Low-frequency electromagnetic scattering from one or more tunnels in a lossy dielectric half-space is considered. The tunnel radii are assumed small compared to the wavelength of the electromagnetic field in the surrounding medium; a tunnel can thus be modeled as a thin scatterer, described by an equivalent impedance per unit length. We examine the normalized backscattering width for cases in which the air-ground interface is either smooth or rough

Casey, K

Pao, H

UNT Digital Library

UCRL-CONF-227353
Low-Frequency Electromagnetic
Backscattering from Tunnels
K. Casey, H.-Y. Pao
January 19, 2007
2007 IEEE AP-S International Symposium
Honolulu, HI, United States
June 10, 2007 through June 15, 2007
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Low-Frequency Electromagnetic Backscattering from Tunnels
Kendall F. Casey*1 and Hsueh-Yuan Pao2
1 SRI International, Menlo Park, CA 94025; kendall.casey@sri.com
2 Lawrence Livermore National Laboratory, Livermore, CA 94550;
pao2@llnl.gov
Introduction
Low-frequency electromagnetic scattering from one or more tunnels in a lossy di-
electric half-space is considered. The tunnel radii are assumed small compared to
the wavelength of the electromagnetic ﬁeld in the surrounding medium; a tunnel
can thus be modeled as a thin scatterer, described by an equivalent impedance per
unit length. We examine the normalized backscattering width for cases in which
the air-ground interface is either smooth or rough.
Tunnel in a Lossy Half-Space
The tunnel is a circular-cylindrical region of free space situated in the half-space
z < 0, of relative permittivity r. The tunnel is parallel to the y-axis; its axis is
located at (x, z) = (0,−d) and its radius is a, which we assume to be small compared
to the wavelength in the surrounding region. By virtue of this assumption, we can
model the tunnel as a thin scatterer described by an equivalent impedance per unit
length Z ′w. It can be shown that this equivalent impedance per unit length is
Z ′w =
jk0Z0
π(k0a)2(r − 1) (1)
in which k0 is the free-space wavenumber and Z0 is the intrinsic impedance of free
space. The time dependence exp(jωt) is assumed. At the exterior wall of the tunnel,
the electric ﬁeld component parallel to the tunnel axis is equal to the impedance per
unit length multiplied by the equivalent current I0 carried by the scatterer. (Because
the tunnel radius is electrically small, the axial electric ﬁeld is essentially uniform
around the tunnel wall.) Our approach to the scattering problem is to determine
the ﬁeld incident on the scatterer and the ﬁeld excited by the (as yet unknown)
equivalent current on it; we then impose the boundary condition at the tunnel wall
to determine the equivalent current, and the solution is complete.
We assume that a perpendicularly-polarized plane electromagnetic wave is incident
on the lower half-space from the free-space region z > 0. The incident electric ﬁeld
in the free-space region and the electric ﬁeld transmitted into the lower medium are
Eyi(x, z) = E0e−jk0(x sin θi−z cos θi) (2)
Eyt(x, z) = T (θi)E0e−jk0(x sin θi−z
√
r−sin2 θi ) (3)
with θi denoting the angle of incidence; the transmission coeﬃcient T (θ) is
T (θ) =
2 cos θ
cos θ +
√
r − sin2 θ
(4)
Next consider the electric ﬁeld created by a ﬁlamentary current I0 at (x, z) = (0,−d).
This ﬁeld can be expressed in the two regions as the Fourier integrals
Ey>(x, z) =
jk0Z0I0
2π
∫ ∞
−∞
e−jkxx−jkz0z−jkzgd
dkx
j(kz0 + kzg)
(5)
Ey<(x, z) =
jk0Z0I0
2π
∫ ∞
−∞
e−jkxx
[
kzg − kz0
kzg + kz0
ejkzg(z−d) − e−jkzg|z+d|
]
dkx
2jkzg
(6)
where kz0 =
√
k20 − k2x and kzg =
√
k20r − k2x. The second term in the integrand on
the RHS of eq. (6) above yields the electric ﬁeld directly radiated by the ﬁlamentary
current, and the ﬁrst term yields the ﬁeld that is reﬂected back into the ground
by the air-ground interface. The electric ﬁeld radiated into the free-space region,
evaluated in the far zone, is
Ey>(ρ, θ) ∼ − jk0Z0I0√8πjk0ρ
T (θ)e−jk0ρ−jk0d
√
r−sin2 θ (7)
We deﬁne the normalized equivalent backscattering width k0 as
k0 = lim
ρ→∞ 2πk0ρ
Sr
Si
∣∣∣∣
θ=θi
=
∣∣∣∣k0I0Z02E0
∣∣∣∣
2 ∣∣∣T (θi)e−jk0d
√
r−sin2 θi
∣∣∣2 (8)
Si and Sr denote the incident and backscattered power densities respectively. Now
imposing the condition Ey(0,−d+ a) = Z ′wI0 and using eqs. (3) and (6), we obtain
I0 =
T (θi)E0e−jk0d
√
r−sin2 θi
jk0Z0ζn
(9)
with the normalized impedance ζn given by
ζn =
Z ′w
jk0Z0
+
1
4j
H
(2)
0 (kga) +
1
2π
∫ ∞
−∞
e−2jkzgd
(
kzg − kz0
kzg + kz0
)
dkx
2jkzg
(10)
with kg = k0
√
r. The normalized scattering width k0 is
k0 =
|T (θi)|4
|2ζn|2
∣∣∣e−jk0d
√
r−sin2 θi
∣∣∣4 (11)
The relative permittivity of the ground is expressed in terms of a high-frequency
relative permittivity r∞ and a low-frequency conductivity σ0 as [1]
r =
(√
r∞ +
√
σ0Z0/(jk0)
)2
(12)
Typical values of r∞ are in the range 6 to 8, and σ0 varies upward from 10−4 S/m
for very dry soils. The value r∞ = 7 was used in the numerical results shown
here. In Figure 1 we show the normalized scattering width of a tunnel of radius
a = 5 m at depth d = 50 m as a function of frequency for incidence angles of 0◦,
30◦, and 60◦. The low-frequency conductivity σ0 = 10−4 S/m. We note that the
normalized scattering widths tend to increase with increasing frequency, up to a
maximum value that occurs at approximately 6.25 MHz, and then drop oﬀ fairly
rapidly as the frequency increases further. The normalized scattering width also
decreases rapidly with increasing ground conductivity.
0 2 4 6 8 10
−70
−65
−60
−55
−50
−45
−40
−35
−30
frequency (MHz)
n
o
rm
a
liz
ed
 b
ac
ks
ca
tte
rin
g 
wi
dt
h 
(dB
)
a = 5 m; d = 50 m; σ0 = 0.0001 S/m; ∆ = 0 m
θ=0° 
θ=30°
θ=60°
Figure 1: Normalized scattering width vs. frequency in very dry soil (σ0 = 10−4
S/m). Depth d = 50 m and radius a = 5 m; incidence angles θi = 0◦, 30◦, and 60◦.
Rough-Surface Eﬀects on Tunnel Backscatter
We model the interface as a random-phase screen. This model is valid only when
the standard deviation of the phase shift imposed by the screen is small, so that
our analysis is valid for a “slightly rough” air-ground interface. The interface is
now taken to be the surface z = ∆(x, y), where the random process ∆(x, y) is
homogeneous and Gaussian, with expected value E{∆(x, y)} = 0 and variance σ2.
Using this model we can show (see, e.g., [2]) that the coherent amplitude of the
electric ﬁeld of a plane wave transmitted across the screen is reduced by the factor
F = e−q
2(kx0,ky0)σ2/2 (13)
in which kx0 and ky0 are the transverse propagation constants of the plane wave and
the function q(·) is given for the present problem by
q(kx0, ky0) = k0
{√
r − sin2 θi − cos θi)
}
(14)
In the problem of backscattering from a buried target, the electromagnetic wave
passes through the rough interface twice, so that the coherent backscattered electric-
ﬁeld amplitude is reduced by the factor F 2 and the backscattering width is reduced
by the factor F 4.
A portion of the equivalent current induced on the buried tunnel results from re-
ﬂection by the air-ground interface of the ﬁeld that is radiated by the tunnel. The
fact that this surface is rough is accommodated by including the following factor in
the integrand of the integral in eq. (10):
exp
[
−2k20σ22
{√
r − k2x/k20
}]
(15)
0 0.5 1 1.5 2 2.5 3 3.5 4
−40
−39
−38
−37
−36
−35
−34
−33
−32
−31
−30
roughness standard deviation (m)
n
o
rm
a
liz
ed
 b
ac
ks
ca
tte
rin
g 
wi
dt
h 
(dB
)
f = 6.25 MHz; a = 5 m; d = 50 m; σ0 = 0.0001 S/m
θi = 90
°
Figure 2: Normalized scattering width vs. σ in very dry soil (σ0 = 10−4 S/m).
Depth d = 50 m and radius a = 5 m; θi = 0◦ and f = 6.25 MHz.
Computations of the backscattering width are performed as before with the mod-
iﬁcation described above made in eq. (10) and the overall result multiplied by the
factor F 4. Figure 2 shows the normalized scattering width as a function of the stan-
dard deviation of the surface roughness for normal incidence when the frequency is
6.25 MHz. When the standard deviation reaches 4 m, the normalized backscattering
width is reduced by approximately 10 dB from its value for a smooth interface.
Concluding Remarks
We have investigated the problem of low-frequency electromagnetic backscattering
from a tunnel located in a lossy ground. The fact that the tunnel diameter is assumed
to be small in comparison to the wavelength allows us to model the tunnel as a thin
scatterer described by an equivalent impedance per unit length. We computed the
electric ﬁeld backscattered from a tunnel for the case of a smooth air-ground interface
as a function of the signal frequency, tunnel depth and radius, incidence angle, and
soil properties. Rough-surface eﬀects were included by using a random phase-screen
model for the rough interface.
References
[1] M. A. Messier, “The propagation of an electromagnetic impulse through soil: in-
ﬂuence of frequency dependent parameters,” Technical Report MRC-N-415, Mission
Research Corporation, Santa Barbara, CA, 1980.
[2] Kendall F. Casey, “Rough-surface eﬀects on subsurface target detection,” Pro-
ceedings of SPIE, Vol. 4394, 2001.


Low-Frequency Electromagnetic Backscattering from Tunnels

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Low-frequency electromagnetic backscattering from tunnels

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Low-Frequency Electromagnetic Backscattering from Tunnels

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