基于积分方程技术的三维电磁法正演模拟研究

发布时间：2018-06-11 13:42

本文选题：三维电磁法 + 积分方程法　；参考：《吉林大学》2015年硕士论文

【摘要】：自上世纪20年代兴起以来，电磁法在矿产资源勘探和地壳构造的研究中一直起着非常重要的作用。长期以来，电磁法的理论研究一直局限在一维情况，但是对于复杂的地质条件，一维理论存在极大的局限性。积分方程在三维数值模拟中存在较大优势，对小型异常体只需要对异常体本身进行剖分，避免了使用微分方程法时在内存上的需求，但是积分方程方法也存在相当的局限性：对于大型异常体，由于积分方程离散所得的是稠密矩阵，对计算机的内存要求急剧增加；矩阵求逆对大型矩阵来说几乎不可行，所以需要采用迭代法来求解，然而对于大型矩阵，迭代时需要进行的矩阵向量乘积所需的计算时间不容忽视；此外，由于格林函数的限制，传统的积分方程只适合于层状介质中相关问题的求解。这些问题都限制了积分方程技术的应用。针对这些问题，本文提出了一些改进方法。通过利用系数矩阵的Toeplitz性质，使传统积分方程方法中遇到的稠密矩阵的存储困境得到了有效地解决。同时，，利用Toeplitz矩阵所具有的特殊性质，用快速傅里叶变换来实现了迭代算法中的矩阵向量乘积，加速了传统积分方程技术中直接求矩阵向量乘积的过程。由于复杂的地质模型都可以转化为多个异常体的问题，本文利用不均匀背景电导率的概念，在考虑异常体间耦合的基础上，给出了对多个异常体进行响应模拟的方法，并将这种方法与不考虑耦合时的响应进行了对比。通过本文的工作，得出以下结论：（1）对积分方程离散的系数矩阵按性质可分解为两项，其中一项满足三重Toeplitz矩阵的特征，另一项满足Hankel-两重Toeplitz矩阵的特征，而Hankel矩阵可以通过简单的置换矩阵转化为Toeplitz矩阵；（2）由于Toeplitz矩阵存在的特殊性质，在矩阵的存储时只需要存储各重矩阵的第一行和第一列，这样，当剖分网格数量很多时，这种存储可以节省大量内存；（3）对Toeplitz矩阵，可以使用快速傅里叶变换实现矩阵与向量的乘积，这种方法可以高效准确地得到解；（4）用考虑耦合效应的方法计算多异常体的响应更适合计算相距较远的多个异常体响应，所得到的响应与不考虑耦合时的响应结果是一致的。
[Abstract]:Electromagnetic method has been playing an important role in the exploration of mineral resources and the study of crustal structure since its rise in 1920s. For a long time, the theoretical study of electromagnetic method has been confined to one-dimensional case, but for complex geological conditions, one-dimensional theory has great limitations. The integral equation has a great advantage in 3D numerical simulation. It only needs to divide the abnormal body itself to avoid the memory requirement when using differential equation method. However, the integral equation method also has some limitations: for large abnormal bodies, because the discrete integral equations are dense matrices, the requirements for computer memory are increased dramatically, and the inverse matrix is almost impossible for large matrices. So it is necessary to use iterative method to solve the problem. However, for large matrices, the computing time required for the product of matrix vectors in iteration cannot be ignored; in addition, due to the restriction of Green's function, The traditional integral equations are only suitable for solving related problems in layered media. These problems limit the application of integral equation technique. In order to solve these problems, some improved methods are proposed in this paper. By using Toeplitz property of coefficient matrix, the storage dilemma of dense matrix in traditional integral equation method is solved effectively. At the same time, by using the special properties of Toeplitz matrix, the matrix vector product in iterative algorithm is realized by using fast Fourier transform, which accelerates the process of directly calculating matrix vector product in traditional integral equation technology. Because complex geological models can be transformed into multiple anomalous bodies, this paper presents a method to simulate the response of multiple anomalous bodies based on the concept of non-uniform background conductivity and considering the coupling between anomalous bodies. By comparing this method with the response without considering coupling, the following conclusion is drawn: 1) the discrete coefficient matrix of integral equation can be decomposed into two terms according to its properties, one of which satisfies the characteristic of triple Toeplitz matrix. The other satisfies the characteristic of Hankel-double Toeplitz matrix, and Hankel matrix can be transformed into Toeplitz matrix by simple permutation matrix.) because of the special property of Toeplitz matrix, only the first row and the first column of each multiple matrix need to be stored in the storage of the matrix. In this way, when the number of meshes is large, this storage can save a lot of memory. The Toeplitz matrix can be used to realize the product of matrix and vector by using fast Fourier transform. This method can be used to calculate the response of multiple anomalous bodies efficiently and accurately. The response of multiple anomalous bodies is more suitable to be calculated by considering the coupling effect. The results obtained are in agreement with those obtained without considering the coupling.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：P631.325

【引证文献】