﻿ 球坐标系下三维大地电磁正演研究
 地球物理学报  2019, Vol. 62 Issue (10): 3885-3897 PDF

1. 成都理工大学地球物理学院, 成都 610059;
2. 四川省蜀通岩土工程公司, 成都 610084;
3. 西南科技大学环境与资源学院, 四川绵阳 621010

Three-dimensional forward modeling of the magnetotelluric method in spherical coordinates
LUO Wei1,2, WANG XuBen1, WANG KunPeng1, ZHANG Gang3, LI DeWei1
1. College of Geophysics, Chengdu University of Technology, Chengdu 610059, China;
2. Sichuan Shutong Geotechnical Engineering Company, Chengdu 610084, China;
3. School of Environment and Resource, Southwest University of Science and Technology, Mianyang Sichuan 621010, China
Abstract: Theoretical research on forward modeling of the magnetotelluric (MT) method has focused on how to improve the calculation efficiency and precision. When the profile is long enough and detection depth is sufficiently deep, it is difficult to fit the curvature of the earth accurately by numerical simulation in a Cartesian coordinate system. This paper studies three-dimensional MT forward modeling with a staggered grid based in a spherical coordinate system. We deduce the formula for staggered-grid finite difference three-dimensional forward modeling and compare it with of one-dimensional analytical solution and the DTM1 three-dimensional modeling to verify correctness of the algorithm. Calculation on a theoretical model shows that the modeling in the spherical coordinates is more reasonable, which can avoid errors caused by the conventional Cartesian coordinate system, so can replace the Cartesian coordinates modeling method. The differences of 3D MT forward modeling responses in theses two coordinate systems are related with frequency, model structure and resistivity. For example, such differences become more notable with lowering frequency. They are close to 10% at the period of tens of thousands of seconds. It implies that the effect of earth curvature on large-scale long-period MT cannot be ignored, so MT forward modeling on large scales should be conducted in spherical coordinates.
Keywords: 3-D magnetotelluric    Spherical coordinates    Earth curvature    Finite difference
0 引言

1 球坐标三维大地电磁正演算法 1.1 控制方程

 (1)

 (2)

1.2 数值离散

 图 1 坐标系统和网格剖分示意图 (a)坐标系统；(b)笛卡尔坐标网格剖分；(c)球坐标网格剖分. Fig. 1 Coordinates system and grid subdivision (a) Coordinates system; (b) Cartesian coordinate grid subdivision; (c) Spherical coordinates grid subdivision.

 图 2 球坐标交错网格 Fig. 2 Staggered grid in spherical coordinates

1.3 方程组及求解

 (3)

 (4)

2 正演验证及分析

 图 3 DTM1模型示意图 Fig. 3 Sketch of DTM1 model

 图 4 DTM1模型(0, 0)点处的响应对比 Fig. 4 Comparison of the obtained DTM1 model responses at site (0, 0)
 图 5 DTM1模型测线x=0的响应对比 Fig. 5 Comparison of the obtained DTM1 model responses at Line x=0
3 球坐标系及地球曲率的影响

 图 6 球坐标系模型示意图 Fig. 6 Sketch of spherical coordinates model

 图 7 球坐标系和笛卡尔坐标系模型响应相对误差 (a) S1模型X=0测线; (b) S2模型X=0测线; (c) S3模型X=0测线; (d) S3模型Y=0测线. Fig. 7 Error between cartesian coordinate solution and spherical coordinate solution (a) Line X=0 of S1 model; (b) Line X=0 of S2 model; (c) Line X=0 of S3 model; (d) Line Y=0 of S3 model.

 图 8 放大尺寸DTM1模型X=0测线响应误差 Fig. 8 Error between cartesian coordinate response and spherical coordinate response at line(X=0) of extension DTM1 model
 图 9 放大尺寸DTM1模型X=0测线响应曲线对比 Fig. 9 Responses comparison of extension DTM1 model at line(X=0)
4 结论

(1) 相较于传统的笛卡尔坐标系模拟方法，球坐标系模拟更直观合理，能避免传统笛卡尔坐标拉伸投影所引入的误差.球坐标系模拟方法完全可代替目前的笛卡尔坐标模拟方法.

(2) 基于球坐标和笛卡尔坐标的三维大地电磁正演响应值随着频率变低差异越明显.

(3) 球坐标和笛卡尔坐标计算结果差异度与频率、模型结构和电阻率有关.本文模型计算结果在数万秒周期处已出现最大接近10%的差异，表明对于较大尺度的长周期大地电磁，地球曲率影响不能忽略.

 (A1)

 (A2)

 (A3)

 (A4)

 (A5)

 (A6)

 (A7)

 (A8)

 (A9)

 (A10)

 (A11)

 (A12)

 (A13)

(A13) 式中各系数的表达式分别为

 (A14)

(A14) 式中各系数的表达式分别为

 (A15)

(A15) 式中各系数的表达式分别为

(A13) 式中的a0、(A14)式中的b0和(A15)式中的c0在模型内部单元离散时值为0，在边界单元由边界条件确立.

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