复杂地下异常体的可控源电磁法积分方程正演

可控源电磁法具有分辨率高及抗干扰能力强等特点，是一种重要的地电磁勘探方法.目前，可控源电磁法的高精度正演计算一直是其核心研究问题之一.传统积分方程法一般采用近似积分公式、简单矩形网格和近似的奇异性体积分计算技术，制约了体积分方程法处理复杂地下异常体的能力，降低了计算精度.针对上述问题，本文基于完全积分公式、四面体非结构化网格和奇异体积分的精确解析解来高精度求解复杂可控源电磁模型的正演响应.首先，从电场积分公式出发，推导了可控源电磁问题满足的积分方程；其次，借助于非结构化四面体网格离散技术，实现了地下复杂异常体的有效模拟.最后，利用散度定理把强奇异值体积分转换为一系列弱奇异性的面积分公式，并通过推导获得了这些弱奇异性的面积分公式的解析解，从而最终实现三维可控源电磁问题的高精度积分求解.以块状低阻体地电模型为测试模型，采用本文提出的积分方程方法获得的数值解与其他公开数值算法解进行对比分析，其对比结果具有高度的吻合性，验证了算法的正确性；同时，设计了球状及复杂地电模型进行算法收敛性测试，进一步验证算法的正确性以及能够处理地下复杂模型的能力.

Three-dimensional forward modeling of the controlled-source electromagnetic problem based on the integral equation method with an unstructured grid

The controlled source electromagnetic method (CSEM) is characterized by high resolution and strong anti-interference ability, which is an important tool in geo-electromagnetic exploration. Inversion is a key step of data processing and interpretation in this method, while the forward modeling is the foundation of inversion. Therefore, searching for a high-accuracy forward algorithm is one of the core research questions to interpret CSEM data. The traditional volume integral equation formula is successfully applied to compute the electromagnetic response of the 3D CSEM as a semi-analytical solution. This method often adopts the approximate integral formula, a regular hexahedron grid and the approximate singular value integral processing technique, which restricts the ability of the volume integral equation method to deal with anomaly bodies with arbitrary complex geometry and reduces its calculation precision. To solve these problems, a new integral strategy is proposed to accurately calculate the 3D controlled-source electromagnetic forward response based on the complete integration formula using a tetrahedral unstructured grid and singularity-free analytical solution for the singular volume integral. Firstly, the integral equation of the CSEM problem is deduced from the formula of the electric field integral. Then, the underground complex abnormal body is discretized by the latest unstructured discrete technique based on a tetrahedral grid. Using the divergence theorem, we transform the strong singular value volume integral into a series of weak singularity integral formulas. And we obtain the analytic solutions of these weak singularity integral formulas by vector-scalar identity, and finally the new singular integral techniques are successfully applied to compute the electromagnetic response of 3D CSEM with high precision. At last, for a conductive block buried in a less conductive half-space with a 100 m grounded wire, the total and secondary electric fields calculated by our algorithm are compared with those calculated by the integral equation method based on the secondary electric field, the finite element method based on magnetic vector potential and the DC resistivity forward modeling code (DCIP3D), respectively. The results show that four numerical solutions coincide well each other, and the algorithm suggested by this work is correct. Meanwhile, we conduct tests of this method on a sphere model and a complex geoelectric model, demonstrating that it is effective and capable of dealing with complicated subsurface anomaly bodies.