基于模型降阶方法的瞬变电磁三维快速反演算法研究

Currently, the main bottlenecks restricting the transient electromagnetic (TEM) 3D inversion to practical application is that the size of 3D inversion system is large and the time-consuming forward must be solved repeatedly. Model reduction can reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. It has been widely applied in numerical simulation and parameter optimization of an enormous variety of large-size systems. Therefore, this research establishes a fast 3D inversion algorithm for TEM data by the model reduction method. The inversion uses the Gauss-Newton algorithm, and introduces the model reduction method to deal with the most time-consuming part of the inversion: Optimization repeat poles are choosing based on the error analysis theory of rational function approximation. The orthogonal basis and projection matrix of rational Krylov subspace is built by using matrix decomposition technique. Then 3D TEM response is obtained. Based on the rational Krylov subspace approximation of the 3D TEM response, with the eigenvalue and eigenvector of projection matrix, and introduces the perturbation theory of quantum mechanics, Jacobian matrix will eventually translate into the form of matrix-vector multiplication, which can avoids the required forward calculation for solving the Jacobi matrix. In the end, a fast algorithm for 3D TEM inversion based on model reduction method is established. The research from this project will promote the development of 3D TEM accurate interpretation technology.

瞬变电磁三维反演的规模庞大，反演迭代中需要求解大量耗时的正演计算，是目前制约瞬变电磁三维反演走向实用化的主要瓶颈。模型降阶方法，将大型系统转化为一个近似的较小系统，从而降低大型系统的理论分析难度和减少数据运算量，目前已被广泛应用于各类大型问题的数值模拟与参数最优化计算中。为此，本项目将模型降阶方法应用于瞬变电磁三维反演。采用高斯牛顿算法，反演中最耗时的部分采用模型降阶方法计算：根据有理函数近似误差分析理论，实现最优化重复极点选择，利用矩阵直接分解技术，构建有理函数Krylov子空间的正交基和投影矩阵，得到瞬变电磁三维正演响应；在此基础上利用投影矩阵的特征值和特征向量，并引入量子力学中的微扰理论，将雅可比矩阵最终转化为矩阵矢量相乘形式，避免了求解雅可比矩阵所需的大量正演计算。最终建立一套基于模型降阶的瞬变电磁三维快速反演算法。研究成果对于推动瞬变电磁三维勘探精细解释技术的发展具有重要意义。

瞬变电磁三维反演的规模庞大，反演迭代中需要求解大量耗时的正演计算，是目前制约瞬变电磁三维反演走向实用化的主要瓶颈。本项目研究采用模型降阶方法实现瞬变电磁三维快速反演。采用高斯牛顿算法，反演中最耗时的正演和雅可比矩阵采用模型降阶方法快速计算。对于瞬变电磁三维正演计算，本项目采用拟态有限体积算法实现了三维复杂各向异性介质的空间离散；引入单个重复极点的位移逆Krylov子空间投影算法结合矩阵直接分解技术实现了瞬变电磁三维正演的模型降阶快速算法，相比之前的方法速度提高了4-5倍；为了扩展算法的实用性，采用电偶极子分段近似实现了任意形状发射源的三维正演；引入指数求积法则首次将三维模型降阶算法扩展到全波形瞬变电磁三维正演；引入频谱分析理论实现了全波形模型降阶正演的快速算法；引入迭代算法实现了大尺度模型的三维正演。进一步将模型降阶正演算法扩展到多频可控源电磁法和大地电磁法的三维正演快速计算。对于瞬变电磁三维反演计算，引入积分方程理论和互易定理将雅可比矩阵的计算简化为两个正演响应的卷积形式，结合模型降阶正演结果，将雅可比矩阵的计算量减少为多次矩阵回代，避免了直接求解雅可比矩阵所需的大量正演计算，大大减小了雅可比矩阵的计算时间。在此基础上实现了瞬变电磁三维高斯牛顿反演的模型降阶快速算法和瞬变电磁三维拟牛顿反演算法。本项目研究成果对于推动瞬变电磁三维勘探精细解释技术的发展和模型降阶思想在地球物理电磁法正反演领域的应用具有重要意义。