大地电磁法正演中多重网格法求解的广义傅里叶谱分析

J4 ›› 2011, Vol. 41 ›› Issue (5): 1587-1595.

大地电磁法正演中多重网格法求解的广义傅里叶谱分析

柳建新^1,2, 郭荣文^1,2,3|童孝忠^1,2|刘颖^1,2|刘鹏茂^1,2

1.中南大学地球科学与信息物理学院|长沙410083;
2.有色资源与地质灾害探查湖南省重点实验室|长沙410083;
3.维多利亚大学地球与海洋学院|加拿大|维多利亚V8W 3P6

收稿日期:2011-03-27 出版日期:2011-09-26 发布日期:2011-09-26
通讯作者: 郭荣文（1981-），男，江西赣江人,博士研究生，主要从事地球物理反演理论研究工作 E-mail:rwguo@uvic.ca
作者简介:柳建新(1962-)|男|湖南岳阳人|教授|博士|主要从事大地电磁理论与研究|E-mail:ljx6666@126.com
基金资助:
国家科技支撑计划项目（2011BAB04B08）；有色资源与地质灾害探查湖南省重点实验室项目（2010TP4012-6）；中国地质调查局科研项目（资\[2011\]03-01-64）;中南大学自由探索计划项目（2011QNZT010）

General Fourier Analysis of Multi-Grid in Magnetotelluric Modeling

LIU Jian-xin^1,2, GUO Rong-wen^1,2,3, TONG Xiao-zhong^1,2, LIU Ying^1,2, LIU Peng-mao^1,2

1.School of Info-physics and Geomatics Engineering, Central South University, Changsha410038, China;
2.Hunan Key Laboratory of Non-ferrous Resources and Geological Hazard Detection| Changsha410038, China;
3.School of Earth and Ocean Sciences, University of Victoria, Victoria B.C |V8W 3P6, Canada

Received:2011-03-27 Online:2011-09-26 Published:2011-09-26

摘要/Abstract

摘要：

为了准确预测和分析多重网格法用于大地电磁法正演计算的收敛效果，对多重网格法的收敛性进行了广义傅里叶谱分析。通常情况下，系数矩阵的傅里叶谱是复数，为了直观地判断、提取收敛信息，将谱转换到实数域。在实数域内，特征向量谱定量解释了二重网格法收敛慢的原因（最粗网格用Gauss-Seidel法求解）。传统的局部傅里叶谱分析没有考虑边界条件和模型参数的变化，在分析二重网格法求解收敛性时得出的渐进收敛估计与数值解偏差大。针对这一问题提出广义傅里叶谱分析，其结果与数值解接近（比如2-V（0，1）的广义傅里叶谱分析为0.706,真实值为0.710）。对五重网格法求解的高重广义傅里叶谱分析结果表明，随广义傅里叶谱分析分析重数的增加，所求渐近收敛估计趋于收敛的数值结果。基于此得出高重广义傅里叶谱分析的经验公式。求得低重广义傅里叶谱分析，通过矫正近似得到高重广义傅里叶谱分析，其结果在本文2个例子有效。

关键词: 多重网格法, 傅里叶谱分析, 最粗网格, 收敛性, 大地电磁法

Abstract:

In this paper, we apply Fourier analysis (including local Fourier analysis and general Fourier analysis) of multi-grid method to predict and analyze the convergence of multi-grid method used to discretize Helmholtz equations with complex-valued entries aroused in magnetotelluric problem. The Fourier spectra, which are usually complex in spectral domain,however, should be converted to real domain when visualizing the convergence behavior. The eigenvector spectra in real domain could be used to answer the slow convergence in two-grid method with Gauss-Seidel solver used on the coarsest grid. As not including boundary conditions and variant coefficients, local Fourier analysis (e.g., one-grid Fourier analysis and two-grid Fourier analysis) is difficult to give a reasonable asymptotic convergence estimate of two-grid method with a direct solver on the coarsest grid, whereas two-grid general Fourier analysis could. The Fourier analysis for five-grid method using a direct solver on coarsest grid shows that when the general higher-grid Fourier analysis is applied, the general Fourier analysis gets close to the numerical asymptotic convergence. Based on the set of general multi-grid Fourier analysis results, an empirical formula approximating the asymptotic convergence for five-grid method is derived,thus we just need to carry out the general low-grid Fourier analysis(two-grid and three-grid) on coarsest grids and approximate the asymptotic convergence with low cost.

Key words: multigrid method, Fourier analysis, coarsest grid, convergence, magnetotellurics

中图分类号:

P631.3

柳建新, 郭荣文, 童孝忠, 刘颖, 刘鹏茂. 大地电磁法正演中多重网格法求解的广义傅里叶谱分析[J]. J4, 2011, 41(5): 1587-1595.

LIU Jian-xin, GUO Rong-wen, TONG Xiao-zhong, LIU Ying, LIU Peng-mao. General Fourier Analysis of Multi-Grid in Magnetotelluric Modeling[J]. J4, 2011, 41(5): 1587-1595.

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