基于耦合均一化多重网格法的大地电磁快速三维正演研究

Fast 3D Magnetotelluric (MT) inversion is essential for MT exploration, largely determined by the efficiency of its forward problem. This proposal will focus on the development of efficient MT forward algorithms based on a uniformed coupling multigrid method. It is well known that variable physical parameters and grid stretching cause discrete anisotropy, which poses serious convergence problem for traditional multigrid method. One remedy is to use block smoothers, within which all unknowns in each block are updated collectively, thus high frequency error smoothed effectively. Another way is to coarsen the grid in the direction or block with strong coupling, called semi-coarsening, that is, grid only coarsened in which the errors are smooth. For MT forward problems, variable conductivity models and the use of stretching grid are very common. To develop a efficient multigrid method, the electromagnetic field discrete anisotropy is the key issue, which will be completely treated in our proposal. Based the degree of coupling of the discrete curl-curl operator, the proposal will develop a semi-coarsening strategy combined with a efficient block smoother to handle the ubiquitous discrete anisotropy. Multigrid methods based potential equations from Helmholtz decomposition of Electrical field will also be explored to some extent. Discrete form of the potential equations possess good numerical properties, e.g., decoupled components and diagonal dominance and is probably an idea choice for the development of a fast multigrid preconditioner. The proposed multigrid method and multigrid preconditioner could speed up the 3D Magnetotelluric forward modeling significantly and thus improve the 3D inversion efficiency.

针对在大地电磁三维（3D）正演中由于计算未知量多，无法采用直接解法进行求解，同时迭代求解方法收敛速度过慢等问题。本项目拟开展基于耦合均一化多重网格法的研究，以提高大地电磁3D正演计算速度，克服传统多重网格法中由于网格剖分大小不均一或模型电性变化而导致收敛变慢问题。拟采用分块平滑算法，对高频误差进行有效消除，根据模型耦合强弱将模型分区，确定具体采用线、面或分块平滑算法；采用半粗网格剖分对网格内耦合较强的节点进行粗化，保证粗化在耦合最强的区域进行，使耦合趋于均一，只对平滑误差进行剖分；在以上技术基础上通过改变方程本身的耦合特性，即将场方程转化为位势方程，由于位势场方程具有分量独立，对角占优，多重网格法具有更好收敛性。高效的多重网格法作为预处理或直接解法将显著提高大地电磁3D正演效率，为大地电磁3D反演的实用化提供条件，对促进大地电磁3D勘探具有实际意义，对快速地球物理正演研究具有理论指导意义。

大地电磁测深法是一种使用天然源电磁场信号的地球物理方法。它的广泛应用离不开高效、精确的大地电磁数据解释，特别是高效、精确的大地电磁正演算法。但是由于电磁双旋度扩散方程具有丰富的零空间，同时当频率比较低时方程无法模拟电荷的积累，传统迭代方法在求解大地电磁正演问题时往往收敛比较慢。针对在大地电磁三维正演中由于计算未知量多，无法采用直接解法进行求解，同时迭代求解方法收敛速度过慢等问题。本项目开展了基于耦合均一化多重网格法的研究，克服了传统多重网格法中由于网格剖分大小不均一或模型电性变化而导致收敛变慢问题。在平滑算法方面，将具有相近耦合性的节点分区，然后在迭代平滑时采用高斯-赛德尔法，将这些相近耦合性的节点同时更新，实现了四色分块平滑等平滑算法，研究结果表明四色分块单元高斯-赛德尔的效率很高。通过地电模型和剖分要求等初步确定节点耦合较强的方向或区域，然后确定半粗化的方向或区域，进行粗化，其它方向或区域保持不变，下一步的粗化以粗化后的网格重复以上过程。粗化方式同时通过局部傅里叶谱分析，以获得近似的收敛值。研究结果表明，结合分块平滑算法和半粗化网格剖分，可以显著减小各自算法的复杂性，而且能够处理更复杂的地电模型。本项目完成了基于Helmholz分解后的标量位和矢量位方程正演研究，利用分解后位势分量相互独立，离散后方程组具有对角占优等特点，开发了基于位势的多重网格法预处理器，该预处理显著提高大地电磁3D正演效率，为大地电磁3D反演的实用化提供条件，对促进大地电磁3D勘探具有实际意义，对快速地球物理正演研究具有理论指导意义。