邱长凯^1,2,,
殷长春^2,,,
刘云鹤²,
张博²,
任秀艳²,
齐彦福³,
蔡晶²
1. 中国地质调查局发展研究中心, 北京 100037
2. 吉林大学地球探测科学与技术学院, 长春 130026
3. 长安大学地质工程与测绘学院, 西安 710054

基金项目: 国家自然科学基金项目（41530320，41774125，41804098），国家重点研发计划重点专项（2018YFC0603800，2016YFC0303100，2017YFC0601900），中科院先导专项（XDA14020102）和中央级公益性科研院所基本科研业务费专项经费（JYYWF20180103）联合资助


详细信息

作者简介: 邱长凯, 男, 1991年生, 博士, 工程师, 主要从事三维电磁正反演和固体矿产勘查技术研究.E-mail:qiuchangkai@163.com

通讯作者: 殷长春, 男, 1965年生, 教授, 国家"海外高端人才", 主要从事电磁勘探理论、特别是航空和海洋电磁方面的研究.E-mail:yinchangchun@jlu.edu.cn

中图分类号: P631

收稿日期:2018-08-27
修回日期:2019-06-19
上线日期:2020-02-05

3D time-domain airborne electromagnetic forward modeling using the rational Krylov method

QIU ChangKai^1,2,,
YIN ChangChun^2,,,
LIU YunHe²,
ZHANG Bo²,
REN XiuYan²,
QI YanFu³,
CAI Jing²
1. Development and Research Center, China Geological Survey, Beijing 100037, China
2. College of Geo-Exploration Science and Technology, Jilin University, Changchun 130026, China
3. School of Geological Engineering and Geomatics, Chang'an University, Xi'an 710054, China


More Information

Corresponding author: YIN ChangChun,E-mail:yinchangchun@jlu.edu.cn

MSC: P631

--> Received Date: 27 August 2018
Revised Date: 19 June 2019
Available Online: 05 February 2020


摘要
摘要:常规的三维时间域航空电磁模拟通常采用隐式步长方法进行时间离散，需要几次矩阵分解和上百次右端源项回带，计算效率较低.为了提高正演计算效率，本文提出使用有理Krylov方法求解时间域电场扩散方程.首先使用非结构四面体网格进行空间离散，采用Nédélec矢量基函数近似四面体单元内的电场；然后基于有限元离散给出矩阵指数和矢量乘积表示的电场显式解；最后采用有理Arnoldi算法构造Krylov子空间内的正交基函数并进一步求解矩阵指数与矢量的乘积，直接得到任意时刻的电场解向量，避免步长离散过程.此外，本文还提出一种指数加权偏移参数优化方法，使得有理Arnoldi近似在瞬变衰减晚期具备更高的精度，从而降低Krylov子空间阶数并提高计算效率.通过和层状模型解析解的对比验证了有理Krylov方法的精度.针对三维异常体模型使用全局网格和局部网格剖分并和其他数值方法比较，进一步说明了有理Krylov方法的有效性.
关键词: 航空电磁/
时间域/
三维正演/
有限元法/
有理Krylov方法

Abstract:Traditional Three-Dimensional (3D) time-domain airborne Electromagnetic (EM) modeling utilizes an implicit time discretization technique to calculate the EM responses in the time domain,which requires several matrix decompositions and hundreds of forward and backward substitutions. This reduces largely the modeling efficiency. To speed up the 3D forward modeling for time-domain Airborne Electromagnetic (AEM),we propose to solve the diffusion equation for the electric field using the rational Krylov method. The spatial discretization is completed by the unstructured tetrahedral grids,where lowest order Nédélec first kind vector basis functions are adopted to approximate the electric field inside each element. Then we derive the electric field solutions directly in terms of the product of a matrix exponential function with a vector. The rational Krylov method is used to solve the matrix function,where the orthogonal basis vectors in the rational Krylov subspace are constructed using the rational Arnoldi algorithm. Finally,the electric field is calculated for any desired time by the rational Arnoldi approximation,which avoids explicit or implicit time stepping. In addition,an exponential weight is incorporated into the rational Arnoldi approximation errors when optimizing the shift parameters,resulting in small approximation errors at late times. This further reduces the size of the rational Krylov subspace and improves the computational efficiency. We verify the correctness and accuracy of the proposed rational Krylov method by comparing the forward modeling results for a homogeneous half-space model with the semi-analytic solution. Furthermore,we prove the effectiveness of the presented rational Krylov method by comparing our solutions with time-domain finite-element and finite-volume solutions for a 3D abnormal model using global and local grids. The simulations of AEM responses for two typical 3D abnormal models demonstrate that the rational Krylov method can speed up the time-domain AEM forward modeling for as up to 7 times while keeping high modeling accuracy.
Key words:Airborne electromagnetic/
Time-domain/
3D forward modeling/
Finite-element method/
Rational Krylov method



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