Modeling the 3D Terrain Effect on MT by the Boundary Element Method
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摘要: 提出了一种用边界元法计算大地电磁场三维地形影响的数值模拟方法.首先用矢量积分理论和电磁场边界条件, 将上半空间(空气)和下半空间(地下介质)两个区域电磁场边值问题变为仅对地形界面的两个矢量面积分方程, 其中一个计算磁场, 称磁场方程; 另一个计算电场, 称电场方程.然后将对地形界面的积分剖分为一系列的三角单元积分.在三角单元积分中, 假设单元中电磁场为水平均匀大地空间电磁场与地形影响的迭加, 并假设地形影响为常项, 这样既保证了计算精度又使得计算方法简便.通过分解和计算, 每一个矢量面积分方程分解为对应3个坐标方向的3个常量线性方程, 这些线性方程组成了对角占优的线性方程组, 可用SSOR方法求解.文中给出了2个三维地形上大地电磁视电阻率曲线的计算结果.Abstract: A numerical method is put forward in this paper, using the boundary element method (BEM) to model 3D terrain effects on magnetotelluric (MT) surveys. Using vector integral theory and electromagnetic field boundary conditions, the boundary problem of two electromagnetic fields in the upper half space (air) and lower half space (earth medium) was transformed into two vector integral equations just related to the topography: one magnetic equation for computing the magnetic field and the other electrical equation for computing the electrical field. The topography integral is decomposed into a series of integrals in a triangle element. For the integral in a triangle element, we suppose that the electromagnetic field in it is the stack of the electromagnetic field in the homogeneous earth and the topography response which is a constant; so the computation becomes simple, convenient and highly accurate. By decomposition and computation, each vector integral equation can be calculated by solving three linear equations that are related to the three Cartesian directions. The matrix of these linear equations is diagonally dominant and can be solved using the Symmetric Successive Over-Relaxation (SSOR) method. The apparent resistivity curve of MT on two 3D terrains calculated by BEM is shown in this paper.
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Key words:
- 3D terrain /
- MT /
- boundary element method /
- numerical modeling
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图 1 区域与边界
Ω1.空气; Ω2.地下均匀岩石; Γ.三维地形; Γ1和Γ2.分别表示上下部无穷远边界; n.表示法向
Fig. 1. Domain and boundary
图 4 三维正地形上MT视电阻率曲线(T=0.1s)
Fig. 4. MT apparent resistivit y plot in t he uplifted t opography
图 5 三维负地形上MT视电阻率曲线(T=0.1 s)
Fig. 5. MT apparent resistivity plot in the 3D depressed topography
表 1 水平均匀大地上MT边界元法计算结果(T=0.1 s)
Table 1. Result obtained by the MT boundary element method in the horizontal homogeneous half-space
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[1] Chouteau, M., Bouchard, K., 1988. Two-di mensional terrain correction in magnetotelluric surveys. Geophysics, 53: 854-862. doi: 10.1190/1.1442520 [2] Ruan, B. Y., Wang, Y. X., 2005. A boundary element modeling method for the electromagnetic field by artificial source in frequency domain with 3-D topography. Chinese J. Geophy. , 48 (5): 1197-1204 (in Chinese with English abstract). [3] Wannamaker, P. E., Stodt, J. A., Rijo, L., 1986. Two-di mensional topographic responses in magnetotelluric model using finite elements. Geophysics, 51: 2131-2144. doi: 10.1190/1.1442065 [4] Xu, S. Z., 1995. The boundary element method in geophysics. Science Press, Beijing (in Chinese). [5] Xu, S. Z., Ruan, B. Y., Zhou, H., et al., 1997a. Numerical modeling of 3-D terrain effect on MT field. Science in China (Series D), 43 (2): 269-275 (in Chinese). [6] Xu, S. Z., Ruan, B. R., Zhou, H., et al., 1997b. Modeling the 2D terrain effect on MT by the boundary element method. Geophysical Prospecting, 45: 931-943. doi: 10.1046/j.1365-2478.1997.610301.x [7] 阮百尧, 王有学, 2005. 三维地形频率域人工源电磁场的边界元模拟方法. 地球物理学报, 48 (5): 1197-1204. doi: 10.3321/j.issn:0001-5733.2005.05.031 [8] 徐世浙, 1995. 地球物理中的边界元法. 北京: 科学出版社. [9] 徐世浙, 阮百尧, 周辉, 等, 1997a. 大地电磁场三维地形影响的数值模拟. 中国科学(D辑), 43 (2): 269-275. https://www.cnki.com.cn/Article/CJFDTOTAL-JDXK199701002.htm -
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