Three-dimensional Modeling of Transient Electromagnetic ... · D TEM modeling: the integral...

Three-dimensional Modeling of Transient Electromagnetic Responses of Water-bearing Structures inFront of a Tunnel Face

Shucai Li1, Huaifeng Sun1,*, XuShan Lu2 and Xiu Li21Geotechnical and Structural Engineering Research Center, Shandong University, Jinan, China 250061

Email: [email protected] of Geology Engineering and Geomatics, Chang’an University, Xi’an, China 710054

*Corresponding author

ABSTRACT

We present a finite-difference time-domain (FDTD) approach for the simulation of three-

dimensional (3-D) transient electromagnetic diffusion phenomena for the detection of water-

bearing structures in front of a tunnel face. The unconditionally-stable du Fort-Frankel

difference discrete method is used and an additional fictitious displacement current is introduced

into the diffusion equations to form explicit difference equations. We establish a new excitation

loop source which considers Maxwell’s equations in source media to overcome the limitations of

the precondition that the near-surface resistivity of the model is uniform in the well-known 3-D

FDTD algorithm demonstrated by Wang and Hohmann in 1993. The algorithm has the abilityto simulate any type of transmitting current waveforms and arbitrarily complicated earth

structures. A trapezoidal wave is used to simulate a step-off source. The fictitious permittivity is

allowed to vary during the computation to ensure the stability and optimize an efficient time

step. Homogeneous full-space models with different resistivities are simulated and compared

with the analytical solutions to demonstrate the algorithm. Transient electromagnetic (TEM)

responses of a tunnel with and without a water-filled vertical fault in front of the tunnel face are

simulated and compared. 3-D models with water-filled fault and karst caves in front of a tunnel

face are simulated with different parameters considered.

Introduction

The transient electromagnetic (TEM) method has

been widely used in near-surface geophysical explora-

tion such as unexploded ordnance detection (Pasion

et al., 2007; Doll et al., 2010; Asten and Duncan, 2012),

mining exploration and monitoring (Xue et al., 2013),

metallic ore exploration (Yang and Oldenburg, 2012;

Xue et al., 2012), mapping contaminant migration

(Pellerin et al., 2010), and groundwater investigation

(Vrbancich, 2009; Ezersky et al., 2011). The TEM

responses of low resistivity targets are thoroughly

researched, including the very mature one-dimensional

(1-D) modeling for layered earth (Nabighian, 1988;

Christiansen et al., 2011), 2.5 dimensional modeling

(Abubakar et al., 2006; Streich et al., 2011; Xiong, 2011)

and three-dimensional (3-D) responses (Adhidjaja and

Hohmann, 1989; Wang and Hohmann, 1993; Zhdanov

and Tartaras, 2002; Newman and Commer, 2005;

Viezzoli et al., 2008) for simple and complex models.

The 3-D modeling of TEM for complex models has been

implemented in the past two decades. Meanwhile,

interpretation methods have also been widely researched

such as the Conductivity Depth Imaging (CDI) or

Conductivity Depth Transform (CDT) (Macnae et al.,

1991; Huang and Rudd, 2008), Born approximation

(Christensen, 1995), 1-D, 2-D, 3-D modeling and

inversion (Zhdanov and Tartaras, 2002; Newman and

Commer, 2005; Haber et al., 2007; Cox et al., 2010;

Yang and Oldenburg, 2012; Burschil et al., 2012), pseudo-

seismic migration of electromagnetic data (Zhdanov and

Portniaguine, 1997; Li et al., 2005; Xue et al., 2007; Li

et al., 2010; Xue et al., 2011) and principal component

analysis (Kass and Li, 2012). A novel application of

TEM for the prediction of water-bearing structures in

front of a tunnel face has been proposed in recent years

(Xue et al., 2007; Sun et al., 2011; Sun et al., 2012). It

is important to ensure safety during tunnel construc-

tion, as unforeseen water in-rush threatens the safety of

construction workers and tunnel structures. However,

the response characteristics of TEM methods in such

cases have not been investigated. The use of TEM in

tunnel-face applications is still based on ground TEM

theories. The application of TEM for the detection of

water-bearing structures in front of a tunnel face is

complex, therefore it is important to understand and

13

JEEG, March 2014, Volume 19, Issue 1, pp. 13–32 DOI: 10.2113/JEEG19.1.13

Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

model the response characteristics to improve TEM

detection capabilities.

Generally, four methods are commonly used in 3-

D TEM modeling: the integral equations (IE) method,

finite-element method (FEM), finite-difference time-

domain (FDTD) and finite volume (FV) method. IE is

widely used in 3-D modeling of magnetotelluric (MT)

and frequency-domain electromagnetic data. For TEM

modeling, SanFilipo and Hohmann (1985) gave a time-

domain IE solution for the TEM response of a 3-D body

in a half-space model. Newman et al. (1986) improved

the former research and defined the responses of a 3-D

body in a layered earth. Subsequent studies simulated

numerous EM problems using the IE method (Mendez-

Delgado et al., 1999; Abubakar et al., 2006) and

developed various improvements of this method (Hur-

san and Zhdanov, 2002; Zhdanov et al., 2006; Singer,

2008; Endo et al., 2008; Avdeev and Knizhnik, 2009;

Zaslavsky et al., 2011). For the FEM method, the

greatest advantage is the capability and flexibility in

modeling arbitrarily complicated earth structures, espe-

cially using tetrahedral unstructured grids. Two-dimen-

sional modeling of EM data using FEM has been

discussed in the literature (Coggon, 1971) and is still a

popular topic (Li and Key, 2007; Li and Dai, 2011). For

modeling in three dimensions, Pridmore et al. (1981)

introduced an investigation of FEM modeling for EM

data. Variants of FEM methods were developed such as

spectral-finite-element (Martinec, 1999), iterative finite-

element time-domain (FETD) method (Um et al., 2012),

adaptive higher order FEM (Schwarzbach et al., 2011),

vector finite-element method (VFEM) (Li et al., 2011)

and parallel algorithm (Puzyrev et al., 2013). FV

methods in modeling EM problems were established

by the Geophysical Inversion Facility of University of

British Columbia (UBC-GIF) (Haber and Ascher, 2001)

and were used in many inversion problems (Haber et al.,

2004; Haber et al., 2007). Finite-difference (FD) or

FDTD methods in EM modeling succeeded in simulating

2-D problems (Goldman and Stoyer, 1983; Oristaglio

and Hohmann, 1984), but failed in simulating 3-D

problems (Adhidjaja and Hohmann, 1989) in the

1980’s. After that, the research on solving 3-D

problems with FDTD was discussed, but developed

very slowly until Wang and Hohmann (1993) presented

a 3-D finite-difference time-domain solution (Wang et

al., 1995; Maaø, 2007), and soon parallel algorithms

were presented (Commer and Newman, 2004). Inver-

sion techniques were also developed based on this

algorithm (Wang et al., 1994; Newman and Commer,

2005). Although the number of iteration steps is large,

it can be solved easily by parallel computing as FDTD

has an intrinsic parallelism. The spectral Lanczos

decomposition method (SLDM) described by Druskin

et al. (1994, 1999) is also a novel method in solving 3-

D EM problems.

We use FDTD in our simulation of 3-D TEM

diffusion phenomena in tunnels. Our work improved

upon the basic theory based on the well-known FDTD

solution by Wang and Hohmann (1993). The uncondi-

tionally stable du Fort-Frankel difference discrete

method is used, and an additional fictitious displace-

ment current is introduced into the diffusion equations

to form explicit difference equations. However, the

solution from Wang and Hohmann (1993) uses the

analytical EM field of a homogeneous half-space model

at a very short time after the transmitting current cut off

as the initial conditions. This implies a precondition that

the near-surface resistivity of the model is uniform. This

applies to most ground TEM surveys with an overbur-

den, but is not applicable to TEM detection in tunnels.

For a tunnel case, the area of concern is less than

100 meters in front of the tunnel face. To obtain higher

resolutions of the conductivity depth profile, a high

frequency is usually used. We are not interested in the

responses at very late time corresponding to the

conductivity far from the tunnel face. Furthermore,

the complexity of a full space with an excavating tunnel

cavity leads to no analytical solution. As a result, the

resistivity near the tunnel face cannot be set uniformly

and the analytical response of the EM field cannot be

used as the initial conditions of our FDTD method. We

introduce the loop current source into Maxwell’s

equations by means of current density according to

Ampere circuital theorem. Primary fields of TEM are

included in the entire modeling and the excitation source

no longer depends on assuming the near surface as a

homogeneous half-space model in the initial conditions.

The algorithm applies to arbitrarily complex models

with non-uniform surface resistivity distributions. A

trapezoidal waveform with a very short ramp time is

used to simulate a step-off source. To maintain stability

and optimize an efficient time step, we define the time

steps using an empirical formula, also from Wang

and Hohmann (1993), and then change the fictitious

permittivity during the computation to satisfy the

Courant-Friedrichs-Lewy (CFL) condition.

To demonstrate our algorithm, we compute

several homogeneous full-space models with 3-D FDTD

and compare the results with analytical solutions. In

addition, we simulate the TEM responses of a tunnel

with and without a water-filled vertical fault in front of

the tunnel face using our 3-D FDTD algorithm. We

then simulate 3-D models with water-filled fault

and karst caves in front of a tunnel face. Different

parameters, such as fault thickness, fault size, and

resistivity contrast between the rock mass and water, are

compared.

14

Journal of Environmental and Engineering Geophysics

Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Methodology

Maxwell’s Equations in a Source-free Medium

In lossy media such as the earth, displacementcurrents can be ignored for TEM exploration. Maxwell’s

equations under the quasi-static approximation in linear,

isotropic, lossy, non-magnetic and source-free media can

be expressed as follows (Kaufman and Keller, 1983):

+|E~{LB

Lt, ð1:aÞ

+|H~sE, ð1:bÞ

+:E~0, ð1:cÞ

+:H~0, ð1:dÞ

where E and H are the electric field intensity and

magnetic field intensity, respectively, B is magnetic

induction, s is conductivity of the earth, and t is time.

However, the conduction current does not exist in

a lossless medium such as the air in the excavated tunnelcavity. Thus, Eq. (1.b) should be modified as:

+|H~e0LE

Lt, ð2Þ

where e0 is the magnetic permeability of a vacuum.

Diffusion equations of E and H can be derived

from Eq. (1):

+2E{msLE

Lt~0, ð3:aÞ

+2H{msLH

Lt~0: ð3:bÞ

While homogeneous wave equations are derived if

considering Eq. (2) as follows:

+2E{meL2E

Lt2~0, ð4:aÞ

+2H{meL2H

Lt2~0: ð4:bÞ

Equations (3) and (4) are approximated from

homogeneous damped wave equations of E and H as

follows:

+2E{meL2E

Lt2{ms

LE

Lt~0, ð5:aÞ

+2H{meL2H

Lt2{ms

LH

Lt~0: ð5:bÞ

The differences lie in ignoring s. However, if omitting ein Maxwell’s equations, a time derivative of the electric

field will not exist. This will cause difficulty in forming

an explicit FDTD discretization.

For the problem of transient electromagnetic

responses of water-bearing structures in front of a tunnel

face, the basic electromagnetic propagation equations are

different in the excavated tunnel cavity and the rock

mass. The coupling of the electromagnetic wave in the

interface of the two media should also be considered.

To form the explicit difference schemes required by

FDTD, we introduce the fictitious displacement current

into Maxwell’s equations. Former research has proved

that the solution of diffusion equations can be replaced

by the solution of damped wave equations under certain

conditions (Oristaglio and Hohmann, 1984).

Equation (1.b) and Eq. (2) are rewritten as follows:

+|H~cLE

LtzsE, ð6Þ

where c is the fictitious permittivity.

Both the conduction current and the displacement

current are considered. The real permittivity is replaced

by the fictitious one. The conductivity should be 0 in the

area of the tunnel cavity. However, it is set to a number

small enough for stability, then we can use a unified

equation to solve the problem.

The difference forms of Maxwell’s equations in

Cartesian coordinates and the Yee grids are the same as

Wang and Hohmann (1993) and we are not going to

repeat them here. Uniform grids have second-order

accuracy, while non-uniform grids only have first-order

accuracy because the electric field would no longer be

located at the center of two adjacent magnetic field points.

However, we choose non-uniform discretization of the

earth and specify grids and coordinates to form a model

large enough with a minimum number of cells (Fig. 1). Yu

et al. (2006) recommended the ratio between two adjacent

cells should be less than 1.2, and the accuracy of the result

can be quite satisfying despite the loss of the second-order

accuracy. The ratio of the maximum length to the

minimum length of grids is limited to less than 20 to

control the error resulting from the effects of the deviation

from a uniform model. However, all the limitations are

specified based on previous studies (Yu et al., 2006).

The low-frequency approximation involving Eq.

(1.d) explicitly in Wang and Hohmann’s (1993) research

calculates Bz using the value of Bx and By step by step

upward from the bottom to the surface of the model.

However, for detection in tunnels, the rock mass behind

the transmitter and receiver loops should also be

considered in the numerical simulation. In the detection

of water-bearing structures in front of a tunnel face by

TEM, the transmitter and receiver loops are put on the

tunnel face, which lies in the middle of the discretization

model as shown in Fig. 2. The excavated tunnel cavity is

from the front to the middle part of the tunnel, which is

surrounded by rock mass. The water-bearing structures

may be located in front of the tunnel face between the

middle to the back in the model.

15

Li et al.: 3-D TEM Modeling in Tunnels

Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

We also use Bx and By to calculate Bz. In our case,

the transmitter and receiver loops are in the middle of

the model in the Z direction (as shown in Fig. 1). We

first step Bz from the back to the middle in the Z

direction, and then step Bz from the front to the middle

to finish a single calculation iteration step of Bz. The

difference iterative equations are different in the two

steps. The first step is the same as that in Wang and

Hohmann’s (1993) research and we give the second step

difference scheme as follows:

Bnz1=2z (iz1=2,jz1=2,kz1)

~Bnz1=2z (iz1=2,jz1=2,k)

{Dzk

Bnz1=2x (iz1,jz1=2,kz1=2)

{Bnz1=2x (i,jz1=2,kz1=2)

Dxi

266664 ð7Þ

z

Bnz1=2y (iz1=2,jz1,kz1=2)

{Bnz1=2y (iz1=2,j,kz1=2)

Dyj

377775:

Maxwell’s Equations in a Source Medium

The approach of adding initial conditions in Wang

and Hohmann’s (1993) research does not apply here, as

there is no analytical solution for the models of

detection in tunnels. We introduce the source into

Maxwell’s equations. The excitation is implemented

through the source current term in Maxwell’s equation.

This type of excitation is broadly used in practice

because it does not cause reflection from the source

region (Yu et al., 2006). Hence, Eq. (6) becomes:

+|H~cLE

LtzsEzJs, ð8Þ

where Js is the current density of the source.

The source position on the Yee grids is shown in

Fig. 3. The source is added on the electrical field nodes.

The grey grids in Fig. 3 are the elements with sources and

should be processed separately. From Faraday’s law and

Ampere’s law, it is easy to integrate the magnetic field at

the center of each element. However, our low frequency

approximation has changed the calculation of Bz, using

Bx and By rather than the electric field.

We rewrite Eq. (8) into Cartesian coordinates as

follows:

LHz

Ly{

LHy

Lz~c

LEx

LtzsExzJs,

LHx

Lz{

LHz

Lx~c

LEy

LtzsEyzJs,

LHy

Lx{

LHx

Ly~c

LEz

LtzsEz:

8>>>>>><>>>>>>:

ð9Þ

The iterative difference scheme of the electric field can

be derived from Eq. (9) using a central difference scheme:

Figure 2. Schematic diagram of TEM detection in

tunnels. The transmitting loop is put on the tunnel face.In-loop or central-loop configuration is usually used

because of the narrow space on the tunnel face. Potential

water-bearing structures may be located in front of the

tunnel face.

Figure 1. Non-uniform discretization of the 3-D model.

The loop source is located at the center of the model and

the grid spacing in the X, Y and Z directions increaseswith fixed magnification coefficients to the neighboring

nearest one. Two times length of the loop at the center of

the model in the three directions are uniform.

16


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

http://library.seg.org/action/showImage?doi=10.2113/JEEG19.1.13&iName=master.img-000.jpg&w=189&h=190


Enz1x iz

1

2,j,k

� �~

2c{sDt

2czsDt:En

x iz1

2,j,k

� �z

2Dt

2czsDt:

Hnz1=2z iz

1

2,jz

1

2,k

� �{Hnz1=2

z iz1

2,j{

1

2,k

� �Dy

2664

{

Hnz1=2y iz

1

2,j,kz

1

2

� �{Hnz1=2

y iz1

2,j,k{

1

2

� �Dz

3775

{2Dt

2czsDtJnz1=2

s ,

ð10Þ

Enz1y i,jz

1

2,k

� �~

2c{sDt

2czsDt:En

y i,jz1

2,k

� �z

2Dt

2czsDt:

Hnz1=2x i,jz

1

2,kz

1

2

� �{Hnz1=2

x i,jz1

2,k{

1

2

� �Dz

2664

{

Hnz1=2z iz

1

2,jz

1

2,k

� �{Hnz1=2

z i{1

2,jz

1

2,k

� �Dx

3775

{2Dt

2czsDtJnz1=2

s :

ð11Þ

To simulate a step-off current source, we use a

trapezoidal waveform with a very short ramp time. A

linear ramp function is commonly used in TEM

acquisition instruments. However, this waveform has

four non-derivable points, marked with hollow circles in

Fig. 4(a). The FDTD computation requires a smooth

excitation function to minimize noise and shock effect

during the EM propagation in the grids (Taflove and

Hagness, 2005). We introduce a switching function to

reshape the current waveform. Raised and anti-raised

Figure 3. Schematic diagram of the position of the transmit-ting loop on the tunnel face. The outside thick solid line is the

edge of the tunnel side wall. The inside thick solid line with large

arrows is the transmitting loop. The grey grids in the figure are

the elements influenced by the source and should be processed

separately. The arrows represent the Ex or Ey as marked in the

figure. The open circles with crosses represent the Hz component.

Figure 4. Schematic diagram of (a) the trapezoidal

transmitting current and (b) the reshaped rising and ramp

edge by switching functions. t1 is the end time of the risingedge, t2 and t3 are the start and end time of the ramp edge,

respectively. The solid line in (b) uses the bottom x-axis

while the dashed line uses the top axis.

17


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/



cosine functions are used to smooth the rising edge and

the ramp, respectively. The two switching functions are:

U tð Þ~0 tv0

0:5 1{cos(pt=t1)½ � 0ƒtvt1

1 t1ƒt

8><>: , ð12Þ

and

U tð Þ~

1 tvt2

0:5 1zcospt

t3{t2

� �� t2ƒtvt3

0 t3ƒt

8>><>>: , ð13Þ

where U(t) is the current at different time t, t1 is the end

time of the rising edge, and t2 and t3 are the start and

end time of the ramp edge, respectively.

The non-differentiable points in the trapezoidal

waveform are reshaped by Eqs. (12) and (13) (see the

raise and ramp edge in Fig. 4(b)). This is a smooth

excitation function. To demonstrate that the frequency

spectrum characteristics do not change with the

switching functions, we transform the time domain

functions into frequency domain to compare the

amplitude and the phase (Fig. 5). The Fourier transform

can be found in Appendix A.

Stability and Boundary Conditions

The Courant–Friedrichs–Lewy (CFL) condition is

the most basic condition for FDTD problems. For a 3-

D problem, the CFL condition is (Taflove and Hagness,

2005):1ffiffiffiffiffifficmp Dtƒ

dffiffiffi3p , ð14Þ

where Dt is the time step and d is the minimum grid length.

Reshaping Eq. (14) yields the restriction of time

steps as follows:

Dtƒd

ffiffiffiffiffifficm

3

r: ð15Þ

We deduce from Eq. (15) that the time steps can be

appropriately increased by adjusting the value of the

Figure 5. Frequency spectrum characteristics compari-

son before and after applying the switching function. (a)

Amplitude and (b) phase. The time parameters are shown

in (a).

Figure 6. Grid meshing and the transmitting loop

position on the tunnel face in the numerical simulation

models. Each grid corresponds to a cube Yee grid. The

transmitting loop is 3-m by 3-m square. The size of the

tunnel face is 6-m by 6-m square. The numbers from 26 to

6 in both the X and Y directions are the numbering of thegrids on the tunnel face. Each grid node can be uniquely

determined by the numberings in the two directions.

18


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/



fictitious permittivity. This will decrease the number of

iterations and save calculation time. In fact, the values

of Dt and c are not independent. The purpose of

introducing fictitious displacement currents is to form

the explicit finite difference equations. The value of the

fictitious permittivity should be small enough to

maintain the diffusion characteristics. We specify the

time steps according to an empirical formula from Wang

and Hohmann (1993) and then calculate the fictitious

permittivity to satisfy the stability condition. In fact, we

can also get the correct solution if using the true

permittivity of the earth. The waving characteristic will

disappear at the very early time, leaving only diffusion

characteristics. However, this will cause the time steps to

be extremely small and the iteration number to be

extremely large. For example, the relative permittivity of

a type of limestone is 7; if the minimum grid length is

0.5 m, the maximum time step is 2.55e-9 seconds under

Courant’s condition. The calculation iterations number

for 1 millisecond in pure secondary field will be 392,157

without considering the refinement of the time meshing

to keep the diffusion phenomena at early times. Suitable

values of the fictitious permittivity are very important to

reduce the time cost.

We apply the Dirichlet boundary condition to

the difference equations. Tangential components of the

electric fields and vertical components of the magnetic

fields at the six faces of the model are set to zero. This

requires that the model must be large enough, requiring

non-uniform grids.

Numerical Examples

To test the feasibility and the effectiveness of our

methodology, we carried out several numerical simula-

tions. A group of homogeneous full-space models are

used to compare the FDTD result with the analytical

results to demonstrate the accuracy and validity of

the method. A model with only the tunnel cavity is

simulated to show the responses of the tunnel cavity,

which contains an anomalous body with high resistivity

in a homogeneous full space. A model with a vertical

water-filled fault in front of the tunnel face is also

simulated to show the responses of TEM to water-

bearing structures in front of a tunnel face.

The transmitting loop configuration and the grid

meshing on the tunnel face are shown in Fig. 6. All of

the related models in the paper use the same configu-

rations. The cross section of the tunnel face is 6-m by

6-m square and the transmitting loop is 3-m by 3-m

square, which is located at the center of the tunnel face.

The areas near the tunnel face are meshed with uniform

Figure 7. Contrast curves of the FDTD solution andanalytical solution for the homogeneous full-space models.

Figure 8. Contrast curves of the FDTD solution and

analytical solution for the homogeneous half-space

models. Left is a comparison between our algorithm andthe analytical solution with step current (solid circles) and

considering ramp time (hollow circles). Right is a

comparison of late time apparent resistivity between our

algorithm and the analytical solution.

19


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/



grids using a 0.5-m cube. We give numberings of the

grids on the tunnel face from 26 to 6 in both X and Y

direction for convenience in our result analysis.

Validation using Homogeneous Full-space Models and

a Half-space Model

In this part, three homogeneous full-space models

with resistivities of 1 V-m, 10 V-m and 100 V-m are

simulated for verification. We use the aforementioned

transmitting loop configurations with no tunnels added.

The applications of TEM in tunnels usually use the

central loop system because of the space limitation

previously described. We compare the LB/Lt at the

center of the loop between the analytical solution and

the FDTD solution in three dimensions (as shown in

Fig. 7). The analytical solution uses a step current while

Figure 9. Representative curves outside and inside the transmitting loop. (a) At the diagonal position and (b) not at the

diagonal position, both outside the transmitting loop. (c) At the diagonal position and (d) not at the diagonal position, both

inside the transmitting loop. The exact locations on the tunnel face are specified in each figure and can be located from the

numbering in Fig. 6. Each figure shows the X, Y and Z components. A dashed line indicates the received LB/Lt is negative.

The outside (a) and inside (c) points are along the diagonal position of the transmitting loop; the X and Y components

should be symmetric, as shown by the results. The X and Y components of points located inside (d) and outside (b) of the

transmitting loop, but off the diagonal are not symmetric.

20


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/


the FDTD modeling uses a trapezoidal current with a

1ms ramp time. The analytical solution and the FDTD

solution are in good agreement at late times.

A homogeneous half-space model with a 100 V-m

background is also used to test the algorithm. A comparison

of late time apparent resistivity is also given in Fig. 8.

Responses of the Tunnel Cavity

We simulate a model with only a tunnel cavity.

The tunnel can be considered as an anomalous body

with high resistivity inside a homogeneous full space.

This model is to show the TEM responses of the tunnel

cavity (as the schematic diagram shows in Fig. 9). The

background resistivity is 100 V-m. The conductivity of

the tunnel should be 0; however, this conflicts with the

stability conditions. In general practice, the conductivity

of the air can be replaced with a small number (Um

et al., 2012). In our simulation, the tunnel resistivity is

105 times the highest resistivity of the other objects,

which is 107 V-m in this case.

We give the decay curve responses of X, Y and Z

components at different positions on the tunnel face (as

shown in Fig. 9). Figure 6 gives the positions of the

receiver points. The X, Y and Z components are all

simulated. The four plots in Fig. 9 are representative

curves. Figures 9(a)–(b) are outside the transmitting

loop, while Figs. 9(c)–(d) are inside the loop. Also,

Figs. 9(a)–(c) are located symmetrically at the diagonal

of the transmitting loop. The X and Y components at

positions both inside and outside the transmitting loop

Figure 10. XZ cross-sections of Ex across the axis in the Y direction of the tunnel at different decay times: (a) 10 ms, (b)

100 ms, (c) 500 ms, and (d) 1 ms after turn off. The white rectangle corresponds to the tunnel location. The electric fields

change rapidly at the interface between the rock mass (resistivity 100 V-m) and the tunnel.

21


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/


should be symmetric. The X and Y components in

Figs. 9(a)–(c) completely overlap; the dashed lines

indicate the data are negative. The plots with X and Y

components not at the diagonal position of the

transmitting loop, such as Figs. 9(b)–(d), are not

symmetric. The Z component response is significantly

larger than the X and Y components. The inside loop

position (Fig. 7(d)) has its Y component all positive,

while the X component has some negative values at

early time. Similar phenomena occur in its symmetrical

position at point (1, 21).

The electric fields XZ cross-sections are shown in

Fig. 10. The contours of Ex at four decay times (10 ms,

100 ms, 500 ms and 1 ms after turn off) are drawn. The

XZ cross-section is across the axis in the Y direction of

the tunnel. The white rectangle in each plot locates the

edge of the tunnel. The diffusion process can be

identified through the four decay times; the shape

changes of the contours in front of the tunnel face and

their values decrease in the computation areas. The

interfaces between the tunnel and the rock mass have

rapid changes of resistivity and the electric fields focus in

Figure 11. Representative curves on the tunnel face. (a) and (b) are outside the transmitting loop, while (c) and (d) are

inside the transmitting loop. The positions of each receiver point are given in the figure and can be located in Fig. 6. Each

plot shows the X, Y and Z components. A dashed line indicates the received LB/Lt is negative. The decay curves’

characteristics are quite different from that without a water-filled fault, as shown in Fig. 9.

22


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/


those areas. The influences of the tunnel cavity on the

TEM survey are obviously demonstrated by Fig. 10.

Modeling of a Water-filled Fault in Front of a

Tunnel Face

The purpose of our methodology is to simulate the

TEM responses of water-bearing structures in front of a

tunnel face. Geophysicists can analyze the response

characteristics of water-bearing structures, such as

water-filled faults and underground rivers, using the 3-

D FDTD modeling method. The geophysical anomalies

of these water-bearing structures have a lower resistivity

than the background rock mass. We present a model

with a background resistivity of 100 V-m, tunnel

resistivity of 107 V-m, and a water-filled fault resistivity

of 1 V-m located 10 meters ahead of the tunnel face, as

shown in Fig. 11. The size of the water-filled fault is 50-

m by 50-m in the X and Y directions and 5-m in the Z

direction.

With the same configurations as aforementioned,

we simulate and give the decay curves at different

receiving points of the three components in Fig. 11.

The decay curves’ characteristics of this model are very

different from the model in Fig. 9 with only the tunnel

cavity—especially the X and Y components. The

dashed lines in the plots indicate negative values.

Figure 12. XZ cross-sections of Ex across the axis in the Y direction of the tunnel at different decay times: (a) 10 ms, (b)

100 ms, (c) 500 ms, and (d) 1 ms after turn off. In each plot, the left white rectangle corresponds to a water-filled fault while

the right white rectangle corresponds to the tunnel location. The electric fields change rapidly at the interface of the rock

mass and the tunnel. The contours are also quite different from that without a water-filled fault, as shown in Fig. 10.

23


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/


Two direction reversals of the LB/Lt response exist both

inside and outside transmitting loop survey points on

the tunnel face. The responses on the diagonal

positions of the transmitting loop are the same, while

the responses not on the diagonal positions are

different for the X and Y components. The low

resistivity fault can be identified clearly by the Z

components. We also give the XZ cross-sections of Ex

across the axis in the Y direction of the tunnel at

different decay times (Fig. 12). The two rectangles in

each plot mark the tunnel cavity (right) and the water-

filled fault (left). The electric fields focus on the low

resistivity fault at early times, and then change to focus

on the interfaces between the rock mass and the fault,

also between the rock mass and the tunnel cavity. The

values change rapidly.

Figure 13. Response curves of the X (a) and Z (b) components at point (21, 21) on the tunnel face with different fault

thicknesses. The distance between the tunnel face and the water-filled fault is 10 m. The dashed line in (a) indicates the

received LB/Lt is negative.


thicknesses. The distance between the tunnel face and the water-filled fault is 20 m. The dashed line in (a) indicates the

received LB/Lt is negative.

24


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/



To investigate the influence of the thickness of the

vertical water-filled fault, we present a group of

comparisons with different thicknesses and different

distances between the tunnel face and the fault. A

schematic diagram is given in Fig. 13. Six models with

three thicknesses and two distances are simulated. We

present the response curves for the X and Z components

at point (21, 21) in Fig. 13 and Fig. 14 corresponding to

distances 10 m and 20 m, respectively. The responses are

quite clear as low resistivity targets. Notice that the X

components have two LB/Lt reverse points in these models

with the water-filled fault in front of the tunnel face.

To investigate the influence of the distance

between the tunnel face and the vertical fault, we

present four different models with a fixed thickness, 5 m,

for the water-filled fault. The distances are 10 m, 20 m,

30 m and 50 m. The horizontal and vertical responses on

the tunnel face are shown in Fig. 15. Notice the LB/Lt

Figure 15. Response curves of the X (a) and Z (b) components at point (21, 21) on the tunnel face with a fixed fault

thickness (5 m) and different distances between the tunnel face and the water-filled fault. The dashed line in (a) indicates

the received LB/Lt is negative.

Figure 16. Response curves of the X (a) and Z (b) components at point (21, 21) on the tunnel face with a fixed distance

(70 m) and different fault size, as given in the figure. The dashed line in (a) indicates the received LB/Lt is negative.

25


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/



reverse phenomenon, and it differs slightly from the

curves in Fig. 13 and Fig. 14.

To investigate the influence of the water-filled fault’s

size on the TEM response, we fix the distance between the

tunnel face and the fault at 70 m and give three models

with different fault size. The horizontal and vertical

responses on the tunnel face are shown in Fig. 16. The X

components have only one LB/Lt reverse phenomenon

within the simulation time. The LB/Lt reverse time is much

later than the models with shorter distances, such as the

first reverse time in Fig. 14 or Fig. 15. Also, the reverse

times are very close to each other for the models with a

distance of 70 m. The Z component responses of different

anomalous body sizes exhibit similar characteristics for

Figure 17. Response curves of the X (a) and Z (b) components at point (21, 21) with different resistivity ratios (5, 10

and 100). In this comparison, the basic model is similar with that in Fig. 13 while the distance is 10 m and the fault size is

50-m by 50-m with a thickness of 5 m. The resistivity of the rock mass and the water are also different than that in Fig. 13,

but use the ratio shown in the legend. The dashed line in (a) indicates the received LB/Lt is negative.


dips. In this comparison, the length of the fault is fixed at 50 m. The distance between the tunnel face and the middle right

edge of the water-filled fault is 50 m; the fault size is 50-m by 50-m with a thickness of 2 m. The dashed line in (a) indicates

the received LB/Lt is negative.

26


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/



the low resistivity target, with the response increasing as

target size increases.

To investigate the influence of resistivity differ-

ences on the TEM response, we compare three groups

with different resistivity ratios (5, 10, and 100). The

distance is 10 m and the fault size is 50-m by 50-m with a

thickness of 5 m. The responses curves are given in

Fig. 17. The anomalous response becomes more obvious

when the resistivity ratio between the rockmass and

water-bearing structure is large. We noticed that when

the resistivity ratio reduces to 5, the Z component

response is very close to the response of the model with

only a tunnel cavity; meanwhile, there still exists notable

differences in the X component.

To investigate the influence of a dipping water-

filled fault on the TEM response, we compare three

different dips (26.5u, 45u and 56.5u) with the distance

fixed at 50 m and thickness fixed at 2 m (Fig. 18). In

Fig. 19, the fault length varies, but the fault projection

is fixed at 50 m (as shown in Fig. 18). The response is

dependent on the size of the fault. When the fault

length is fixed (Fig. 18), the LB/Lt reverse phenomenon

in the X component disappears when the dip angle is

less than 45u. However, when the fault projection is

fixed (Fig. 19), the responses in the X component are

quite different with different dips. Differences in the X

and Z components in both Fig. 18 and Fig. 19 are very

small.


dips. In this comparison, the length of the fault varies while its projection is fixed at 50 m, as shown in Fig. 18. The distance

between the tunnel face and the middle right edge of the water-filled fault is 50 m; the fault thickness is also 2 m. The

dashed line in (a) indicates the received LB/Lt is negative.

Figure 20. Schematic diagram of (a) water-filled and (b) semi-water-filled karst cave in front of a tunnel face.

27


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/



Modeling of a Water-filled Karst Cave in Front of a

Tunnel Face

In karst areas, a karst cave is frequently encoun-

tered during tunnel construction. We present the

responses of water-filled and semi-water-filled karst

caves with different parameters. The schematic diagrams

of water-filled and semi-water-filled karst caves in front

of a tunnel face are shown in Fig. 20.

To investigate the influence of karst cave size on

the TEM response, we first simulate the water-filled

karst cave models (Fig. 21), and then simulate a

comparison model with a semi-water-filled karst cave

Figure 21. TEM response curves for different karst cave sizes in front of the tunnel face. The distance between the tunnel

face and the right side of the karst cave is 30 m. We present two different karst cave sizes for comparison, a 10-m cube and

30-m cube. The dashed line in (a) indicates the received LB/Lt is negative. The Z component (b) for the 10-m cube karst

cave is quite weak and cannot be distinguished from the tunnel-only model decay curve on a log-log coordinate plot.

Figure 22. TEM responses curves of a water-filled and semi-water-filled karst cave in front of the tunnel face. The karst

cave size is a 30-m cube. The distance between the tunnel face and the right side of the karst cave is 30 m. The dashed line

in (a) indicates the received LB/Lt is negative. The Z component (b) responses of a semi-water-filled karst cave is weakerthan a full water-filled karst cave. However, the abnormal response of the X component (a) is evident.

28


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/



(Fig. 22). The Z component responses are nearly

identical to the tunnel-only cavity model when the

water-filled karst cave size is 10-m by 10-m by 10-m or

smaller (Fig. 21), and the semi-water-filled karst cave

size is 30-m by 30-m by 30-m or smaller (Fig. 22). A

more distinctive response is seen in the X component,

where the reverse phenomenon is not observed in either

water-bearing structure until the cave size reaches 30-m

by 30-m by 30-m. Hence, we deduce that the LB/Lt

reverse phenomenon is related to the size ratio of the

water-bearing structures in the X and Z components.

To investigate the influence of resistivity differ-

ences on the TEM response for a karst model, we

present four comparisons with resistivity ratios of 100,

10, 5 and 2.5. The simulation results are given in Fig. 23.

For karst cave models, the Z component response

curves are almost the same as the tunnel-only cavity

model when the resistivity ratio is less than 10; however,

differences in the X component can still be clearly

identified, even with a small resistivity ratio of 2.5 (see

inset in Fig. 23(a)).

Results and Discussion

Our modified FDTD algorithm for modeling 3-D

TEM problems in tunnels considers both source and

source-free media. The diffusion phenomenon of TEM

is simulated. From the results of the numerical modeling

(Figs. 13-17 and Figs. 18-23), we obtain some useful

TEM characteristics for identifying water-bearing struc-

tures in front of a tunnel face as follows: 1) the X

component responses have two LB/Lt reverse phenom-

ena when the water-bearing structure in front of a tunnel

face is sufficiently large and the distance between the

tunnel face and the structures is sufficiently small; 2) the

Z component responses have clear characteristics if

the low resistivity targets are sufficiently large; and 3)

the response of the X component is more sensitive than

that of the Z component if the target’s size is not very

large or the target is located far from the tunnel face.

If we take the influence of the tunnel cavity into

consideration, the conductivity of air should be set to zero.

However, this will bring instability into our equation even

if an extremely short time step is given. We use a resistivity

large enough to replace the infinite air conductivity to

conquer this instability. Thus, the equations can be solved.

However, the time step must be very short as the phase

velocity of the electromagnetic wave in air is much faster

than that in a very lossy medium, and therefore it causes a

very long computing time. We suggest two possible ways

to overcome this problem. First, a parallel algorithm is

recommended based on the current methodology. Second,

the alternating-direction implicit finite-difference time-

domain (ADI-FDTD) algorithm for time domain electro-

magnetic modeling may be helpful to save time in

computation (Taflove and Hagness, 2005; Yu et al., 2006).

Figure 23. Response curves of the X (a) and Z (b) components at point (21, 21) on the tunnel face with different

resistivity ratios. The karst cave size is a 30-m cube. The distance between the tunnel face and the right side of the karst

cave is 30 m. The dashed line in (a) indicates the received LB/Lt is negative. The Z component responses are quite weak and

cannot be distinguished from the tunnel-only model decay curve on a log-log coordinate plot when the resistivity difference

ratio is less than 10. In contrast, the X component responses are much larger than the tunnel-only model, even when theresistivity difference ratio is up to 2.5.

29


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/


Conclusions

We have developed a finite difference time domain

(FDTD) approach for the simulation of 3-D TEM

diffusion phenomena in tunnels. The method applies to

TEM detection in tunnels with water-bearing structures

in front of a tunnel face. Arbitrarily complex models can

be simulated by the methodology described in this

paper. All three components of the TEM response both

inside and outside the transmitting loop can be

simulated. This method can also be applied in airborne

TEM modeling for arbitrary complex models.

Acknowledgments

The authors would like to thank the helpful suggestions

and discussions from Tsili Wang during the development of

the modified FDTD modeling algorithm in three dimensions.

The authors want to thank the Associate Editor, Antonio

Menghini, and two anonymous reviewers for their valuable

comments and useful suggestions that helped to improve the

presentation of this paper. This research is funded by the

National Program on Key Basic Research Project (973 Program)

under the grants 2013CB036002 and 2014CB046901, the

National Natural Science Foundation of China (NSFC) under

the grants 51139004 and 41174108.

References

Abubakar, A., Van Den Berg, P.M., and Habashy, T.M.,

2006, An integral equation approach for 2.5-dimension-

al forward and inverse electromagnetic scattering:

Geophysical Journal International, 165(3) 744–762.

Adhidjaja, J.I., and Hohmann, G.W., 1989, A finite-difference

algorithm for the transient electromagnetic response of a

three-dimensional body: Geophysical Journal Interna-

tional, 98(2) 233–242.

Asten, M.W., and Duncan, A.C., 2012, The quantitative

advantages of using B-field sensors in time-domain EM

measurement for mineral exploration and unexploded

ordnance search: Geophysics, 77S(4) B137–B148.

Avdeev, D., and Knizhnik, S., 2009, 3D integral equation

modeling with a linear dependence on dimensions:

Geophysics, 74(5) F89–F94.

Burschil, T., Wiederhold, H., and Auken, E., 2012, Seismic

results as a-priori knowledge for airborne TEM data

inversion — A case study: Journal of Applied Geophys-

ics, 80, 121–128, doi:10.1016/j.jappgeo.2012.02.003.

Christensen, N., 1995, 1D imaging of central loop transient

electromagnetic soundings: Journal of Environmental

and Engineering Geophysics, 1(A) 53–66.

Christiansen, A.V., Auken, E., and Viezzoli, A., 2011,

Quantification of modeling errors in airborne TEM

caused by inaccurate system description: Geophysics,

76(1) F43–F52.

Coggon, J., 1971, Electromagnetic and electrical modeling by

the finite element method: Geophysics, 36(1) 132–155.

Commer, M., and Newman, G., 2004, A parallel finite-difference

approach for 3D transient electromagnetic modeling with

galvanic sources: Geophysics, 69(5) 1192–1202.

Cox, L.H., Wilson, G.A., and Zhdanov, M.S., 2010, 3D

inversion of airborne electromagnetic data using a moving

footprint: Exploration Geophysics, 41(4) 250–259.

Doll, W.E., Gamey, T.J., Holladay, J.S., Sheehan, J.R.,

Norton, J., Beard, L.P., Lee, J.L.C., Hanson, A.E.,

and Lahti, R.M., 2010, Results of a high-resolution

airborne TEM system demonstration for unexploded

ordnance detection: Geophysics, 75(6) B211–B220.

Druskin, V., and Knizhnerman, L., 1994, Spectral approach to

solving three-dimensional Maxwell’s diffusion equations

in the time and frequency domains: Radio Science, 29(4)

937–953.

Druskin, V., Knizhnerman, L., and Lee, P., 1999, New spectral

Lanczos decomposition method for inductionmodeling

in arbitrary 3–D geometry: Geophysics, 64(3) 701–706.

Endo, M., Cuma, M., and Zhdanov, M.S., 2008, A multigrid

integral equation method for large-scale models with

inhomogeneous backgrounds: Journal of Geophysics

and Engineering, 5(4) 438–447.

Ezersky, M., Legchenko, A., Al-Zoubi, A., Levi, E., Akkawi,

E., and Chalikakis, K., 2011, TEM study of the

geoelectrical structure and groundwater salinity of the

Nahal Hever sinkhole site, Dead Sea shore, Israel:

Journal of Applied Geophysics, 75(1) 99–112.

Goldman, M.M., and Stoyer, C.H., 1983, Finite-difference

calculations of the transient field of an axially symmetric

Earth for vertical magnetic dipole excitation: Geophys-

ics, 48(7) 953–963, doi:10.1190/1.1441521.

Haber, E., Oldenburg, D.W., and Shekhtman, R., 2007, Inversion

of time domain three-dimensional electromagnetic data:


Haber, E., and Ascher, U., 2001, Fast finite volume simulation

of 3D electromagnetic problems with highly discontin-

uous coefficients: Siam Journal On Scientific Comput-

ing, 22(6) 1943–1961.

Haber, E., Ascher, U.M., and Oldenburg, D.W., 2004,

Inversion of 3D electromagnetic data in frequency and

time domain using an inexact all-at-once approach:

Geophysics, 69(5) 1216–1228.

Huang, H., and Rudd, J., 2008, Conductivity-depth imaging

of helicopter-borne TEM data based on a pseudolayer

half-space model: Geophysics, 73(3) F115–F120.

Hursan, G., and Zhdanov, M.S., 2002, Contraction integral

equation method in three-dimensional electromagnetic

modeling: Radio Science, 37(6) 1089 pp.

Kass, M.A., and Li, Y., 2012, Quantitative analysis and inter-

pretation of transient electromagnetic data via principal

component analysis: Geoscience and Remote Sensing,

IEEE Transactions, 50(5) 1910–1918, doi:10.1109/TGRS.

2011.2167978.

Kaufman, A.A., and Keller, G.V., 1983, Frequency and

transient soundings: Elsevier Science Ltd.

Li, J.H., Zhu, Z.Q., Liu, S.C., and Zeng, S.H., 2011, 3D

numerical simulation for the transient electromagnetic

field excited by the central loop based on the vector

30


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

finite-element method: Journal of Geophysics and

Engineering, 8(4) 560 pp.

Li, X., Qi, Z.P., Xue, G.Q., Fan, T., and Zhu, H.W., 2010,

Three dimensional curved surface continuation image

based on TEM pseudo wave-field: Chinese Journal of

Geophysics, 53(12) 3005–3011, doi:10.3969/j.issn.0001-

5733.2010.12.025.

Li, X., Xue, G.Q., Song, J.P., Guo, W.B., and Wu, J.J., 2005,

An optimize method for transient electromagnetic field

to wave field conversion: Chinese Journal of Geophys-

ics, 48(5) 1185–1190.

Li, Y., and Key, K., 2007, 2D marine controlled-source

electromagnetic modeling: Part 1–An adaptive finite-

element algorithm: Geophysics, 72(2) A51–A62.

Li, Y., and Dai, S., 2011, Finite element modelling of marine

controlled-source electromagnetic responses in two-

dimensional dipping anisotropic conductivity structures:


Maaø, F., 2007, Fast finite-difference time-domain modeling

for marine-subsurface electromagnetic problems: Geo-

physics, 72(2) A19–A23.

Macnae, J.C., Smith, R., Polzer, B.D., Lamontagne, Y., and

Klinkert, P.S., 1991, Conductivity-depth imaging of

airborne electromagnetic step-response data: Geophys-

ics, 56(1) 102–114, doi:10.1190/1.1442945.

Martinec, Z., 1999, Spectral–finite element approach to three-

dimensional electromagnetic induction in a spherical

earth: Geophysical Journal International, 136(1) 229–250.

Mendez-Delgado, S., Gomez-Trevino, E., and Perez-Flores,

M.A., 1999, Forward modelling of direct current and low-

frequency electromagnetic fields using integral equations:


Nabighian, M.N., 1988. Electromagnetic Methods in Applied

Geophysics — Theory (Volume 1), Society of Explora-

tion Geophysicists, Tulsa, OK.

Newman, G.A., and Commer, M., 2005, New advances in

three dimensional transient electromagnetic inversion:


Newman, G.A., Hohmann, G.W., and Anderson, W.L., 1986,

Transient electromagnetic response of a three-dimen-

sional body in a layered earth: Geophysics, 51(8)

1608–1627, doi:10.1190/1.1442212.

Oristaglio, M.L., and Hohmann, G.W., 1984, Diffusion of

electromagnetic fields into a two-dimensional earth: A

finite-difference approach: Geophysics, 49(7) 870–894.

Pasion, L.R., Billings, S.D., Oldenburg, D.W., and Walker,

S.E., 2007, Application of a library based method to

time domain electromagnetic data for the identification

of unexploded ordnance: Journal of Applied Geophys-

ics, 61(3–4) 279–291, doi:10.1016/j.jappgeo.2006.05.006.

Pellerin, L., Beard, L., and Mandell, W., 2010, Mapping

Structures that Control Contaminant Migration using

Helicopter Transient Electromagnetic Data: Journal of

Environmental and Engineering Geophysics, 15(2) 65–75.

Pridmore, D., Hohmann, G., Ward, S., and Sill, W., 1981, An

investigation of finite-element modeling for electrical

and electromagnetic data in three dimensions: Geophys-

ics, 46(7) 1009–1024.

Puzyrev, V., Koldan, J., de la Puente, J., Houzeaux, G.,

Vazquez, M., and Cela, J.M., 2013, A parallel finite-

element method for three-dimensional controlled-source

electromagnetic forward modelling: Geophysical Jour-

nal International, 193(2) 678–693.

SanFilipo, W.A., and Hohmann, G.W., 1985, Integral equation

solution for the transient electromagnetic response of a

three-dimensional body in a conductive half-space:

Geophysics, 50(5) 798–809, doi:10.1190/1.1441954.

Schwarzbach, C., Borner, R., and Spitzer, K., 2011, Three-

dimensional adaptive higher order finite element simu-

lation for geo-electromagnetics-a marine CSEM exam-

ple: Geophysical Journal International, 187(1) 63–74.

Singer, B.S., 2008, Electromagnetic integral equation approach

based on contraction operator and solution optimiza-

tion in Krylov subspace: Geophysical Journal Interna-

tional, 175(3) 857–884.

Streich, R., Becken, M., and Ritter, O., 2011, 2.5D controlled-

source EM modeling with general 3D source geometries:

Geophysics, 76(6) F387–F393.

Sun, H., Li, X., Li, S., Qi, Z., Su, M., and Xue, Y., 2012,

Multi-component and multi-array TEM detection in

karst tunnels: Journal of Geophysics and Engineering,

9(4) 359–373.

Sun, H., Li, S., Su, M., Xue, Y., Li, X., Qi, Z., Zhang, Y., and

Wu, Q., 2011, Practice of TEM tunnel prediction in

Tsingtao subsea tunnel: in Expanded Abstracts, 81st

Annual International Meeting, Society of Exploration

Geophysicists, 761–765.

Taflove, A., and Hagness, S.C., 2005. Computational electro-

dynamics: The finite-difference time-domain method, 3rd

ed.: Artech House, Boston.

Um, E.S., Harris, J.M., and Alumbaugh, D.L., 2012, An

iterative finite element time-domain method for simulat-

ing three-dimensional electromagnetic diffusion in earth:


Um, E.S., Alumbaugh, D.L., Harris, J.M., and Chen, J.P.,

2012, Numerical modeling analysis of short-offset

electric-field measurements with a vertical electric dipole

source in complex offshore environments: Geophysics,

77(5) E329–E341.

Viezzoli, A., Christiansen, A.V., Auken, E., and Sorensen, K., 2008,

Quasi-3D modeling of airborne TEM data by spatially

constrained inversion: Geophysics, 73(3) F105–F113.

Vrbancich, J., 2009, An investigation of seawater and sediment

depth using a prototype airborne electromagnetic instru-

mentation system — A case study in Broken Bay,

Australia: Geophysical Prospecting, 57, 633–651, doi:10.

1111/j.1365-2478.2008.00762.x.

Wang, T., and Hohmann, W.G., 1993, A finite-difference, time-

domain solution for three-dimensional electromagnetic mod-

eling: Geophysics, 58(6) 797–809, doi:10.1190/1.1443465.

Wang, T., Tripp, A.C., and Hohmann, G.W., 1995, Studying the

TEM response of a 3-D conductor at a geological contact

using the FDTD method: Geophysics, 60(4) 1265–1269.

Wang, T., Oristaglio, M., Tripp, A., and Hohmann, G., 1994,

Inversion of diffusive transient electromagnetic data by a

conjugate-gradient method: Radio Sci., 29(4) 1143–1156.

31


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Xiong, B., 2011, 2.5D forward for the transient electromag-

netic response of a block linear resistivity distribution:

Journal of Geophysics and Engineering, 8(1) 115 pp,

doi:10.1088/1742-2132/8/1/014.

Xue, G., Bai, C., and Li, X., 2011, Extracting the virtual reflected

wavelet from TEM data based on regularizing method:

Pure and Applied Geophysics, 169(7) 1269–1282.

Xue, G.Q., Cheng, J.L., Zhou, N.N., Chen, W.Y., and Li, H.,

2013, Detection and monitoring of water-filled voids

using transient electromagnetic method: A case study

in Shanxi, China: Environmental Earth Sciences, 1–8,

doi:10.1007/s12665-013-2375-2.

Xue, G.Q., Qin, K.Z., Li, X., Li, G.M., Qi, Z.P., and Zhou, N.N.,

2012, Discovery of a large-scale porphyry molybdenum

deposit in Tibet through a modified TEM exploration

method: Journal of Environmental and Engineering

Geophysics, 17(1) 19–25, doi:10.2113/JEEG17.1.19.

Xue, G.Q., Yan, Y.J., and Li, X., 2007, Pseudo-seismic wavelet

transformation of transient electromagnetic response in

engineering geology exploration: Geophysical Research

Letters, 34, L16405, doi:10.1029/2007GL031116.

Xue, G.Q., Yan, Y.J., Li, X., and Di, Q.Y., 2007, Transient

electromagnetic S-inversion in tunnel prediction: Geo-

physical Research Letters, 34, L18403, doi:10.1029/

2007GL031080.

Yang,D.,andOldenburg,D.W.,2012,Three-dimensionalinversion

of airborne time-domain electromagnetic data with applica-

tions to a porphyry deposit: Geophysics, 77(2) B23–B34.

Yu, W., Mittra, R., Su, T., Liu, Y., and Yang, X., 2006. Parallel

finite-difference time-domain method, Artech House, USA.

Zaslavsky, M., Druskin, V., Davydycheva, S., Knizhnerman,

L., Abubakar, A., and Habashy, T., 2011, Hybrid finite-

difference integral equation solver for 3D frequency

domain anisotropic electromagnetic problems: Geo-

physics, 76(2) F123–F137.

Zhdanov, M.S., and Tartaras, E., 2002, Three-dimensional

inversion of multitransmitter electromagnetic data based

on the localized quasi-linear approximation: Geophys-

ical Journal International, 148(3) 506–519.

Zhdanov, M.S., and Portniaguine, O., 1997, Time-domain

electromagnetic migration in the solution of inverse prob-

lems: Geophysical Journal International, 131(2) 293–309.

Zhdanov, M.S., Lee, S.K., and Yoshioka, K., 2006, Integral

equation method for 3D modeling of electromagnetic

fields in complex structures with inhomogeneous back-

ground conductivity: Geophysics, 71(6) G333–G345,

doi:10.1190/1.2358403.

APPENDIX A

FOURIER TRANSFORM OF THE CURRENT

WAVEFORM BEFORE AND AFTER SWITCHING

PROCESS

The current waveform functions before and after the

switching process can be expressed as follows, respectively:

U1(t)~

0 tv0t

t10ƒtvt1

1 t1ƒtvt2

t{t3

t2{t3t2ƒtvt3

0 t§t3

8>>>>>>>><>>>>>>>>:

, ðA:1Þ

and

U2 tð Þ~

0 tv0

0:5 1{cos(pt=t1)½ � 0ƒtvt1

1 t1ƒtvt2

0:5 1zcospt

t3{t2

� �� t2ƒtvt3

0 t3ƒt

8>>>>>>><>>>>>>>:

: ðA:2Þ

We take Eq. (A.2) as an example. Its Fourier image

function is as follows:

F2(v)~

ðz?

{?U2(t)e{ivtdt: ðA:3Þ

Substituting piecewise function (A.2), we obtain:ðz?

{?U2(t)e{ivtdt~

1

2

ðt1

0

1{cos(pt=t1)½ �e{ivtdtz

ðt2

t1

e{ivtdt

z1

2

ðt3

t2

1zcospt

t3{t2

� �� e{ivtdt:

ðA:4Þ

Solve Eq. (A.4) using the quadrature rule to obtain thefrequency domain expression:

F2(v)~

2t21v

2 sin(vt1)zicos(vt1)½ �{p2 sin(vt1){i 1{cos(vt1)½ �f g2 v3t2

1{p2v� �

z2e{ivt1{2e{ivt2{e{ivt3ze{ivt2

2iv

{

ivcospt

t2{t3

� �{psin

pt

t2{t3

� �.t2{t3ð Þ

2p2

t2{t3ð Þ2{v2

" #eivt

t3

t2

:

ðA:5Þ

Similarly, we get the frequency domain Eq. for (A.1) as

follows:

F1(v)~1{eivt1zivt1

v2t1eivt1z

e{ivt1

iv{

e{ivt2{e{ivt3

v2(t2{t3): ðA:6Þ

32


Dow

nloa

ded

03/1

5/14

to 2

19.2

31.1

56.1

27. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Date post:	15-Jul-2020
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

Three-dimensional Modeling of Transient Electromagnetic ... · D TEM modeling: the integral...

Documents