基于高斯牛顿法的频率域可控源电磁三维反演研究

2016-11-16彭荣华胡祥云韩波

地球物理学报 2016年9期

关键词：模型

彭荣华，胡祥云，韩波

1 中国地质大学(武汉)地球物理与空间信息学院，武汉 430074 2 不列颠哥伦比亚大学地球、海洋与大气科学学院，温哥华V6T 1Z4，加拿大 3 中国海洋大学海洋地球科学学院，青岛 266100

基于高斯牛顿法的频率域可控源电磁三维反演研究

彭荣华1,2，胡祥云1*，韩波3

三维反演解释是电磁法勘探发展的重要趋势，而如何提高三维反演的可靠性、稳定性和计算效率是算法开发者们目前的研究重点.本文实现了一种频率域可控源电磁(CSEM)三维反演算法.其中正演基于拟态有限体积法离散化，利用直接矩阵分解技术来求解大型线性系统方程，不仅准确、稳定，而且特别有利于含有大量发射场源位置的CSEM勘探情况；对目标函数的最优化采用高斯牛顿法(GN)，具有近似二次的收敛性；使用预条件共轭梯度法(PCG)求解每次GN迭代所得到的法方程，避免了显式求解和存储灵敏度矩阵，减小了计算量.以上这些方法的结合应用，使得本文的三维反演算法准确、稳定且高效.通过陆地和海洋CSEM勘探场景中的典型理论模型的反演测试，验证了本文算法的有效性.

频率域可控源电磁法；三维反演；高斯牛顿法；直接矩阵分解法；预条件共轭梯度法

Quantitative interpretation of large-scale CSEM data in the frequency domain requires efficient and stable 3D forward modeling and inversion codes. Considerable efforts have been contributed to developing numerical algorithms concerning 3D inversion of CSEM data that can accurately and efficiently recover subsurface electrical structure over last decade. Most existing 3D inversion algorithms employ Krylov subspace iterative methods as their forward solvers. Iterative techniques usually need little memory due to only matrix-vector products storage required, and they are fast for computation of single field solution. However, there are two main issues when using iterative solvers for 3D CSEM problems: (1) The ill-conditioning of linear systems arising from discretization of Helmholtz equation can lead to poor behavior of iterative solvers and even divergence in some cases. (2) Iterative solvers are very time-consuming for multi-source problems. These difficulties become major impediments when solving large-scale multi-source CSEM problems using iterative solvers.

Given the availability of more powerful workstations or computer clusters, direct methods have been increasingly used for solving 3D CSEM problems. Compared to iterative methods, direct solvers have several distinct advantages: (1) They provide more accurate solutions. (2) They are less prone to ill-conditioning of matrix, making them more robust for highly heterogeneous models or non-uniform grids. And (3) they separate the solving of linear system into an expensive matrix factorization and comparably inexpensive forward-backward substitution steps, which make them more suitable for multi-source CSEM surveying.

In this paper, we present an efficient inversion algorithm for 3D inversion of multi-frequency and multi-source CSEM data, which is based on a direct solver for solving the forward problem. The Gauss-Newton (GN) optimization algorithm is applied considering its high convergence rate, thus limiting the number of expensive matrix factorization required when using a direct solver. A preconditioned conjugate gradient solver (PCG) is used to solve the normal equation resulted from linearization at each GN iterate, in order to avoid computing and storing sensitivity matrix explicitly. This scheme only requires matrix-vector products of Jocabian and its transpose with vectors, which are equivalent to one forward and one adjoint problem. In addition, the matrix factorization obtained when solving forward problem can be used in subsequent PCG iterates, which dramatically speeds up PCG iteration and reduces computational costs. Numerical experiments on synthetic data from land and marine CSEM surveys show that our inversion algorithm has an excellent convergence rate and only tens to several tens of iterates are needed to reach desired misfit, demonstrating its efficiency and stability.