three-dimensional modeling of transient electromagnetic ... · d tem modeling: the integral...
TRANSCRIPT
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
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
/
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
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
/
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
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
/
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
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
/
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
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
/
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
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
/
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
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
/
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
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
/
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.
Figure 14. 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 20 m. The dashed line in (a) indicates the
received LB/Lt is negative.
24
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
/
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
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
/
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.
Figure 18. Response curves of the X (a) and Z (b) components at point (21, 21) on the tunnel face with different fault
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
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
/
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.
Figure 19. Response curves of the X (a) and Z (b) components at point (21, 21) on the tunnel face with different fault
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
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
/
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
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
/
(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
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
/
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:
Geophysical Journal International, 171(2) 550–564.
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
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
/
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:
Geophysical Journal International, 185(2) 622–636.
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:
Geophysical Journal International, 137(2) 336–352.
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:
Geophysical Journal International, 160(1) 5–32.
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:
Geophysical Journal International, 190(2) 871–886.
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
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
/
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
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
/