瞬变电磁场中的数学问题及应用
发布时间：2019-04-26 15:50
【摘要】：近年来，瞬变电磁测深法（Transient Electromagnetic Method）在实际工程地质勘探中得到广泛的应用。它是一种时间域电磁勘探方法，利用阶跃波或其它脉冲电流场源激励，在大地产生过渡过程场，断电瞬间在大地中形成涡旋交变电磁场，测量这种由地下介质产生的二次感应电磁场随时间变化的衰减特性，从测量得到的异常信号中分析出地下介质体的导电性能和位置，从而达到解决地质问题的目的。并且随着将地震学中一些已成熟的方法和技术应用到瞬变电磁测深方法中，人们越来越关注从波场的角度，把瞬变电磁场转换为人们所熟悉的波动场以获取更多、更准确的地下地质体信息。然而以往都是将磁性源或电性源看做偶极子，通过比拟得到谐变场的频域表达式，再经频域到时域的变换得到时域解析公式。 本文以时变点电荷假设代替偶极子假设，直接在时间域求解瞬变电磁场解析式。首先总结回顾了全空间有源波动场的时域Green函数解，并运用分离变量方法推导出全空间齐次阻尼波动方程的直接时域解析式。 其次，在全空间直接时域解析式的基础上，采用分离变量法给出均匀半空间齐次阻尼波动方程的时域解析式，通过有源Maxwell方程对应的电磁场矢量方程，推导出上半空间一次有源波动场和反射波（无源二次波动场）时域解析式，进而给出了均匀半空间瞬变电磁场直接时域解析式。 然后，在均匀半空间直接时域解析式的基础上，利用边界条件，推导出线性、均匀水平层状介质中的瞬变电磁场直接时域响应式。 这种在时间域内严格推导瞬变电磁场响应式的方法，避免了传统引入位函数以及偶极子假设所带来的误差，并为从事地质勘探的工作者和瞬变电磁探测理论的改进提供新的理论依据。 最后，，采用连续正则化方法求解波场变换反演问题。构造展平泛函，基于已经正则化的变分问题用差分法作有限维逼近。利用偏差原理和Newton三阶迭代收敛格式选出最优的正则化参数，实施数值求解。通过对数值计算结果和已知的波场函数之间的比较，证明了该方法的有效性和可行性。与离散正则化算法相比，这种连续正则化算法具有保结构和收敛速度快等优点。
[Abstract]:In recent years, transient electromagnetic sounding (Transient Electromagnetic Method) has been widely used in practical engineering geological exploration. It is a kind of time domain electromagnetic exploration method, which uses step wave or other pulse current field source to generate transition process field in the earth, and produces vortex alternating electromagnetic field in the earth at the moment of loss of electricity. The attenuation characteristics of the secondary induction electromagnetic field produced by the underground medium with time are measured, and the electrical conductivity and position of the underground dielectric body are analyzed from the abnormal signals obtained from the measurement, so as to achieve the purpose of solving the geological problems. And with the application of some mature methods and techniques in seismology to transient electromagnetic sounding methods, people pay more and more attention to converting transient electromagnetic fields into familiar wave fields from the angle of wave field in order to obtain more. More accurate underground geological information. However, in the past, magnetic or electrical sources were regarded as dipoles. The frequency domain expressions of the harmonic field were obtained by analogy, and then the time domain analytical formulas were obtained by the transformation from frequency domain to time domain. In this paper, the time-varying point charge hypothesis is used to replace the dipole hypothesis, and the analytical expression of transient electromagnetic field is solved directly in the time domain. Firstly, the time-domain Green function solutions of the full-space active wave field are reviewed, and the direct time-domain analytical formula of the full-space homogeneous damping wave equation is derived by using the method of separating variables. Secondly, on the basis of the full-space direct time-domain analytic formula, the time-domain analytic expression of homogeneous damping wave equation in half-space is given by using the separated variable method, and the electromagnetic field vector equation corresponding to the active Maxwell equation is obtained. The time domain analytical expressions of primary active wave field and reflected wave (passive quadratic wave field) in upper half space are derived, and then the direct time domain analytical expressions of transient electromagnetic field in uniform half space are given. Then, on the basis of the direct time-domain analytical formula in uniform half-space, the direct time-domain response of transient electromagnetic field in linear and uniform horizontal layered media is derived by using the boundary conditions. This method of strictly deducing the transient electromagnetic field response in the time domain avoids the errors caused by the traditional introduction of the potential function and the dipole hypothesis. It also provides a new theoretical basis for the workers engaged in geological exploration and the improvement of the theory of transient electromagnetic detection. Finally, the continuous regularization method is used to solve the inversion problem of wave field transformation. The flattened functional is constructed and the finite-dimensional approximation is made by the finite difference method based on the regularized variational problem. The optimal regularization parameters are selected by using the deviation principle and the Newton third-order iterative convergence scheme, and the numerical solution is carried out. The effectiveness and feasibility of the proposed method are proved by comparing the numerical results with the known wave field functions. Compared with the discrete regularization algorithm, the continuous regularization algorithm has the advantages of structure-preserving and fast convergence.
【学位授予单位】：辽宁工业大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：O441.4
本文编号：2466202
[Abstract]:In recent years, transient electromagnetic sounding (Transient Electromagnetic Method) has been widely used in practical engineering geological exploration. It is a kind of time domain electromagnetic exploration method, which uses step wave or other pulse current field source to generate transition process field in the earth, and produces vortex alternating electromagnetic field in the earth at the moment of loss of electricity. The attenuation characteristics of the secondary induction electromagnetic field produced by the underground medium with time are measured, and the electrical conductivity and position of the underground dielectric body are analyzed from the abnormal signals obtained from the measurement, so as to achieve the purpose of solving the geological problems. And with the application of some mature methods and techniques in seismology to transient electromagnetic sounding methods, people pay more and more attention to converting transient electromagnetic fields into familiar wave fields from the angle of wave field in order to obtain more. More accurate underground geological information. However, in the past, magnetic or electrical sources were regarded as dipoles. The frequency domain expressions of the harmonic field were obtained by analogy, and then the time domain analytical formulas were obtained by the transformation from frequency domain to time domain. In this paper, the time-varying point charge hypothesis is used to replace the dipole hypothesis, and the analytical expression of transient electromagnetic field is solved directly in the time domain. Firstly, the time-domain Green function solutions of the full-space active wave field are reviewed, and the direct time-domain analytical formula of the full-space homogeneous damping wave equation is derived by using the method of separating variables. Secondly, on the basis of the full-space direct time-domain analytic formula, the time-domain analytic expression of homogeneous damping wave equation in half-space is given by using the separated variable method, and the electromagnetic field vector equation corresponding to the active Maxwell equation is obtained. The time domain analytical expressions of primary active wave field and reflected wave (passive quadratic wave field) in upper half space are derived, and then the direct time domain analytical expressions of transient electromagnetic field in uniform half space are given. Then, on the basis of the direct time-domain analytical formula in uniform half-space, the direct time-domain response of transient electromagnetic field in linear and uniform horizontal layered media is derived by using the boundary conditions. This method of strictly deducing the transient electromagnetic field response in the time domain avoids the errors caused by the traditional introduction of the potential function and the dipole hypothesis. It also provides a new theoretical basis for the workers engaged in geological exploration and the improvement of the theory of transient electromagnetic detection. Finally, the continuous regularization method is used to solve the inversion problem of wave field transformation. The flattened functional is constructed and the finite-dimensional approximation is made by the finite difference method based on the regularized variational problem. The optimal regularization parameters are selected by using the deviation principle and the Newton third-order iterative convergence scheme, and the numerical solution is carried out. The effectiveness and feasibility of the proposed method are proved by comparing the numerical results with the known wave field functions. Compared with the discrete regularization algorithm, the continuous regularization algorithm has the advantages of structure-preserving and fast convergence.
【学位授予单位】：辽宁工业大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：O441.4
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