APPLIED GEOPHYSICS, Vol.10, No.2 (June 2013), P. 134-144, 10 Figures. DOI: 10.1007/s11770-013-0376-2
Numerical modeling of the 2D time-domain transient electromagnetic secondary field of the line source of the current excitation* Liu Yun1, Wang Xu-Ben2♦, and Wang Yun1 Abstract: To effectively minimize the electromagnetic field response in the total field solution, we propose a numerical modeling method for the two-dimensional (2D) timedomain transient electromagnetic secondary field of the line source based on the DuFortFrankel finite-difference method. In the proposed method, we included the treatment of the earth-air boundary conductivity, calculated the normalized partial derivative of the induced electromotive force (Emf), and determined the forward time step. By extending upward the earth–air interface to the air grid nodes and the zero-value boundary conditions, not only we have a method that is more efficient but also simpler than the total field solution. We computed and analyzed the homogeneous half-space model and the flat layered model with high precision—the maximum relative error is less than 0.01% between our method and the analytical method—and the solution speed is roughly three times faster than the total-field solution. Lastly, we used the model of a thin body embedded in a homogeneous half-space at different delay times to depict the downward and upward spreading characteristics of the induced eddy current, and the physical interaction processes between the electromagnetic field and the underground low-resistivity body. Keywords: Time-domain transient electromagnetics, secondary field, DuFort-Frankel finitedifference method, numerical modeling.
Introduction Numerical modeling of the transient electromagnetic field are mostly in the frequency domain, and the electromagnetic field modeling are based on varying the single-frequency harmonics. Then, the results of the frequency domain are converted to the time domain by
using the inverse Fourier transform (Stoyer, 1976; Wang, 2003; Xiong, 2004, 2006; Liu, 2010; Piao, 1990). In frequency domain analysis, the frequency bandwidth of the step-pulse excitation is theoretically infinite, which requires to calculate more than the frequency of the electromagnetic field response to achieve high accuracy. In analog multidimensional problems, the computation cost is especially high. Furthermore, the conversion from
Manuscript received by the Editor August 25, 2012; revised manuscript received April 2, 2013. *The research is supported by the National High Technology Research and Development Program (863 Program) (2009AA06Z108). 1. State Key Laboratory of Ore Deposit Geochemistry, Institute of Geochemistry Chinese Academy of Sciences, Gui Yang 550002, China. 2. School of Geophysics, Chengdu University of Technology, Chengdu 610059, China. ♦Corresponding Author: Wang Xu-Ben (Email:
[email protected]) © 2013 APPLIED GEOPHYSICS. All rights reserved.
134
Liu et al. the frequency domain to time domain, often uses the G– S transform or the Guptasarma algorithm (Guptasarma, 1982; Ruan, 1996), both of which are numerical filtering algorithms and it is not easy to select a universal set of filtering coefficients. Therefore, it is necessary to study the direct time-domain transient electromagnetic response of numerical algorithms. Numerical modeling of the time-domain transient electromagnetic response of the line source is based on the fixed-source loop-transient electromagnetic method. Oristaglio (1984), Nabighian (1992), Niu (1992) and Wang (1993) obtained explicit solutions based on the two- and three-dimensional DuFort–Frankel finitedifference method and extended the surface field to the air boundary conditions to avoid midair processing, while obtaining analytical solutions for the deep underground boundary conditions of the homogeneous half-space field. In addition, Yan (2002) modeled and described the physical characteristics of the interaction of a transient electromagnetic field and an underground anomaly. Ruan (1995, 1996) transformed the twodimensional problem into a series of one-dimensional problems to solve and improve the calculation accuracy and speed. Nonetheless, when powered off, the early transient field of a constant current power line near the source is singular. Yan (2002) and Ruan (1995, 1996) used a cubic spline to smooth the transient field distortion, including the time delay. The error caused by the interpolation was included in each time step. Adhidjaja (1985) proposed a secondary field solution method for a two-dimensional medium and directly solved the underground secondary field produced by the inhomogeneous medium to avoid the singularity problem near the source of the primary field of the early transient electromagnetic field. This paper is based on the work of Adhidjaja (1985) and the papers of Oristaglio (1984) and Ruan (1995, 1996), which solve the line-source two-dimensional time domain transient electromagnetics of the secondary field by using the DuFort–Frankel finite-method. We also defined the weighted average abnormal electrical conductivity, and improved the treatment of the boundary conditions, the calculation of the partial derivatives of the induced electromotive force (Emf), and the advanced time step. The comparison of the analytical solution with the uniform half-space model shows that the maximum relative error is less than 0.01%. In the 200 × 100 grid model calculations, the delay time was 20 ms after 3051 time step iterations. The calculation time was 27 s on a 2.1 GHz PC.
Diffusion equation of the line source of a 2D transient electromagnetic secondary field Basic equation In the quasi-static approximation, the time-domain Maxwell equations are:
u E
P
wH , u H wt
Jl V E ,
(1)
where E is the electric field vector, H is the magnetic field vector, σ is the medium conductivity, μ is the magnetic permeability, and J l is the applied current density. For the line-source two-dimensional problem, the applied current source is along the y direction. In this case, the electromagnetic waves are in the TE mode, the electric field only has a y-direction component, and the magnetic field possesses x- and z-direction components; thus, E = Ey j, H = Hxi+Hzk, Jl = Jl j, where i, j, and k are the unit vectors of the x, y, and z directions, respectively. Consequently, the scalar diffusion equation of the electric field in the y direction is
2 E PV
wE wt
P
wJ l , wt
(2)
w2 w2 2 is the two-dimensional Laplace 2 wx wz operator. Equation (2) is the electric field diffusion equation for solving the two-dimensional problem of the line source using the total field solution method (Oristaglio et al., 1984). In the two-dimensional space of an inhomogeneous medium, σ' and σ0 represent the abnormal and the nearsource conductivity of the medium, respectively. The total conductivity of the medium is σ = σ0 + σ'. The total electric field is E = Ep + Es, where Ep is the electric field of the uniform half-space; that is, the primary field. Es is the secondary electric field that comes from the underground inhomogeneous medium. For the 2D uniform half-space of the line source, the primary electric field satisfies the equation
where 2
2 E p PV 0
wE p wt
P
wJ l . wt
(3)
From equations (2) and (3), the secondary electric field Es satisfies 2 Es PV
wEs wt
PV '
wE p wt
,
(4) 135
Transient electromagnetic secondary field where equation (4) is the diffusion equation for solving the two-dimensional problem of the line source using the secondary field solution method. Equation (4) cannot describe the field source for retrieving the secondary field Es that is directly coming from the underground inhomogeneous medium. The analytical solution of the primary electric field of the uniform half-space of the time partial derivatives is discussed in appendix A. To get the numerical solution of the diffusion equation (4), the 2D secondary field boundary and initial conditions of the line source are required.
the anomaly medium, the secondary field generated by the uneven dielectric body is zero. Therefore, the choice of the initial time to is t0 d
2 Es
0,
(5)
and the solution is
Es ( x, z , t )
z
S
xmax
³
xmin
Es ( x,0, t ') dx ' , z ( x x ')2 2
(6)
Initial conditions At the point of power failure, the early transient field conductivity near the line source and the uniform halfspace field conductivity are approximate equal. At the initial time to, when the primary field has not spread to 136
(7)
The DuFort–Frankel finite difference equation can readily and stably establish the time–space contact, while the requirements of the grid scale are not high (Oristaglio,1984; Nabighian, 1992; Ruan, 1995). The steps for using the DuFort-Frankel finite-difference equation are the following (for simplicity, Es is taken as E).
Gridding As shown in Figure 1, the two-dimensional study area is represented with a rectangular mesh. The target area forms a uniform grid, and the upper and lower borders gradually diffuse to a nonuniform grid. The surface grid extends to the air forming the air grid. Two infinite line sources are placed along the y direction to make the twodimensional loop that does not touch the ground and from which passes a positive and negative current. *
x y
Adhidjaja (1985) used adjacent time gates to absorb the boundary conditions, which resulted in lack of measurements for several early time gates. When the boundaries Γ2, Γ3, and Γ4 are sufficiently far away from the line source and the target area, we propose that the secondary electric field of the target area that is generated in the inhomogeneous medium has little effect on the boundaries. Thus, for all delay times, the values at the far boundary can be set to zero.
[min( 'x, 'z )]2 ,
DuFort–Frankel finite-difference equation
where z = z0, and x' [xmin, xmax]. The integral is obtained numerically (He, 2002).
Side boundaries Γ2 (x = xmin) and Γ4 (x = xmax), and bottom boundary Γ3 (z = zmax)
2
where ∆x and ∆z are the grid scales near the source (Ruan, 1995).
Boundary and initial conditions Air boundary Γ0 (z0 = -z2) As shown in Figure 1, the regional grid is extended to the air boundary Γ0, and the electric field of the ground –air boundary Γ1 is calculated (z1 = 0) using a five-point DuFort–Frankel explicit difference scheme. The electric field values on the air boundary Γ0 can be obtained from the values of the ground–air boundary Γ1. In the case of the quasi-static approximation, Es in the air satisfies the Laplace equation
PV 0
z
+I
-I
*
Air Earth
*
z0 z=0 z
zi * x(xmin)
x
xj
*
zmax xmax
Fig.1 Finite-difference mesh of the study area.
Spatial difference The second-order finite-difference numerical solution of partial differential equations is in essence a secondorder central difference approximation. In this paper, we use the definition of the second-order central difference in the calculations relative to the conventional formula of the Green dimensionality reduction method (Oristaglio, 1984; Nabighian,1992; Yan, 2002; Ruan et al., 1996). The delay time of the nth time step is n(∆t) and the node Ei, j is E ni, j. Using equation (4), the grid equation is
Liu et al. rewritten as
w 2 Ein, j
wx 2
w 2 Ein, j wz 2
PV ' i , j (
wE p wt
)in, j
PV i , j
wEin, j wt
.
(8)
In Figure 2, the node Eni, j and the four adjacent nodes E , E ni+1, j, E ni, j–1, and E ni, j+1 constitute the five-point DuFort–Frankel explicit difference scheme.
wEin, j
Ein, j 1 Ein, j 1
wt
2't
ı i , j +1
ı i, j ı i', j E
ı i', j +1
n i , jí1
Ein, j ı i +1, j +1
¨zi
wx
wE wz
n i, j 2
P min(V i , j ) 4
Ein, j 1 Ein, j 'x j 1
and the first-order backward
Ein, j Ein, j1 'xj
; thus, the second-order
2'zi 1 2'zi 1 Ein1, j Ein1, j 2Ein, j ). ( 'zi 'zi 1 'zi 'zi 1 'zi 'zi 1 (10)
For P
(13)
2ri ,xj 'x j 1 n 'x 1 4r i , j n 1 Ei , j ( Ei , j 1 j Ein, j 1 ) 1 4r i , j 1 4r i , j 'x j 'x j
Ein, j 1
2 ( Ein, j Ein, j ) 'x j 'x j 1
n i, j
[min( 'x, 'z )]2 .
Fig.2 Five-point difference grid position.
2'x j 1 2'x j 1 ( Ein, j 1 Ein, j 1 2Ein, j ). 'x j 'x j 1 'x j 'x j 1 'x j 'x j 1 (9) Similarly, we also obtain 2
(12)
By substituting the equations (8) - (11) into the equation (7), we get the secondary field of the time-domain transient electromagnetic field based on the DuFort–Frankel differential equation (DuFort and Frankel, 1953; Birtwistle, 1968) as
central difference is wE
.
The solution of the DuFort-Frankel difference equation
difference is Ein, j
n i, j 2
2
¨zi +1
ı i'+1, j +1
The first-order forward difference is expressed as
2
't d
n i , j+1
¨x j E n ¨x j +1 i +1, j
Ein, j
Ein, j 1 Ein, j 1
Ein, j
The time step ∆t gradually increases with time delay and is calculated with the following equation
E
ı i +1, j ı i'+1, j
(11)
Meanwhile, The E field of time t is obtained by linear interpolation (Yan, 2002). That is
n i–1, j
Einí1, j
.
(
wE p
n i, j
) , of equation (8), it can be calculated wt by solving in equation (A-1) (Appendix A). For calculating the conductivity σ i, j and the abnormal conductivity σ' i, j of node E ni, j, the reader can refer to appendix B.
Time difference In the DuFort–Frankel discrete problem, the time derivation of E ni, j is carried out using the first-order central difference (Oristaglio, 1984; Yan, 2002)
2ri ,z j 1 4r i , j
(
'zi 1 n 'z Ei 1, j i Ein1, j ) 'z i 'z i
2 't V ' i , j n Pi , j , 1 4r i , j V i , j
(14)
where
'x j
ri ,xj
and r i , j
'x j 'x j 1 2
,
'z i
't , ri ,z j PV i , j 'x j 'x j 1
ri ,xj ri ,z j 2
'zi 'zi 1 , 2 't , PV i , j 'zi 'zi 1
.
The calculation steps of the DuFort–Frankel differential equation for solving the secondary field of the 2D time-domain transient electromagnetic field are the following: Step 1: We built a grid for the model space, and input the grid conductivity to determine the initial time and the initial time step. Step 2: Set the initial boundary value to zero when t = t0. Step 3: t = t + ∆t. 137
Transient electromagnetic secondary field Step 4: Iterate the DuFort–Frankel differential equation and calculate the field values of the odd (i + j = odd) grid points at time t. Step 5: When the time delay meets the output conditions, insert in the cubic spline the odd grid point field values at time t and calculate the output grid point field values. Step 6: Extend upward the known surface values of the odd node points at time t to generate the air boundary, the lateral boundary, and the bottom boundary with zero value. Step 7: t = t + ∆t. Step 8: Iterate the DuFort–Frankel differential equation and calculate the field values of the even (i + j = even) grid points at time t. Step 9: When the time delay meets the output conditions, insert the field values of the grid points at time t in the cubic spline and output the grid point field values. Step 10: Extend upward the known surface values of the even grid points at time t to form the air boundary, the lateral boundary, and the bottom boundary with zero value. Step 11: Return to step 3.
The calculation of induced electromotive for the total field and the secondary field Using equation (14), we derive the secondary field Es of the line source of the 2D time-domain transient electromagnetic field. Then, we calculate the primary field E p , the total field E, the normalized induced electromotive force Emfp, the normalized secondaryfield-induced electromotive force Emf s and the normalized total-field-induced electromotive force Emf. The normalized induced electromotive force is defined as the first derivative of the vertical magnetic induction intensity with respect to time
Emf
wBz wt
wE . wx
(15)
The line source of the 2D time-domain primary field E p at the uniform half-space can be calculated analytically. When the power is off, from equation (3), the primary field Ep of the uniform half-space follows wE p the diffusion equation 2 E p PV 0 0 and the wt analytical solutions can be obtained with equation (A-2） in appendix A. The total field of the line source of the 2D time-domain transient electromagnetic field is E = Ep + Es. The normalized primary-field-induced electromotive force Emfp has a numerical solution and an analytical solution. If 138
Ni , j 1
Ni, j
x j 1 x j
, ( x j 1 x j )( x j 1 x j 1 ) x j 1 x j 1 2 x j , ( x j x j 1 )( x j x j 1 )
and
N i , j 1
x j 1 x j ( x j 1 x j )( x j 1 x j 1 )
,
then the numerical solution of the first-field-induced electromotive force is
Emf s
N i , j 1 Ei , j N i , j Ei , j N i , j 1 Ei , j 1 ,
(16)
where E = Ep. The analytical solution of Emfp at the surface (z = 0) is given by equation (A-3) of appendix A. Furthermore, the analytical solution is typically used for verifying the accuracy and reliability of the numerical solution. The numerical solution of the normalized secondaryfield-induced electromotive force is
Emfs = Ni, j–1Ei, j + Ni, jEi, j + Ni, j+1Ei, j+1,
(17)
where E = Es. The numerical solution of the normalized total-fieldinduced electromotive force is Emf = Emfp + Emfs.
Model calculations Firstly, we compare the numerical simulation algorithm with the theoretical analytical method and the total field method of uniform half-space. Secondly, we calculate the total field of the horizontally layered model. Lastly, we use the algorithm to model the low-resistivity anomaly and describe the physical characteristics of the interaction of the electromagnetic fields and the low-resistivity body. In the following model calculations, the infinite difference gridding is shown in Figure 1 and the specific parameters are listed in Table 1. The initial time t0 and the initial time step ∆t0 are given by equations (7) and (13)， respectively. The time step is increasing with the time delay. In this paper, for 1 –1000 iterations, ∆t = ∆t0; for 1001–2000 iterations, ∆t = 10∆t0; and for 2001 iterations and higher, ∆t = 50 ∆t0.
The uniform half-space model Figure 3 shows the uniform half-space section model, where the resistivity is 100 Ωm, the positive and negative
Liu et al. current lines are located at −200 m and 200 m from the surface. The current intensity is 1 A, the dot-space is 10 m, and the time delay is 0.1 ms, 1 ms, and 3 ms.
field DuFort–Frankel method in the model simulations. Figure 4 shows the comparison of the results calculated with the analytical method and the 2D numerical method. The purpose of this exercise is to study the change in the electric field and the normalized induced electromotive force at the surface. The horizontal axis shows the position of the measuring points. The vertical axis in Figures 4a and 4c is the electric field, and in Figures 4b and 4d is the normalized induced electromotive force. From Figure 4, we see that the results of the analytical method and the secondary field method agree. In the model calculations, we compare the fitting accuracy of the three methods at the same conditions and the same time step. For a time delay of 0.1 ms, after 315 iterations, the total field method converged after 1.7 s in contrast to the 1.5 s of the secondary field method, and the maximum relative error between the total and the analytical method is 15%; however, between the secondary field method and the analytical method is less than 0.01%. When the iterations increased to 1218 for a time delay of 1 ms, the total method converged after 4.2 s with a maximum relative error 17%, whereas, the secondary field method needed 3.9 s and had a relative error less than 0.01%. Moreover, when increasing the time step in the secondary field method, the calculation speed increased three times more than the total field method and had a relative error less than 1%.
Table 1 Difference meshing parameters Vertical (z)
Step (m) 240 120 60 30 15 10 15 30 60 120 240
Node (i) 0~70 70~78 78~84 84~89 89~93 93~100
Step 10 15 30 60 120 240
400 m
100 ȍ m
Fig.3 Uniform half-space model.
We use the line-source 2D uniform half-space analytical method, the total field DuFort–Frankel method (Oristaglio et al. 1984), and the secondary
1000 500 0 -500
0 -4
-1000
0 X (m)
1000
-8 -2000
2000
0.16
E 1 ms Analytical Total Secondary
40 20 0 -20
-1000
0 X (m)
1000
2000
Emf 1 ms Analytical Total Secondary
0.12 0.08 0.04 0 -0.04
-40
-1000 -1500 -2000
60
Emf 0.1 ms Analytical Total Secondary
4 Emf (9Âm-2)
-60 -2000
-1000
0 X (m)
1000
2000
-0.08 -2000
-1000
0 X (m)
1000
2000
(a) The electric field curve for a (b) The normalized induced electromotive (c) The electric field curve of a (d) The normalized induced electromotive time delay of 0.1 ms. force for a time delay of 0.1 ms. time delay of 1 ms. force for a time delay of 1 ms.
Fig.4 Comparison of the results of the analytical and the numerical method.
-z (m)
-500 -15
-1000
-2000 -2000 -1500 -1000 -500
-1000 -1500
-1500 0 X (m)
500
1000 1500 2000
Emf t = 1 ms
0 01 -0.
-500
(b)
0.03 0
E t = 1 ms
-z (m)
0
15 0
(a)
01 -0.
0
E (9Âm-1)
8
E 0.1 ms Analytical Total Secondary
E (9Âm-1)
1500
Emf (9Âm-2)
Horizontal (x) Node (j) 1~10 10~14 14~19 19~25 25~33 33~169 169~177 177~183 183~188 188~192 192~201
-2000 -2000 -1500 -1000 -500
0 X (m)
500
1000 1500 2000
139
Transient electromagnetic secondary field E t = 3 ms
(d)
3 0 -3
-500 -3
3
-1000 -1500
-1500 -2000 -2000 -1500 -1000 -500
0 500 X (m)
0.003 0 -0.003
03 0.0
-1000
0
-z (m)
-500
Emf t = 3 ms
0
-0.00 3
0
-z (m)
(c)
-2000 -2000 -1500 -1000 -500
1000 1500 2000
0 500 X (m)
1000 1500 2000
Fig.5 Contour maps of the uniform half-space and normalized induced electromotive force at different time delays. (a) and (c) show the electric field at 1 ms, respectively, and (b) and (d) show the normalized induced electromotive force at 3 ms, respectively.
10 m and the time delay time is 10-2–102 ms, and the measuring point P is located in the middle of the loop.
Horizontally layered model Figure 6 is the model of the H-type geoelectric section. The conductivity of the first layer is 100 Ωm with a height of 200 m and the conductivity of the secondary layer is 20Ω m with a height of 200 m. The conductivity of the third layer is 100 Ωm. The positive and negative current lines are located at −200 m and 200 m from the surface. The current intensity is 1 A, the dot-space is
We used the total field method and the secondary field method to simulate the H-type geoelectric model section. The calculation results are shown in Figure 7, where the horizontal axis is the time delay. The vertical axis is the normalized total field of the induced electromotive force in Figure 7a, the normalized primary field of the induced electromotive force in Figure 7b, and the normalized secondary field in Figure 7c.
100
10
10
1
1
0.1 0.01 0.001
400 m
xP U2 100ȍm
Total Secondary
1E-005
Fig.6 Model of the H-type geoelectric section.
0.1
0
0.1 0.01 0.001
-0.2
Total Secondary
1E-006 0.01
0.1
1
10
100
-0.1
Total Secondary
1E-005
1E-006
h2 = 200 m
U3 100ȍm
0.0001
0.0001
h1 = 200 m
200 m
U2 200ȍm
Emfs (9Âm-2)
100
Emfp (9Âm-2)
Emf (9Âm-2)
Nabighian (1992) studied the induced eddy diffusion characteristics of the ground transient electromagnetic field. He pointed out that the induced eddy current forms “smoke rings” downward and outward as a function of time delay. Figure 5 shows cross-sections of the contour maps of the uniform half-space electric field and the normalized induced electromotive force. The horizontal axis is the position of the measuring points and the vertical axis is the depth. From this figure we can see that the transient field spreads downward and out from the line source with time delay agreeing with the previous conclusions.
-0.3 0.01
0.1
1
10
t (ms)
t (ms)
(a) Total field.
(b) Primary field.
100
0.01
0.1
1
10
100
t (ms)
(c) Secondary field.
Fig.7 Comparison of the results calculated by the total field method and secondary method for the H-type model.
As shown in Figure 7, the total field curve and the secondary field curve reflect the anomaly characteristics of the H-type model. After 8147 iterations at 100 ms, the 140
secondary field method converged after 47 s and the total field method after 76 s. Besides, the respective relative errors were less than 5%.
Liu et al.
Anomalous body model
400 m
Figure 8 shows a model of the anomaly, where the background resistivity is 100 Ω m and the resistivity of the anomaly is 1 Ωm. The top surface of the body of the anomaly is 200 m from the ground and 800 m from the bottom with a width 20 m. The positive and negative current lines are located −200 m and 200 m from the surface. The current intensity is 1A, the dot-space is 10 m, and the delay times are 1 ms, 5 ms, and 20 ms. We used the secondary field method in the numerical simulations. After 3051 iterations at 20 ms, the calculation time was 27 s. The calculation results are shown in Figure 9. The horizontal axis is the position of the measuring points and the vertical axis is the depth. Figures 9a, 9c, and 9e show the contours of the total electric field. Figures 9b, 9d, and 9f show the contours of
-Z (m) -Z (m)
-Z (m)
0.8
0 500 1000 1500 2000 X (m)
(d) Secondary electric field contours at 5 ms.
-2000 -2000 -1500 -1000 -500
-0. 1
-z (m)
-z (m)
(b) Secondary electric field contours at 1 ms.
-1000 -1500
0.8
-2000 -2000 -1500 -1000 -500
-500
5 02 -0.
0 500 1000 1500 2000 X (m)
0 500 1000 1500 2000 X (m)
Es t = 20 ms
0 1 -0.
5
-2000 -2000 -1500 -1000 -500
(e) Total electric field contours at 20 ms.
0 0.8
5
-z (m)
0 500 1000 1500 2000 X (m)
Es t = 5 ms
-1000 -1500
-0.4
-1000 -1500
2
-500
-1500
-0.1 0
-500
-2.5
0 1
E t = 20 ms
0
(c) Total electric field contours at 5 ms.
Es t = 1 ms
-2000 -2000 -1500 -1000 -500
E t = 5 ms
-2000 -2000 -1500 -1000 -500
0 500 1000 1500 2000 X (m)
1
the secondary electric field.
-1000 -1500
-1000
20 m
Fig.8 Diagram of the geoelectric section of the anomalous body.
-500
(a) Total electric field contours at 1 ms.
-500
U0 100ȍm
0.2
-15
-1500
0
600 m
0
-1000
-2000 -2000 -1500 -1000 -500
U1 1ȍm
0 2
-500
E t = 1 ms 10 0
0
200 m
200 m
-0.1
5 02 -0.
0 500 1000 1500 2000 X (m)
(f) secondary electric field contours at 20 ms.
Fig.9 Second electric field method calculations for different delay times.
Comparing Figures 9a, 9c, and 9e with Figures 5a and 5c, we see that when there is a low-resistivity anomaly in a uniform medium, the diffusion of the electric field is distorted. The gradient contours densify near the lowresistivity anomaly as it becomes larger, which suggests that the low-resistivity anomaly concentrates the electric field and the diffusion speed of the induced eddy current slows down. With time delay and the attenuation of the primary field, the induced secondary field of the lowresistivity anomaly gradually spreads outward from the center of the anomaly body. These numerical modeling results are anticipated to be very useful in practical applications. Figure 10 shows the normalized induced electromotive force versus the
location of the measuring points. In Figures 10a, 10b, and 10c, the solid line represents the ground measured curves of Emf and the dashed line is the forward curve of the primary field Emfp. Figure 10d shows the surface of the normalized induced electromotive force at different moments and the dashed line is negative. From Figure 10 we see that because of the time delay in the transient electromagnetic response of the lowresistivity body, the intersection of the solid and dashed lines corresponds to the horizontal center position of the low-resistivity body that does not move with time delay. Thus, in practical applications, the combined analysis of the surface and forward curves can determine the position of the low-resistivity body. 141
Transient electromagnetic secondary field 1 ms
0.04 0
5 ms
0.012 Emf Emfp
-1000
0 X (m)
1000
2000
0.004
-0.004 -2000
(c)
20 ms
0.008
0
-0.04 -0.08 -2000
(b)
0.008 Emf (9Âm-2)
Emf (9Âm-2)
0.08
0.012 Emf Emfp
0.01 Emf Emfp
0.004 0
-1000
0 X (m)
1000
2000
-0.004 -2000
(d)
0.001 Emf (9Âm-2)
(a)
Emf (9Âm-2)
0.12
0.0001 1E-005
-1000
0 X (m)
1000
2000
1E-006 -2000
15 ms 20 ms 30 ms 50 ms
-1000
0 X (m)
1000
2000
Fig.10 The normalized induced electromotive force curves at different time delays. (a) The normalized induced electromotive force curves at 1 ms. (b) The normalized induced electromotive force curves at 5 ms. (c) The normalized induced electromotive force curves at 20 ms, and the multiple time channel normalized induced electromotive force curves.
Conclusions In this paper, we improved the theoretical framework and numerical simulation techniques of the line source of the two-dimensional time-domain transient electromagnetic method. Because the calculation of the primary field can be directly obtained by the analytical method, the secondary field simulation method is faster and more accurate than the total field simulation method, and avoids the singularity problems near the source. By numerically simulating the inhomogeneous medium, we can directly analyze the diffusion characteristics of the total and the secondary field, which improves data analysis and facilitates exploration efforts. Furthermore, the line-source 2D transient electromagnetic problem is based on the response of the infinite field source. The direct time-domain transient electromagnetic response will be addressed in a follow-up study.
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Appendix A Oristaglio (1982) presented an analytical solution for the uniform half-space primary field based on the partial derivatives of time:
V i, j
2
Ep
Ni, jV i, j Ni1, jV i1, j Ni, j 1V i, j 1 Ni1, j 1V i1, j 1 (A-1)
where z is the depth, I is the current intensity, T
t
is
PV 0 the normalized time, and F (u) is the integral of Dawson and its solution is
F(u ) |
u a3u 3 a5u 5 a7u 7 , 1 b2u 2 b4u 4 b6u 6 b8u 8 1 ,a 7 7
37 , a5 84
b6
17 , and b8 105
13 , b2 105
31 ,b 28 4
2 ze
z2 T
S R2
[
1 T
1 2
2 x F( xT
1 2
½ 1 1 ° )( 2 )]¾ T R ° ¿
1 I x z2 2 E rfc( zT ). SV 0 R 4 2
(A-2)
Assuming that the coordinates of the measuring points are (x j, z i), the coordinates of the line source are (x0, z0), and x = xj – x0 and z = zi – z0. Erfc (u) is the complementary error function that can be solved numerically (He, 2002). Oristaglio (1984) also presented the analytical solution for the normalized primary field electromotive force at the surface (z = 0) of the uniform half-space
for
a3
R 2 2 2 I ° z x 2z e T ) ®( 2 T R2 SV 0 ° R ¯
43 , 70
26 . 105
2
The primary field Ep of the uniform half-space satisfies wE the following diffusion equation 2 E p PV 0 p 0, wt The analytical solution was given by Oristaglio (1982)
Emf p
x PI 4 x2 T [ e (1 ) 1]. 2S x PV 0 x 2 T
(A-3)
The analytical solution can be typically used to verify the accuracy and reliability of the numerical solution.
Appendix B n
The conductivity σ i, j of node Ei, j is the weighted average value of the conductivities σi, j, σi+1, j, σi, j+1, and σi+1, j+1 of the four surrounding nodes. The anomalous conductivity σ' i, j is the weighted average value of the conductivities σi, j, σi+1, j, σi, j+1, and σi+1, j+1 of the four surrounding nodes. If
Ni, j
'i, j '
,
N i 1, j
' i 1, j '
,
N i , j 1
' i , j 1 '
, N i 1, j 1
' i 1, j 1 '
,
where
'i, j
' i , j 1
'zi 'x j ,
' i 1, j
'zi 'x j 1 , ' i 1. j 1
'zi 1'x j ,
'zi 1'x j 1 ,
then, 143
Transient electromagnetic secondary field
V i, j
Ni, jV i, j Ni1, jV i1, j Ni, j 1V i, j 1 Ni1, j 1V i1, j 1,
V ' i, j
Ni 1, jV i'1, j Ni 1, j 1V i'1, j 1.
(B-3)
(B-1)
V ' i, j Ni, jVi', j Ni1, jVi'1, j Ni, j1Vi', j1 Ni1, j1Vi'1, j1, (B-2) When the nodes are located at the surface, the conductivity of the air and the surface is not discontinuous. Then, the surface grid conductivity needs to be accordingly modified
' 'zi1('xj 'xj1), V i, j and
144
Ni1, jV i1, j Ni1, j 1V i1, j 1,
Liu Yun is Assistant Professor at the State Key Laboratory of Ore Deposit Geochemistry. His research mainly focuses on numerical modeling and inversion imaging technology of geoelectromagnetic fields.
