瞬变电磁Crank-Nicolson FDTD三维正演

孙怀凤, 柳尚斌, 杨洋. 2021. 瞬变电磁Crank-Nicolson FDTD三维正演. 地球物理学报, 64(1): 343-354, doi: 10.6038/cjg2021O0229
引用本文: 孙怀凤, 柳尚斌, 杨洋. 2021. 瞬变电磁Crank-Nicolson FDTD三维正演. 地球物理学报, 64(1): 343-354, doi: 10.6038/cjg2021O0229
SUN HuaiFeng, LIU ShangBin, YANG Yang. 2021. Crank-Nicolson FDTD 3D forward modeling for the transient electromagnetic field. Chinese Journal of Geophysics (in Chinese), 64(1): 343-354, doi: 10.6038/cjg2021O0229
Citation: SUN HuaiFeng, LIU ShangBin, YANG Yang. 2021. Crank-Nicolson FDTD 3D forward modeling for the transient electromagnetic field. Chinese Journal of Geophysics (in Chinese), 64(1): 343-354, doi: 10.6038/cjg2021O0229

瞬变电磁Crank-Nicolson FDTD三维正演

  • 基金项目:

    山东省自然科学基金(ZR2019MD20), 国家自然科学基金(42074145)资助

详细信息
    作者简介:

    孙怀凤, 1982年生, 男, 博士, 副教授, 博士生导师, 主要从事瞬变电磁正反演与应用方面的教学与科研工作.E-mail:sunhuaifeng@gmail.com

  • 中图分类号: P631

Crank-Nicolson FDTD 3D forward modeling for the transient electromagnetic field

  • 时域有限差分(FDTD)方法使用Yee网格剖分电磁场的空间采样, 通过时间步迭代实现电磁场数值模拟, 具有内存消耗低、计算简单等特点, 常用于瞬变电磁三维正演.然而, 常规FDTD方法的时间迭代步长Δt受Courant-Friedrich-Lewy(CFL)条件严格限制, 过多的迭代次数以及过密的采样往往导致计算速度慢、累积误差不断增大.本文提出一种不受CFL条件约束的无条件稳定隐式差分算法Crank-Nicolson FDTD(CN-FDTD)用于瞬变电磁三维正演.基于Crank-Nicolson差分方法对Maxwell方程组重新离散, 空间网格仍然采用Yee元胞, 时间步进采用在整时间步电场、磁场同时采样的策略, 建立无条件稳定FDTD格式, 突破CFL条件限制.与常规FDTD交替采样相比, CN-FDTD电场、磁场同时采样的策略构成的隐式差分格式, 需要求解大型稀疏矩阵方程组.通常, 瞬变电磁三维正演模型中产生的矩阵阶数往往较大, 需要占用大量内存和求解时间.为解决上述问题, 采用Crank-Nicolson-cycle-sweep-uniform(CNCSU-FDTD)方法近似求解CN-FDTD方程, 在保证求解精度的同时, 计算效率大幅提高.在边界条件处理上, 采用双线性变换推导了复频率参数完全匹配层(CFS-PML)吸收边界.采用均匀半空间模型、四类三层模型进行精度验证, 发现CN-FDTD三维正演结果与解析解、线性数字滤波解吻合较好.之后, 与接触带上的低阻复杂模型进行对比, 结果显示CN-FDTD正演结果与矢量有限元、有限体积法以及FDTD计算结果吻合较好.在此基础上, 研究了时间步放大对CN-FDTD计算精度的影响, 发现最大时间步放大到常规FDTD的3200倍时才会在晚期出现较明显的误差.在一台CPU为Intel Core i5-7300HQ的笔记本电脑单线程计算条件下, 模拟到关断后30 ms仅需要50 min.在进行并行化后, 将有望实现复杂模型分钟级的三维正演, 从而为三维反演提供可靠、快速的正演方法.

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  • 图 1 

    CN-FDTD使用的Yee网格

    Figure 1. 

    Yee grid

    图 2 

    常规FDTD与CN-FDTD电场、磁场时间采样分布对比

    Figure 2. 

    Comparison of original FDTD and CN-FDTD sampling in time for electric and magnetic fields

    图 3 

    均匀半空间模型CN-FDTD与解析解对比

    Figure 3. 

    Comparison of CN-FDTD numerical solution and analytical solution for Homogeneous half space model

    图 4 

    典型三层模型CN-FDTD数值解与线性数字滤波解对比

    Figure 4. 

    Comparison of digital filter solution and CN-FDTD for typical there-layered models

    图 5 

    三维垂直接触带复杂模型(Li et al., 2017)

    Figure 5. 

    Three-dimensional complex model with a vertical contact zone

    图 6 

    三维垂直接触带复杂模型计算结果对比

    Figure 6. 

    The numerical results of three-dimensional complex model

    图 7 

    CN-FDTD在不同CFLNs下结果的对比

    Figure 7. 

    Comparison of CN-FDTD numerical results under different CFLNs

    图 8 

    CN-FDTD在不同CFLNs下Δt的变化对比

    Figure 8. 

    Changes of Δt in the CN-FDTD under different CFLNs

    表 1 

    层状模型参数

    Table 1. 

    Parameters of the layered models

    模型 ρ1(Ωm) ρ2(Ωm) ρ3(Ωm) h1(m) h2(m)
    A 20 50 500 40 100
    H 100 10 100 60 40
    K 100 1000 100 60 40
    Q 100 10 1 40 100
    下载: 导出CSV

    表 2 

    不同CFLNs下CN-FDTD算法内存开销及计算时间

    Table 2. 

    Memory overhead and computing time of CN-FDTD algorithm under different CFLNs

    CFLN 总迭代次数 计算时间(min) 内存(Mb)
    200 16220 70 192
    400 13091 56 192
    800 12038 51 192
    1600 11656 50 192
    3200 11528 49 192
    下载: 导出CSV
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收稿日期:  2020-06-17
修回日期:  2020-12-02
上线日期:  2021-01-10

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