基于有限元方法的频率域弹性波全波形反演

doi:10.12288/szjs.2020.4.315

数值计算与计算机应用 ›› 2020, Vol. 41 ›› Issue (4): 315-336.DOI: 10.12288/szjs.2020.4.315

• 论文 • 上一篇

基于有限元方法的频率域弹性波全波形反演

张文生, 张丽娜

中国科学院数学与系统科学研究院, LSEC, 计算数学与科学工程计算研究所, 北京 100190;数学科学学院, 中国科学院大学, 北京 100049

收稿日期:2019-11-11 出版日期:2020-12-15 发布日期:2020-12-15
基金资助:
国家自然科学基金（11471328，51739007）资助.

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张文生, 张丽娜. 基于有限元方法的频率域弹性波全波形反演[J]. 数值计算与计算机应用, 2020, 41(4): 315-336.

Zhang Wensheng, Zhang Lina. ELASTIC WAVE FULL-WAVEFORM INVERSION BASED ON THE FINITE-ELEMENT METHOD IN THE FREQUENCY DOMAIN[J]. Journal on Numerica Methods and Computer Applications, 2020, 41(4): 315-336.

ELASTIC WAVE FULL-WAVEFORM INVERSION BASED ON THE FINITE-ELEMENT METHOD IN THE FREQUENCY DOMAIN

Zhang Wensheng, Zhang Lina

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received:2019-11-11 Online:2020-12-15 Published:2020-12-15

本文基于有限元方法，研究和发展了频率域弹性波方程的全波形反演方法.首先详细阐述了频率域弹性波方程的有限元正演方法，包括矩形单元上的有限元全离散格式、完全匹配层吸收边界条件以及有限元计算中震源的处理，并进行了正演模拟数值计算.其次，推导了矩形单元的全波形反演公式及其离散格式，也包括预条件最速下降法和正则化方法的结合应用.对均匀模型和国际标准的Overthrust复杂构造模型进行了并行全波形反演计算，反演中假定密度已知，通过反演Lamé参数来反演纵波和横波速度.反演基于频率多尺度策略，从低频至高频逐级进行，对数据含噪情况的也进行了计算，均得到了较好的反演结果.数值计算验证了文中方法、格式和算法的正确性和有效性.

关键词： 弹性波方程, 频率域, 有限元, 全波形反演, 正则化, 预条件, 数值优化, PML吸收边界条件

In this paper, the frequency domain full-waveform inversion for the elastic wave equations based on the finite-element method is studied and developed intensively. Firstly, the finite-element forward method for the elastic wave equations in the frequency domain is expounded in detail. It includes the full discrete finite-element schemes on the rectangle element, the perfectly matched layer (PML) absorbing boundary conditions and the source treatment in the finite-element computations. The numerical computations of forward modelling are also accomplished. Secondly, we derive the full-waveform inversion formulae and the corresponding discrete schemes in detail based on the rectangle element. Moreover, the preconditioned steepest descent method and the regularization method are applied. Finally, the parallel numerical computations of full-waveform inversion for a homogeneous model and the international benchmark Overthrust model are completed. In the inversion, the density is assumed to be known and the compressional wave velocity and the shear wave velocity are obtained through the inversion results of Lame parameters. The inversion is based on the frequency multiscale strategy and is carried out from low frequency to high frequency step by step. Good inversion results are obtained including the inversion for the noisy data. Numerical computations show the correctness and effectiveness of the method, schemes and algorithms presented in this paper.

Key words: elastic wave equations, finite element, frequency domain, full-waveform inversion, regularization, precondition, numerical optimization, PML absorbing boundary conditions

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基于有限元方法的频率域弹性波全波形反演

ELASTIC WAVE FULL-WAVEFORM INVERSION BASED ON THE FINITE-ELEMENT METHOD IN THE FREQUENCY DOMAIN

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