边界单元法全空间无解析奇点重磁场正演

Abstract

The forward problem is to calculate the distribution attribute (abnormal feature and size) of the field in the observation point by the known properties of the field source (physical property and geometric size), and the inverse problem is to calculate the property of the source by the distribution attribute of the known field in the observation point. Forward is the basis of inversion, so it is essential for forward in the gravity and magnetic survey.

There are two kinds of complex object forward methods in the gravity and magnetic field, which mainly are the finite element method and the boundary element method. The finite element method has its obvious advantages for the inhomogeneous physical property in the gravity and magnetic field, and the boundary element method has its obvious advantages to the homogeneous forward in the gravity and magnetic field. Because of the less external surface of the field source, compared with the finite element method, it has the characteritics of less calculation, high velocity. In the traditional analytic formula of forward, the "analytic singularity" problem exists in both the finite element method and the boundary element method. In recent years, many scholars have studied more about the forward problem without "analytic singularity" of the finite element method in the gravity and magnetic field, and have obtained the forward formula without "analytic singularity" in the point element method and the surface element method in the gravity and magnetic field. However, there is less research on the forward calculation formula without "analytic singularity" of the boundary element method in the whole space. This paper will study this problem.

The basic idea of the forward of the boundary element method in the gravity and magnetic field is to convert the volume integral of complex object into surface integral by Gauss formula, and then transform the surface of the field into parallel with the coordinate plane through coordinate rotation, and use the trapezoid model to the solve surface integral directly. Finally, the whole gravity and magnetic field can be obtained by cumulative sum. By analyzing the forward formula of the traditional boundary element method in the gravity and magnetic field, this paper finds out the cause of "analytic singularity" in the boundary element method. Through the study of the value domain of the natural logarithmic function and the arc

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tangent function, give the forward formula without "analytic singularity" in the point element method and the surface element method in the gravity and magnetic field,and give the formula for calculating the first derivative of gravity. Through theoretical model test, we get the correct calculation results, and achieve the goal of this paper.

Key word: gravity, magnetic anomalies; the whole space; analytical singularity, polyhedron, boundary element method.

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目录

第一章绪论 (1)

1.1 研究背景和意义 (1)

1.2 研究现状 (1)

1.3 技术路线 (2)

1.4 主要成果 (3)

第二章边界单元法全空间无“解析奇点”重、磁场正演 (5)

2.1 重、磁异常正演基本公式 (5)

2.1.1 重力异常正演公式 (5)

2.1.2 磁异常正演公式 (7)

2.2 传统边界单元法重力异常正演公式 (9)

2.2.1 传统边界单元法重力异常正演公式及“解析奇点”分析 (9)

2.2.2 传统边界单元法重力异常正演公式推导 (13)

2.3 边界单元法全空间无“解析奇点”磁异常正演公式 (15)

2.3.1 传统边界单元法磁异常正演公式及“解析奇点”分析 (15)

2.3.2 边界单元法全空间无“解析奇点”磁异常正演公式 (18)

2.4 边界单元法重力一阶方向导数公式 (29)

第三章模型测试 (34)

3.1 边界单元法重、磁模型测试 (34)

3.1.1 直立六面体模型 (34)

3.1.2 倾斜模型 (43)

3.1.3 复杂多面体模型 (46)

结论和建议 (52)

参考文献 (54)

攻读学位期间取得的研究成果 (56)

致谢 (57)

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