空间-波数域三维大地电磁场积分方程法数值模拟

地球物理勘探中, 地下复杂地质体电磁场三维数值模拟计算量大、存储要求高.针对这一问题, 本文提出一种空间-波数域三维电磁场数值模拟方法, 该方法利用电磁场积分方程为卷积的特点, 将积分沿水平方向进行二维傅里叶变换, 将电磁场三维空间域卷积问题转换为不同波数相对独立的多个一维积分问题, 由此计算量和存储需求大大减少, 易于实现并行计算.采用有限单元法中的形函数进行一维数值积分, 一维积分离散为多个单元积分之和, 每个单元采用二次形函数表征电流变化, 可得出单元积分的解析表达式; 保留垂向为空间域, 优势之一在于可根据实际情况合理调整单元疏密程度, 准确模拟任意复杂地形和导电率异常, 兼顾计算精度与计算效率, 优势之二是用形函数拟合求得积分的解析解, 计算精度和效率高; 最后引入压缩算子, 采用迭代求得电磁场的数值解.本文方法充分利用不同波数之间一维积分高度并行性、一维形函数积分的高精度及快速傅里叶变换的高效性, 实现电磁场三维高效高精度数值模拟.设计模型将本文方法数值模拟结果和软件INTEM3D的数值模拟结果对比, 验证了方法的正确性; 设计高阻和低阻异常体, 研究了异常电导率与背景电导率差异对迭代收敛速度的影响; 改变计算规模, 随着计算网格增多, 算法耗时与存储呈近似线性增长; 设计复杂模型与目前主流数值模拟方法对比, 本文算法速度快1个数量级以上, 且计算规模越大, 算法效率的优势越明显.研究结果表明, 本文提出的空间-波数域三维电磁场数值模拟方法理论和方法正确, 计算效率高, 对计算机存储要求低, 算法高度并行, 适合任意复杂条件大规模三维电磁场高效、高精度数值模拟.

Three-dimensional numerical simulation of magnetotelluric field in the space-wavenumber domain

Three-dimensional numerical simulation of electromagnetic field under complex geology requires mass calculation and memory space in geophysical exploration. Aiming at this situation, we have developed a 3D electromagnetic field forward method based on integral method in space wavenumber domain. With the help of the horizontal 2D Fourier transform, 3D convolutions of electromagnetic integral equation can be transformed into vertical 1D integrations of different wavenumbers. In this way, the computation and memory requirements can be greatly reduced, 1D integrations of different wavenumbers are a high degree of parallelism. We use quadratic interpolating shape function to calculate 1D integration. The 1D integration is discretized into some integrations of element, fit scattering current of each element with quadratic shape function and we can derive the analytic expression of the integration. The method preserves the vertical component in the space domain, one of the advantages is that a mesh in vertical direction for modeling can be adjusted flexibly according to actual conditions, like complex terrain or anomalies, which keep balance between computational accuracy and efficiency; another advantage is that the use of analytic solution of the integrations in vertical direction improves the calculation accuracy and efficiency. The contraction operator is applied to the iteration method to solve the electromagnetic field in spatial domain. This study fully utilizes high parallelism of 1D integration between different wavenumbers, high accuracy of 1D integration and high efficiency of fast Fourier transform for 3D electromagnetic filed modeling. We verify the correctness of the proposed method by comparing the numerical results with those of INTEM3D software. Tests show that the convergence of the proposed method is mainly affected by the conductivity contrast between anomalies and background. With the increase of calculating scale, time and memory consumption of the proposed method increase approximately linearly. Compared with the other numerical methods, the efficiency of the proposed method is several orders of magnitude higher, and the larger the scale is, the higher advantage the method is. The results indicate that the theory of the proposed method is correct, the method is low-memory, efficient and highly parallel, and it is suitable for the high-efficiency calculation of large-scale complicated 3D MT models.