuses on four important aspects as follows. The first one is the establishment of variational quations which provides the mathematics theories for the finite element method of HMT; the second one is for the study of scalar/node and vector/edge finite element method; the third one is about how to solve faster for the large-sacled matrix system of scalar/node and vector/edge finite element; and the last one is achieving the forward modeling with high precision of HMT method. The author hopes that this work can provide a reference for further research and could do well for reducing the multiplicities of geophysical interpretation for the exploration and practice.
The following is the main contents and results of the paper:
1. Based on the work of predecessors, this paper derives the variational equations of HMT field by generalized variational principle and the weighted residual method from the Maxwell's equations which guarantees the mathematical accuracy and the stability of the finite element method.
2. When solving the HMT field boundary value problems by traditional scalar/node or vector/edge finite element, we should transform unknowns into the scalar field problems, and then solve it. But such scalar basis function is far away from efficiency when dealing with vector electromagnetic field problems. This is because it can result in fake solution or some solution which can not be interpreted by using physical conceptions. And also it is difficult to impose the essential boundary conditions on the interface of abnormal body. Lastly, it is inconvenient to deal with the angle and margin of the conductive bodies. In this paper, a new vector basis function is used to describe the unknowns, and then the degree of freedoms will be assigned to the edges rather than the nodes of the element by which the problems of the traditional scalar/node finite element are solved. The work has proved that the vector finite element method is remarkably efficient in the HMT forward modeling.
3. In this paper, we study an efficient method for solving the large sparse linear equations—the ameliorating Wilkinson method. It has been proved not only for accelerating the convergence rate of the finite element linear equations, but also for remarkably reducing the time to solve the matrix equation, especially in electromagnetic field with very ill-conditional matrix. The conventional numerical methods generally can not convergence to the cut-off error, but the ameliorating Wilkinson method is able to achieve good results.
4. This paper simulates the 3D response characteristics of HMT with scalar/node and vector/edge finite element method, analyzing the numerical precision and speed. By analyzing the characteristics of the valley and ridge terrains and other complex conditions, we sum up the law of the HMT field.
In the last part of this paper, the author give the main conclusions and proposals which point out the deficiency in the paper and some things which are needed to be perfected in the future.
本文的主要内容和创新成果如下:
1.在前人工作的基础上,从麦克斯韦电磁场双旋度方程出发,分别利用广义变分原理和加权余量法推导了高频大地电磁场在有耗介质中边值问题的稳定泛函,建立了变分方程,为有限元的计算的稳定性和准确性提供了数理保证。
2.传统的标量有限元在解决矢量电磁场边值问题时,需要将未知量转化为标量场问题,然后进行求解。这种基于标量基函数处理矢量电磁场问题时,会造成非物理解或伪解问题的出现、异常体表面强加边界条件的不方便以及处理介质或导体边缘及角的困难性。本文利用了一种新型的矢量插值基函数来近似未知函数,将自由度赋给单元的棱边而不是节点,避免了传统标量有限元的困难,所做的工作证明,矢量有限元在高频大地电磁数值模拟中应用的非常成功,得到了较好的结果。
3.论文研究了一种新型的大型病态线性方程组的求解方法——改进的威尔金森方法。应用这种方法进行求解不仅加速了有限元线性方程组的迭代收敛速度,而且大大降低了矩阵方程的求解时间,特别是在求解大地电磁场矩阵方程的情况下,此时矩阵方程的性态极差,一般的数值求解方法很难收敛到比较满意的截断门限,但该方法依然能够取得较好的效果。
4.以典型的模型,计算并详细分析了三维情况下,标量有限元和矢量有限元高频大地电磁的响应特征、精度和速度,分别探讨了山谷地形、山脊地形和复杂地形条件下电磁异常的特点和变化规律。
在本文的结论部分还指出了一些不足和今后工作的建议。
With the development of geophysical prospecting in 1000 meters below surface, the High-Frequency Magnetotelluric (HMT) method is applied more and more extensively in our country, represented by EH-4 conductivity imaging system which is made in USA. HMT method belongs to the passive source electromagnetic method which collects natural electromagnetic signals. And the required frequency range is from 10Hz to 100 KHz, correspondingly, the prospecting depth varies from a few meters to one kilometer. The paper not only has academic value but also has some effect in guiding the practice of the engineering. Aiming at the target of HMT filed, this paper successfully implements the HMT forward numerical modeling with scalar/node and vector/edge finite element method respectively. This paper focuses on four important aspects as follows. The first one is the establishment of variational quations which provides the mathematics theories for the finite element method of HMT; the second one is for the study of scalar/node and vector/edge finite element method; the third one is about how to solve faster for the large-sacled matrix system of scalar/node and vector/edge finite element; and the last one is achieving the forward modeling with high precision of HMT method. The author hopes that this work can provide a reference for further research and could do well for reducing the multiplicities of geophysical interpretation for the exploration and practice.
The following is the main contents and results of the paper:
1. Based on the work of predecessors, this paper derives the variational equations of HMT field by generalized variational principle and the weighted residual method from the Maxwell's equations which guarantees the mathematical accuracy and the stability of the finite element method.
2. When solving the HMT field boundary value problems by traditional scalar/node or vector/edge finite element, we should transform unknowns into the scalar field problems, and then solve it. But such scalar basis function is far away from efficiency when dealing with vector electromagnetic field problems. This is because it can result in fake solution or some solution which can not be interpreted by using physical conceptions. And also it is difficult to impose the essential boundary conditions on the interface of abnormal body. Lastly, it is inconvenient to deal with the angle and margin of the conductive bodies. In this paper, a new vector basis function is used to describe the unknowns, and then the degree of freedoms will be assigned to the edges rather than the nodes of the element by which the problems of the traditional scalar/node finite element are solved. The work has proved that the vector finite element method is remarkably efficient in the HMT forward modeling.
3. In this paper, we study an efficient method for solving the large sparse linear equations—the ameliorating Wilkinson method. It has been proved not only for accelerating the convergence rate of the finite element linear equations, but also for remarkably reducing the time to solve the matrix equation, especially in electromagnetic field with very ill-conditional matrix. The conventional numerical methods generally can not convergence to the cut-off error, but the ameliorating Wilkinson method is able to achieve good results.
4. This paper simulates the 3D response characteristics of HMT with scalar/node and vector/edge finite element method, analyzing the numerical precision and speed. By analyzing the characteristics of the valley and ridge terrains and other complex conditions, we sum up the law of the HMT field.
In the last part of this paper, the author give the main conclusions and proposals which point out the deficiency in the paper and some things which are needed to be perfected in the future.
引文
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