基于电场Helmholtz方程的回线源瞬变电磁法三维正演

Three-dimensional forward calculation for loop source transient electromagnetic method based on electric field Helmholtz equation

正演是电磁法勘探野外工作参数选取、室内资料处理与解释的基础,精确、稳定、高效的三维正演算法尤为重要.本文采取先求解拉普拉斯域电场、再由Gaver-Stehfest算法获得时间域磁场的思路,基于电场异常场Helmholtz方程实现了交错网格有限差分法和有限体积法对回线源瞬变电磁法的三维正演.通过对比低阻块状体的积分方程法、时域有限差分法、矢量有限单元法和SLDM法的数值解,验证了交错网格有限差分法和有限体积法的正确性.由于交错网格有限差分法、有限体积法和基于矩形块单元的矢量有限单元法将待求电场均定义在矩形块单元棱边上,因此三种数值算法可采用相同方法进行电场待求量编码、计算背景场和后处理.然而,与矢量有限单元法相比,交错网格有限差分法和有限体积法的系数矩阵更加稀疏,求解效率更高.通过对水平低阻板状体三维模型的数值模拟,我们发现本研究中交错网格有限差分法比有限体积法精度更高;再利用一维解析法求解相应三层层状地电模型的感应电动势,我们还发现两种数值算法和一维解析法计算的感应电动势等值线形状吻合程度高,只是数值范围略有差异.

Forward and interpretation ; is very important. field firstly, then calculation is the base of parameters selection in field work, data processing, particularly the accurate, stable and efficient three-dimensional forward solver In this paper, based on the strategy that solving the Laplace-domain electrical acquiring the time-domain magnetic field by Gaver-Stehfest algorithm, weemployed staggered finite difference （SFD） and finite volume （FV） to discretize the electrical field Helmholtz equation for three-dimensional forward calculation of loop source transient electromagnetic method. Comparing with the numerical solutions of integral equation （IE）, finite difference in time domain （FDTD）, vector finite element method （VFEM） and spectral Lanczos decomposition method （SLDM） for low resistivity body in homogeneous half-space, we validated the SFD and FV algorithms. Because the unknown electrical field is arranged on the edge of brick for SFD, FV and VFEM based on brick element, so these three numerical algorithms could utilize the same method for coding the unknown electrical field, computing the background field and post processing. However, comparing with the VFEM, the coefficient matrixes produced by SFD and FV are sparser, so these two algorithms are more efficient. For low resistivity horizontal sheet model, we use SFD, FV and 1D analytical method to compute the induced electromagnetic force （EMF）. The results show that SFD has higher accuracy than FV, and the contours of EMF for these three algorithms coincide with each other very well although the range of EMF has fine distinctions.