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物探与化探  2020, Vol. 44 Issue (6): 1387-1398    DOI: 10.11720/wtyht.2020.0343
  方法研究·信息处理·仪器研制 本期目录 | 过刊浏览 | 高级检索 |
基于矢量有限元的大地电磁快速三维正演研究
顾观文1,2(), 武晔1,2, 石砚斌1,2
1.防灾科技学院 地球科学学院,河北 廊坊 065201
2.河北省地震动力学重点实验室,河北 廊坊 065201
Research on fast three-dimensional forward algorithm of magnetotelluric sounding based on vector finite element
GU Guan-Wen1,2(), WU Ye1,2, SHI Yan-Bin1,2
1. School of Earth Sciences, Institute of Disaster Prevention, Langfang 065201, China
2. Hebei Key Laboratory of Earthquake Dynamics, Langfang 065201, China
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摘要 

本文采用直接求解器PARDISO且无需散度校正的正演方案,求解矢量有限元法对应的大型线性方程组,获得不同地形(水平和起伏)条件下地电模型的大地电磁响应,提高了大地电磁三维正演计算的速度。在中等规模计算条件下,通过本文的计算方法与带散度校正的迭代求解法对比,计算速度可提高十倍以上。
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顾观文
武晔
石砚斌
关键词 ： 大地电磁,  矢量有限元法,  三维正演,  PARDISO    
Abstract：

The finite element method has the characteristics of strong adaptability in simulating the electromagnetic response of rugged topography and complex geological bodies. In recent years, it has been widely used in the three-dimensional (3D) forward modeling of magnetotelluric (MT) sounding. However, the finite element method also has some shortcomings in terms of computational efficiency. The large amount of calculation and long running time of the method are the main factors that lead to the lag of the practical process of the 3D MT inversion technology based on the finite element method compared with the 3D MT inversion technology based on the finite difference method. In order to improve the 3D forward speed of MT, the authors adopt the forward modeling scheme which uses the direct solver PARDISO and does not need divergence correction to solve the large-scale linear equations corresponding to the vector finite element method, and obtain the MT response of the geoelectric model under such different terrain conditions as flat and rugged topography. Under the conditions of medium-scale calculation, through the comparison between the direct solution method without divergence correction and the iterative solution method with divergence correction, the authors have detected that the direct solution method without divergence correction has advantages in calculation accuracy and calculation time, especially in the calculation. In terms of time, the ratio of the calculation speed of the direct solution and the iterative solution is raised by more than ten times.
Key words： magnetotelluric    vector finite element method    3D forward    PARDISO
收稿日期: 2020-07-02      出版日期: 2020-12-29
:  P631  
基金资助:中央高校创新团队项目(ZY20180102);自然资源部“十五”重点科技项目(20010211)
作者简介: 顾观文(1975-),男,教授级高级工程师,从事电磁法正反演研究及其软件研制工作。 Email:sun_ggw@163.com
引用本文:   
顾观文, 武晔, 石砚斌. 基于矢量有限元的大地电磁快速三维正演研究[J]. 物探与化探, 2020, 44(6): 1387-1398.
GU Guan-Wen, WU Ye, SHI Yan-Bin. Research on fast three-dimensional forward algorithm of magnetotelluric sounding based on vector finite element. Geophysical and Geochemical Exploration, 2020, 44(6): 1387-1398.
链接本文:  
https://www.wutanyuhuatan.com/CN/10.11720/wtyht.2020.0343      或      https://www.wutanyuhuatan.com/CN/Y2020/V44/I6/1387
Fig.1  带地形三维MT数值模拟区域剖面示意[11]
Fig.2  矢量有限元法的区域剖分示意
a—区域剖分示意; b—电场分量位置示意
频点
/Hz		 视电阻率(ρxy,ρyx)/(Ω·m) 相位φ/(°)
矢量有限元解 解析解 误差/% 矢量有限元解 解析解 误差/%
10000 99.4934 100 0.5066 44.9824 45 0.039111
8000 99.5168 100 0.4832 44.9800 45 0.044444
5000 99.5580 100 0.442 44.9782 45 0.048444
2000 99.6149 100 0.3851 44.9802 45 0.044
1000 99.6266 100 0.3734 44.9895 45 0.023333
500 99.7002 100 0.2998 44.9601 45 0.088667
200 99.8467 100 0.1533 45.0661 45 0.14689
100 99.4868 100 0.5132 45.1051 45 0.23356
50 99.2563 100 0.7437 45.0452 45 0.10044
10 99.4329 100 0.5671 44.9565 45 0.096667
5 99.5295 100 0.4705 44.9596 45 0.089778
2 99.6083 100 0.3917 44.9717 45 0.062889
1 99.6484 100 0.3516 44.9776 45 0.049778
0.5 99.6599 100 0.3401 44.9970 45 0.006667
0.1 99.7267 100 0.2733 44.9301 45 0.155333
0.05 99.9013 100 0.0987 44.9393 45 0.134889
0.01 100.0090 100 0.009 44.9873 45 0.028222
0.005 100.0070 100 0.007 44.9949 45 0.011333
0.001 100.0010 100 0.001 44.9995 45 0.001111
0.0005 100.0000 100 0 44.9998 45 0.000444
0.0001 100.0000 100 0 45.0000 45 0
Table 1  均匀半空间模型矢量有限元解与解析解对比
Fig.3  均匀半空间模型三维正演视电阻率(a)和相位(b)与解析解对比
层参数 第一层 第二层 第三层
电阻率/(Ω·m) 100 10 1000
层厚/m 370 268 ∞
Table 2  H型层状模型参数
频点
/Hz		 视电阻率(ρxy,ρyx)/(Ω·m) 相位φ/(°)
矢量有限元解 解析解 误差/% 矢量有限元解 解析解 误差/%
10000 99.4944 100 0.505638 44.9825 45.00002 0.03894
8000 99.5156 99.99967 0.484074 44.9807 45.00007 0.00043
5000 99.564 100.0036 0.439562 44.9729 44.9985 0.000569
2000 99.1812 99.72298 0.543282 45.0089 45.02392 0.000334
1000 100.138 100.1248 0.01314 44.3103 44.43167 0.002732
500 108.223 107.9785 0.22646 44.9127 44.67756 0.00526
200 112.433 113.6883 1.104134 51.6513 51.46143 0.00369
100 94.4807 95.60484 1.175822 59.265 59.07526 0.00321
50 63.587 64.76429 1.817807 63.9834 63.85949 0.00194
10 28.0673 28.37355 1.079333 45.849 46.13936 0.006293
5 32.2082 32.33247 0.384363 32.3095 32.54673 0.007289
2 55.1794 55.21716 0.06839 21.4561 21.55953 0.004797
1 89.6415 89.67347 0.035652 18.7262 18.78408 0.003081
0.5 143.375 143.3732 0.00122 19.0537 19.09105 0.001956
0.1 348.002 347.851 0.04341 25.4995 25.50588 0.00025
0.05 457.716 457.5438 0.03763 29.036 29.03635 1.19E-05
0.01 692.088 691.9579 0.0188 36.1384 36.13554 7.9E-05
0.005 769.11 769.008 0.01327 38.3803 38.37769 6.8E-05
0.001 888.395 888.3428 0.00588 41.8079 41.80643 3.5E-05
0.0005 919.642 919.6043 0.0041 42.7015 42.70037 2.7E-05
0.0001 963.197 963.179 0.00187 43.9466 43.94614 1.1E-05
Table 3  H型层状模型矢量有限元解与解析解对比
Fig.4  H型层状模型三维正演视电阻率(a)和相位(b)数值解与解析解对比
层参数 第一层 第二层 第三层
电阻率/(Ω·m) 100 1000 10
层厚/m 370 268 ∞
Table 4  K型层状模型参数
频点
/Hz		 视电阻率(ρxy,ρyx)/(Ω·m) 相位φ/(°)
矢量有限元解 解析解 误差/% 矢量有限元解 解析解 误差/%
10000 99.4928 99.99995 0.507154 44.9823 44.99998 0.039291
8000 99.5179 100.0004 0.482464 44.9795 44.99994 0.04542
5000 99.5522 99.9958 0.443615 44.9822 45.00155 0.042987
2000 99.9938 100.3013 0.306591 44.9731 44.9962 0.05134
1000 98.5514 99.11594 0.569571 45.4799 45.46765 0.02695
500 94.8705 94.84442 0.0275 43.8641 44.11071 0.55908
200 111.446 109.5436 1.73666 42.1518 41.35312 1.93136
100 124.19 128.152 3.091671 47.226 45.89309 2.90438
50 115.34 122.0064 5.46397 54.8037 54.506 0.54618
10 55.4147 56.83843 2.504864 64.9559 65.42769 0.721085
5 38.6174 39.2693 1.660063 64.5842 64.96375 0.584241
2 25.6211 25.88921 1.03559 61.7858 62.03061 0.394665
1 20.052 20.20837 0.773778 58.9626 59.13016 0.28337
0.5 16.5801 16.68206 0.611172 56.1309 56.24364 0.200444
0.1 12.6035 12.65607 0.415342 50.8822 50.92886 0.091626
0.05 11.7748 11.82107 0.391403 49.346 49.36433 0.037132
0.01 10.7493 10.77999 0.284731 46.9822 47.06061 0.166611
0.005 10.5349 10.54576 0.102932 46.4109 46.47605 0.140175
0.001 10.2412 10.24057 0.00616 45.6579 45.67163 0.030065
0.0005 10.17 10.16953 0.00462 45.4711 45.47688 0.012716
0.0001 10.0755 10.07547 0.00035 45.2136 45.21445 0.001873
Table 5  K型层状模型矢量有限元解与解析解对比
Fig.5  K型层状模型三维正演视电阻率(a)和相位(b)数值解与解析解对比
Fig.6  COMMEMI3D-2 模型示意
Fig.7  本文矢量有限元正演算法的计算结果与IE方法的计算结果对比
a—Zxy模式正演视电阻率;b—Zyx模式正演视电阻率;c—Zxy模式正演阻抗相位; d—Zyx模式正演阻抗相位
Fig.8  三维地形及其网格剖分示意
a—三维地形示意; b—地形网格剖分和MT测点分布示意
Fig.9  二维山峰地形示意
Fig.10  三维矢量有限元算法(PARDISO)计算的二维地形影响与二维有限元结果对比
a—正演视电阻率; b—正演阻抗相位
Fig.11  无需散度校正的直接解法与带散度校正的迭代解法计算结果对比曲线
a—正演视电阻率; b—正演阻抗相位
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