Three-dimensional FDTD modeling of TEM excited by a loop source considering ramp time
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摘要: 从麦克斯韦旋度方程出发可以直接导出瞬变电磁场扩散方程,然而扩散方程不含电场对时间的一阶导数,不能构成显式的时域有限差分方程,借鉴du Fort-Frankel有限差分离散方法引入虚拟位移电流项构建显式时域有限差分方程.对Wang和Hohmann的经典时域算法进行了两点改进:第一,通过将矩形回线源电流密度加入麦克斯韦方程组的安培环路定理方程,实现回线源瞬变电磁激发源加入;第二,在计算中考虑关断时间.第一点改进使时域有限差分方程考虑了一次场的计算,并且源的计算不再依赖均匀半空间模型响应作为初始条件,使算法能够适应表层电阻率不均匀时的三维复杂模型.由于实际观测中不可能出现阶跃电流的关断形式,第二点改进可以方便设置发射电流下降沿.采用改进的三维时域有限差分正演算法对均匀半空间模型、四类三层模型、均匀半空间中含有低阻块体模型进行了计算并分别与解析解、线性数字滤波解、积分方程解和Wang的三维时域有限差分解进行了对比验证.以H模型为例,采用建立的三维时域有限差分正演算法计算了不同关断时间的斜阶跃脉冲回线源瞬变电磁中心点感应电动势衰减曲线.以实际地质资料为基础,构建包含两层采空区的三维复杂模型,以1 μs的极短关断时间进行了复杂模型定回线源瞬变电磁响应计算,并计算了该复杂模型的视电阻率曲线.Abstract: Transient electromagnetic (TEM) diffusion equations can be directly derived from the Maxwell equations. However, the diffusion equations do not contain the first derivative of the electric field to time, thus the explicit finite difference equations in the time domain cannot be derived. In this work, the du Fort-Frankel method solving finite difference equations is referenced and an additional fictitious displacement current is introduced into the diffusion equations to form explicit difference equations. Two modifications are made based on the famous finite difference TEM modeling method in the time domain by Wang and Hohmann. Firstly, the loop current is introduced into Maxwell equations by means of current density according to Ampere circuital theorem. A loop source is added in modeling. Secondly, the ramp time is considered in modeling. By the first modification, primary fields of TEM are included in modeling and the excitation source is no longer dependent on a homogeneous half-space model as the initial condition. The algorithm can be applied to very complex models with non-uniform surface resistivity distributions. In real TEM data acquisition, the ramp current is not a step and the ramp time is not zero. The ramp time can be considered in modeling by the second modification. The modified TEM 3D FDTD modeling method is checked by several numerical examples, the analytical solution of a homogeneous half-space model, digital filter solutions of four types of three-layer earth, integral-equation solution and Wang's FDTD solution for a low resistivity 3D body in a homogeneous half-space. The central loop induced electromotive force of an H model with different ramp times are simulated by the 3D FDTD modeling method to show the influences of ramp times. Finally, a complex 3D model with two mined out water bodies are built simplified from real geological data and fixed loop TEM responses are simulated using a 1 μs ramp time. The apparent resistivity curves and contours are also drawn.
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Key words:
- Transient electromagnetic /
- 3D modeling /
- FDTD /
- Ramp time
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