Case 2: np
D f s f y f
n s 0,1,2,3, . . .
y x
2 5
Ž .
D f 0 for CW Df - 0 for CCW.
Pipes and other objects scatter energy prefer- entially, depending on the incident polarization.
The polarization and orientation of the transmit antenna is thus important to ensure sufficient
energy is scattered from subsurface targets to allow measurement by the receive antenna.
Preferential scattering may result in depolariza- tion of the incident field. Depolarization occurs
when the amplitude or phase of the incident
Ž Ž ..
field components Eq. 1 are modified such
that the scattered field results in a different polarization. The ability of the receive antenna
to measure these scattered fields is determined not only by the power of the scattered fields,
but also by the polarization match between the scattered fields and the receive antenna. The
polarization of the field incident on the receive antenna is determined by the polarization of the
field radiated by the transmit antenna and the degree of depolarization experienced by scatter-
ing from subsurface objects. It is thus important to understand the polarization properties of GPR
antennas and scattering from subsurface objects.
Most commercial GPR antennas are dipole or bow-tie antennas that radiate linearly polarized
energy with the majority of the radiated electric field oriented along the long axis of the dipole
or bow-tie. For a description of dipole fields
Ž .
over a half-space, consult Annan 1973 , Annan Ž
. Ž
. et al.
1975 , Arcone 1995 , Engheta et al.
Ž .
Ž .
1982 , Smith 1984
and Radzevicius et al. Ž
. 2000b . A complete polarization mismatch us-
ing dipole antennas results when the scattered field and polarization of the receive antenna are
Ž Ž ..
both linearly polarized Eq. 2 and oriented at
right angles to each other. For example, rotating ideal dipole antennas orthogonal to each other
Ž .
crossed-dipoles results in a complete polariza- tion mismatch. Spiral, or other circularly polar-
ized antennas, are also used for pipe detection. A complete polarization mismatch using circu-
larly polarized antennas results when the scat- tered field and receive antenna are both circu-
Ž Ž ..
larly polarized Eq. 3 , but have electric fields Ž
with opposite rotation directions left and right .
circular .
3. Normal incidence plane wave scattering by circular cylinders fundamental theory and
conclusions from analytical solutions
Cylinders represent an important class of ob- jects for GPR since they represent important
environmental and engineering targets and also because their scattering properties are strongly
polarization dependent. A brief discussion of the importance of polarization on the scattering
of plane waves normally incident on both di- electric and conductive circular cylinders is now
described. The reader is referred to Balanis Ž
. Ž
. 1989 and Ruck et al. 1970 for a more de-
tailed explanation of equations and for oblique incidence.
Two linearly
independent basis
vectors Ž
. Ž Ž ..
polarizations Eq.
1 are necessary to de-
scribe scattering from both dielectric and per- fectly conducting cylinders. It is convenient to
choose these polarization vectors such that one vector is oriented along the long axis of the
Ž cylinder
E parallel or transverse-magnetic Ž
.. TM
and the other vector oriented orthogonal Ž
to the long axis of the cylinder E perpendicular Ž
.. Ž .
or transverse electric TE Fig. 1 . TM polar-
ization is achieved when the long axis of the transmit and receive dipole antennas are ori-
ented parallel to the long axis of the cylinder and the survey direction is orthogonal to the
Ž .
long axis of the cylinder Fig. 2a . TE polariza- tion is achieved when both antenna axes are
oriented orthogonal to the long axis of the cylin- der and the survey direction is orthogonal to the
Ž .
long axis of the cylinder Fig. 2b . The scattered field is a function of the electri-
cal properties of the cylinder and surrounding
Ž .
Fig. 1. Definitions of E parallel TM and E perpendicular Ž
. TE polarizations relative to a cylinder, as defined in
Ž .
Balanis 1989 . TM polarization occurs when the electric field is parallel to the long axis of the cylinder. TE
polarization occurs when the electric field is perpendicular to the long axis of the cylinder. E and H represent the
electric and magnetic fields respectively, while a repre- sents the cylinder radius, b represents the propagation
vector, f represents the scattering angle, r represents the radial distance from the cylinder, and x, y, z represent the
coordinates of a cartesian coordinate system.
material, the distance from the cylinder, and the scattering angle f, as defined in Fig. 1. A
scattering angle of 1808 represents backscatter- ing. Since circular cylinders have a cylindrical
shape, their scattering properties are conve- niently described using Hankel and Bessel func-
tions because they represent cylindrical waves. Incident and scattered fields for conductive and
dielectric cylinders are described in terms of cylindrical coordinates in Appendix A.
Ž .
The radar cross-section RCS represents a convenient way to describe the strength of scat-
tered fields observed in the far-field. The RCS is defined as the the area intercepting the amount
of power that when scattered isotropically, pro- duces at the receiver a density that is equal to
Ž the density scattered by the actual target Bal-
. anis, 1989 .
s 2
E
2
RCS s lim 4p r 6
Ž .
2 i
r ™`
E For 2D objects such as an infinite cylinder, the
Ž .
RCS becomes the scattering width SW
or
Fig. 2. Examples of GPR surveys using linearly polarized Ž
. coincident common-offset with small separation antennas
to image cylinders. The GPR survey direction is into the Ž .
page for all the cases. a Dipoles oriented parallel to the long axis of the cylinder are best for imaging conductive
and small diameter, low impedance, dielectric cylinders. Ž .
b Dipoles oriented orthogonal to the long axis of the cylinder are best for imaging high impedance dielectric
Ž . cylinders. c Crossed-dipoles at 458 are good for imaging
both conductive and dielectric cylinders.
Ž alternatively the RCS per unit length Balanis,
. 1989 .
s 2
E SW s lim 2p r
7
Ž .
2 i
r ™`
E All scattering widths of cylinders in this
manuscript are normalized with respect to the wavelength of the incident field and Figs. 3–5
are plots of normalized scattering widths as a function of scattering angle for different radius
to incident wavelength ratios. ´ and s will be used to represent dielectric permittivity and con-
ductivity.
Several universal features are observed in the scattering widths as a function of scattering
angle for both conductive and dielectric cylin- ders. As the radius of the cylinder becomes
Ž . small compared to wavelength: 1 scattering
width amplitudes for both polarizations oscillate Ž .
less, 2 TM polarization scattering widths be- come nearly constant as a function of scattering
Ž . angle, and 3 TE polarization scattering widths
form a single, low amplitude null, at a scattering angle of 908.
Most GPR surveys used to image the subsur- face are conducted in common-offset mode us-
ing closely spaced antennas. This results in a Ž
scattering angle of approximately 1808 back- .
scattered , depending on antenna separation and target depth. It is thus important to further
investigate backscattered scattering widths as a function of pipe radius. Fig. 6a represents
backscattering from metallic pipes as a function of pipe radius. The TM polarization backscatter-
ing width is greater than TE backscattering width for most radius-to-wavelength ratios. The
TE component oscillates about the TM compo- nent and TE converges toward TM as the radius
becomes large compared to wavelength. TM polarization is the preferred polarization for the
detection of metallic pipes, as illustrated by Figs. 2a and 6a.
Ž High impedance dielectric cylinders cylin-
ders with a permittivity less than the surround- .
ing soil represent such targets as PVC pipes filled with air or hydrocarbons and surrounded
by higher permittivity soil. Fig. 6b is a plot of the backscattering widths for dielectric pipes
embedded in a medium having a permittivity seven times greater than the pipe. This repre-
sents air filled PVC pipes surrounded by a typical
moist sand.
The TE
polarization backscattering width is greater than the TM
backscattering width for small diameter, high impedance, dielectric pipes, and Fig. 2b repre-
sents the best survey geometry. Fig. 6c is a plot of backscattering widths for dielectric pipes em-
bedded in a medium having a permittivity seven times less than the pipe. The TM polarization
backscattering width is greater than the TE backscattering width for small diameter, low
impedance, dielectric cylinders, and Fig. 2a rep- resents the best survey geometry. It is often
informative to describe scattering by a scatter- ing matrix defined as:
i s
yi k r
S S
E E
e
x x x y
x x
s 8
Ž .
s i
E S
S r
E
y y x
y y y
where subscripts x and y denote an orthogonal set of coordinates and superscripts i and s de-
note the incident and scattered components of the electric E fields, k is the wavenumber, r is
the distance from the target to observation point. The scattering matrix is useful in describing
Ž scattering from thin metal pipes radius small
. compared to wavelength . The S
term domi-
x x
nates for thin metal pipes oriented with their long axis of symmetry along the x axis, whereas
thin dielectric pipes with an impedance greater than the surrounding soil have S S .
y y x x
Ž .
Cross-pole antennas Fig. 2c are useful for improving antenna isolation and can be used to
reduce clutter under appropriate field conditions and when stratigraphy is not the objective of the
Ž .
GPR survey Radzevicius et al., 2000a . Objects are visible using linear, crossed-dipole antennas
Ž . Fig. 3. Scattering widths for metallic cylinders normalized by the wavelength l of the incident field. As the radius of the
cylinder becomes small compared to wavelength, TM scattering widths become nearly constant as a function of scattering angle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TM polarization
backscattering widths are greater than the TE polarization scattering widths for most cylinders.
when they scatter electric field components or- thogonal to the field components radiated by the
transmit antenna. Scattered cross-components are produced by scattering from rough planes or
Ž most small objects. Cross-components compo-
. nents not present in incident field are not intro-
duced by scattering of plane waves normally incident on infinite circular cylinders, as seen in
Ž . Fig. 4. Scattering widths for high impedance dielectric cylinders, normalized by the wavelength l of the incident field. As
the radius of the cylinder becomes small compared to wavelength, TM scattering widths become nearly constant as a function of scattering angle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TE
polarization backscattering widths are greater than the TM polarization scattering widths for small diameter cylinders.
Ž . Ž
. Ž
. Eqs.
12 – 23 . Balanis 1989
describes the case of a plane wave obliquely incident on both
dielectric and conducting cylinders. Balanis ob- served that scattering from a perfectly conduct-
ing infinite cylinder does not introduce addi- tional components in the scattered field that are
not present in the incident field. This is not the case for dielectric cylinders, which introduce
orthogonally polarized
components under
oblique wave incidences. Cylinders having a
Ž . Fig. 5. Scattering widths for low impedance dielectric cylinders, normalized by the wavelength l of the incident field. As
the radius of the cylinder becomes small compared to wavelength, TM scattering widths become nearly constant as a function of scattering angle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TM
polarization backscattering widths are greater than the TE polarization scattering widths for small diameter cylinders.
finite length also produce depolarization from edge scattering.
It is not necessary for a target to produce a scattered cross-component to be visible with
cross-pole antennas. A long metallic pipe, while Ž
. a strong depolarizer Fig. 3 does not produce
field components that were not originally pre- sent in the incident field. The linear geometry of
pipes results in field components that are ori- ented both parallel and orthogonal to the long
axis of the pipe. The reflected and transmitted fields for thin pipes are related by the following
Ž .
relationships Daniels et al., 1988 :
yi k r
S S
e
x x x y
E s E ysinu cosu
s t
ž
S S
r
y x y y
= cosu
9
Ž .
sinu E
e
yi k r s
s S y S
Ž .
ž
y y x x
ž
E 2 r
t
=sin2u y S sin
2
u q S cos
2
u 10
Ž .
x y y x
where E and and E are the scattered and
s t
transmitted fields, respectively, and u is the angle between the long axis of the transmit
dipole and the long axis of the cylinder. For linear targets S
and S are small compared
x y y x
to other components and thus E
e
yi k r s
s S y S
sin2u 11
Ž .
Ž .
y y x x
ž
E 2 r
t
The ratio E rE is maximized when u s 458 for
s t
both dielectric and conductive pipes and thus crossed-dipole antennas at 458 with respect to
cylinders represent the best antenna geometry to Ž
. image cylinders Fig. 2c .
4. Linear polarization and scattering from pipes physical model examples
