Research on divergence correction method in 3D numerical modeling of 3D controlled source electromagnetic fields

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1 GLOBAL GEOLOGY Vol. 33 No. 4 Dec P A doi /j. issn Research on divergence correction method in 3D numerical modeling of 3D controlled source electromagnetic fields ZHU Cheng LI Tonglin FAN Cuisong. College of Geoexploration Science and Technology Jilin University Changchun 3006 China. Tianjin Center China Geological Survey Tianjin China Abstract In the 3D numerical modeling of controlled source electromagnetic fields the convergence rate of numerical simulation became slower in extremely low frequency which led to the time for getting precisive results became longer. Therefore the application of divergence correction of electric current density at low frequencies has been introduced to help the case improved. Based on the study of 3D controlled source electromagnetic staggered grid finite difference algorithms the authors derived divergence correction formula by using the current density of the divergence condition as zero and implemented divergence correction algorithm with iterative method. Model tests showed that divergence correction can reduce the time and improve accuracy effectively in the low frequency range and the effects were more obvious with largerscale anomalies and lower frequencies. Key words electromagnetic field numerical modeling electric current density divergence correction YQ lilaoshizh@ 63. com

2 4 97 Krylov 8 Smith khz 0 khz 0 Fig. Sampling location of electric field component u sing staggered grid.. e iωt i = 槡 E r = iωμ( r H( r Newman H( r = σ( r E( r J p r E H μ σ J p E s ( r = iωμ 0 H s ( r H s ( r = σ( r E s ( r J s r 3 4 J s r = ( σ( r σ p ( r E p ( r 5 E p σ p J s 3 Fig. Diagram of 3D meshing division E s r = iωμ 0 H s r 6 4 6

3 98 33 E s r iωμ 0 σ r E s r = iωμ 0 J s r 7 ( σ φ = ψ 3 7 Ax = b x b E c s = E s φ 3 A 7 E s E c s σ E QMR 3 x ( σ φ = [ σe s ( σ σ E ] p p 4. 3 φ E φ E H = σe J p 8 J p H s = σe s J a p 9 J a p J a p = ( σ σ E p p 9 3 Fig. 3 Relationship between potential and electric field on sampling point [ σe s ( σ σ E ] = 0 p p 0. QMR 0 QMR ψ ψ ψ = [ σe s ( σ σ E ] p p φ m x m x 0 r m = b Ax m r 0

4 4 99 QMR 3 m x m r m 3 Intel Visual Fortran 0 Intel Core TM i GHz 4GB 3 Windows m 0 A m X 5 Ex Fig. 5 Variety of real part of Ex with frequencies 4 Fig. 4 Diagram of layer model Ex 5 6 Hz 0. 0 Hz Hz Ex Fig. 6 Variety of imaginary part of Ex with frequencies Hz 0. 0 Hz 7

5 Fig Hz Hz Fig. 8 Variety of residual norm with iterations for layer model 0. 0 Hz Variety of residual norm with iterations for layer model Hz Ω m 000 Ω m 00 Ω m 00 m 00 m 00 m 9 00 m 0 m 400 m Fig. 9 Diagram of block model 00 m 0 m 400 m 0 m 0 A m 0. 0 Hz Hz 45 Hz X Hz 0. 0 Hz 0 Hz 45 Hz

6 Hz Hz Fig. 0 Variety of residual norm with iterations for block model 0. 0 Hz Fig. Variety of residual norm with iterations for block model Hz 4 45 Hz 3 Fig. A 45 Hz Variety of residual norm with iterations for block model 45 Hz.

7 9 33 D. J. Global Geology Smith J T. Conservative modeling of 3D electromagnetic DENG Juzhi. Studies of three dimensional staggeredgrid fields Part II biconjugate gradient solution and an accelerator finite difference for controlled source audiofrequen cy magnetotelluric numerical simulation Ph. D dissertation D. Beijing School of Geophysics and Information Technology of China University of Geosciences 0. Newman G A Alumbaugh D L. Frequencydomain modeling of airborne electromagnetic responses using staggered finite differences J. Geophysical Prospecting J. Geophysics J SHEN Jinsong. Modeling of 3D electromagnetic responses in frequency domain by using staggeredgrid finite difference method J. Chinese Journal of Geophysics D. FCSEM J TANG Wenwu LIU Jianxin TONG Xiaozhong. Finite element forward modeling on line source of FCSEM for continuous conductivity J. Journal of Jilin University Earth Science Edition LIU Yunhe. Research on 3D controlled source electromagnetic method inversion using nonlinear conjugate gradients Ph. D dissertation D. Changchun College of Geoexploration Science and Technology of Jilin University J ZHANG Hui LI Tonglin DONG Ruixia. 3D Electromagnetic inversion by volume integral equation method based on current dipole source J. Journal of Jilin University Earth Science Edition J CHEN Hui DENG Juzhi TAN Handong et al. Study on divergence correction method in three dimensional magnetotelluric modeling with staggeredgrid finite difference method J. Chinese Journal of Geophysics J YANG Haibin LIANG Xiao CHANG Kai et al. Topography effects on HEM response and the correction method in frequency domain J. Global Geology J YANG Xiaohong HE Jishan. Numerical simulation of frequencydomain IP with FEM J. Journal of Jilin 6. University Earth Science Edition J XU Kaijun LI Tonglin. The forward modeling of threedimensional geoelectric field of vertival finite line source by finitedifference method J. Journal of Jilin University Earth Science Edition Freund R W Nachtigal N M. An implementation of the QMR method based on coupled twoterm recurrences J. SIAM Journal on Scientific Computing QMR D J LEI Linlin LIU Sixin FU Lei et al. Threedimensional WANG Fang. Restarted QMR methods for solving nonsymmetrical system master's degree thesis D. Hangzhou College of Science of Zhejiang University 009. forward simulation of seabed oil and gas exploration

8 4 93 A 4 rhs A. A. ( σe = s [ ( σ σ E ] p x ( i y ( j ( j k z ( k ( ( ( j k σ i j k Ex i j k σ i j k Ex i j k x i σ i j k Ey i j k σ i j k Ey i j k y j σ i j k p = Ez i σ i z k Ez i j k A. [ σ( i j k σ p ( i j k ] Exp( i j k [ σ( i j k σ p ( i j k ] Exp( i j k x ( i x ( i [ σ( i j k σ p ( i j k ] Eyp( i j k [ σ( i j k σ p ( i j k ] Eyp( i j k y ( j y ( j [ σ( i j k σ p ( i j k ] Ezp ( i j k [ σ( i j k σ p ( i j k ] Ezp ( i j k z ( k z ( k A. 4 ( σ φ = σ φ x x σ φ y y σ φ z z A. 3 A. 3 C 0 φ( i j k C φ( i j k C φ( i j k C 3 φ( i j k C 4 φ( i j k C 5 φ ( i j k C 6 φ ( i j k = rhs A. 4 [ ( ] σ i C 0 = j k [ x( i x( i ] x( i x i [ ] { } σ i j k [ x( i x( i ] x( i x i

9 94 33 C = C 3 = C 5 = [ ( ] ( j k [ ( ( ] σ i j k [ y( j y( j ] y( j y j [ ( ] σ( i j [ z( k ( ] { } σ i [ z( k z( k ] z k z k σ i j k [ y( j y( j ] y( j y j { } [ ( ( ] σ i j k [ x( i x( i ] x i x i [ ( ( ] σ i j k [ y( j y( j ] y j y j ( [ ( ( ] σ i j k [ z( k z( k ] z k z k C = C 4 = C 6 = [ z( k z( k ] [ x ] σ i j k z k [ ( ( ] i x i x i x i [ y ] [ ( ( ] σ i j k j y j y j y j [ z ] ( [ ( ( ] σ i j k k z k z k z k [ ] [ ] [ ] A. 4 x i x i y j y j z k z k A. 4 QMR

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