33 4 04 GLOBAL GEOLOGY Vol. 33 No. 4 Dec. 04 004 5589 04 04 096 09. 3006. 30070 P63. 35 A doi 0. 3969 /j. issn. 0045589. 04. 04. 0 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 30070 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 3 0 4 5 6 7 04063 040 0YQ05006009. 96. Email lilaoshizh@ 63. com
4 97 Krylov 8 Smith 8 996 9 0. 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
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 99 QMR 3 m x m r m 3 Intel Visual Fortran 0 Intel Core TM i3 3. 30GHz 4GB 3 Windows 7 3. 4 00 m 0 A 4 000 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 7 8 6 Ex Fig. 6 Variety of imaginary part of Ex with frequencies Hz 0. 0 Hz 7
90 33 7 Fig. 7 0. 0 8 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 3. 9 0 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 500 4 000 m 0. 0 Hz Hz 45 Hz 7 8 0 X 0 0. 0 Hz 0. 0 Hz 0 Hz 45 Hz
4 9 0 0. 0 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.
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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
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