the velocity of the high-velocity layer and demonstrate its effectiveness with an example.
2. Background
GPR is most frequently used to perform re- flection profiles, which are commonly referred
to as zero offset profiles. Such profiles are acquired by keeping the two GPR antennas an
equal distance apart while taking measurements
Ž at equal spacing along a traverse Davis and
. Annan, 1989 . When viewing a zero offset pro-
file, one is often looking at the topography of a layer or for anomalies in the reflection signal,
such as hyperbolas associated with small buried objects. Due to the interpretation non-unique-
ness of radar images, errors may occur in inter- preting the source of the buried layer or anomaly.
Is the hyperbola caused by a buried barrel or by a boulder? If one is looking at a horizontal
reflector, how does one determine whether the anomaly is a water-table boundary or clay-layer
boundary? The difficulty in interpreting GPR data is due to many factors such as attenuation,
dispersion, scattering and radiation patterns Ž
. Annan, 1996 . Using travel time analysis in
conjunction with a zero offset profile can reduce some of the problems associated with these
factors. Velocity travel-time analysis used in conjunc-
tion with the CMP data acquisition method is the traditional technique for determining the
Ž composition of a reflector Annan and Cosway,
. 1992 . However, there are limitations associated
with this method. To use this technique, one must look at radar wave velocities using travel
times for the reflected waves. Consequently, the velocity of the medium below the lowest reflec-
tor cannot be determined.
3. Theory
The theoretical basis for GPR is found in Maxwell’s equations. However, it is not the
intent of this paper to provide an in-depth math- ematical derivation of the EM wave equation
from Maxwell’s equations. For more detailed information on the derivation, Keller and Zh-
Ž .
Ž .
danov 1994 and Wait 1982 contain thorough discussions of the subject. The success of GPR
is based on EM waves operating in the frequency range where displacement currents
dominate and losses associated with conduction
Ž .
currents are minimal Annan, 1996 . For the purposes of this paper, the assumption is made
that we are dealing only with displacement cur- rents and that the medium is lossless. The justi-
fication for this assumption will be discussed later.
The wave equation, in the propagation regime for electric displacement currents, is given in
Ž . Eq. 1 :
E
2
E
2
= E s m´ ,
1
Ž .
2
Et
where E is the electric field, m is the magnetic permeability and ´ is the permittivity. The per-
mittivity can be defined as ´ s ´ ´ , where ´
o r
o
is the permittivity of free space and ´ is the
r
relative dielectric constant. Using phasor nota- Ž .
tion, Eq. 1 can be represented as shown in Eq. Ž .
2 , where v is the angular frequency: =
2
E s yv
2
m´ E.
2
Ž .
The velocity for an EM wave in a dielectric is given by:
1 y s
, 3
Ž .
1 2
2
m´ s
1 q q 1
2 2
ž
2 v ´
where s represents conductivity. At high fre- Ž .
quencies andror very low conductivity, Eq. 3 reduces to:
1 y s
. 4
Ž .
m´ It is obvious for lower radar frequencies that
the dielectric properties and conductivity play a dominant role in determining the velocity of a
medium. For insulating materials such as dry rocks, dielectric properties alone determine the
velocity of the EM wave. The effect that dielec- tric properties can have is seen in Figs. 1 and 2.
Fig. 1 plots the velocity of an EM wave as a function of conductivity and frequency with a
relative dielectric constant of 4. It can be seen in Fig. 1 that for frequencies greater than 100
Ž . MHz, Eq. 4 is a good approximation of the
velocity. For frequencies below 100 MHz, the Ž .
use of Eq. 4 will depend on the conductivity of the medium. Fig. 2 plots frequency vs. rela-
tive dielectric constant with a constant resistiv- ity of 50 V m. It can be determined from Fig. 2
Ž . and Eq.
4 that for frequencies above 100
MHz, velocity is essentially independent of fre- quency and dependent only on the dielectric
constant and the magnetic permeability. Earth materials rarely have a magnetic per-
meability appreciably different from unity ex- Ž
. cept for a few magnetic minerals see Table 1 .
Magnetic permeability generally has a notice- able effect only when large quantities of Fe O
2 3
Ž .
are present Telford et al., 1990 . Therefore, changes in velocity must be due to changes in
dielectric constant or changes in resistivity of the medium. Consequently, for many earth ma-
terials, at high frequencies or high resistivity,
Fig. 1. Velocity of an EM wave plotted as a function of the soil resistivity with a relative dielectric constant of 4.
Fig. 2. Velocity of an EM wave plotted as a function of the relative dielectric constant with a resistivity of 50 V m.
the velocity of an EM wave is determined only by the relative dielectric constant of the medium.
Table 2 gives a list of relative dielectric con- stants and conductivities for a variety of earth
materials encountered when using GPR.
When EM waves are obliquely incident on an interface between two media, it is necessary to
consider two different cases. The first case ex- ists when the electric field vector is perpendicu-
lar to the plane of incidence. This is commonly referred to as perpendicular polarization. The
plane of incidence is the plane containing the incident ray and is normal to the surface. The
second case exists when the electric field vector is parallel to the plane of incidence. This case is
commonly referred to as parallel polarization. For schematic representations of these two cases,
refer to Fig. 3a and b.
Table 1 List of relative magnetic permeabilities of various minerals
Ž .
adapted from Telford et al., 1990 Mineral
Permeability Magnetite
5 Pyrrhotite
2.55 Hematite
1.05 Rutile
1.0000035 Calcite
0.999987 Quartz
0.999985
Table 2 List of relative dielectric constants and velocities for some
Ž typical earth materials adapted from Annan and Cosway,
. 1992
Ž .
Material Relative dielectric
Velocity mrns constant
Air 1
0.30 Sea water
80 0.01
Dry sand 3–5
0.15 Saturated sand
20–30 0.06
Limestone 4–8
0.12 Silts
5–30 0.07
Granite 4–6
0.13 Ice
3–4 0.16
EM waves may become polarized when radi- ated from an antenna. The antenna design deter-
mines the type of polarization, which can be linear, circular, or elliptical. Most GPR antennas
emit waves that are linearly polarized, which are further subdivided into perpendicular or par-
allel polarized. By changing the orientation of the antennas relative to each other and relative
to the traverse, the polarization detected by the radar will change. Polarization type will affect
how the wave is reflected from a planar surface or a buried object. The reflection differences
show up in the reflection coefficients of the two cases. For a thorough discussion on the subject
of polarization of radar waves, refer to Roberts
Ž .
and Daniels 1996 . The perpendicular polarized reflection coeffi-
cient equation is:
2
´ cos u y ´ y ´ sin u E
1 1
2 1
1 r
s ,
5
Ž .
2
E
´ cos u q ´ y ´ sin u
i 1
1 2
1 1
and the parallel polarized reflection coefficient Ž .
equation, Eq. 6 , gives the ratio of reflected signal to incident signal at a horizontal planar
boundary:
´ ´
2 2
2
cos u y y sin u
1 1
ž ž
´ ´
E
1 1
r
s .
6
Ž .
E ´
´
i 2
2 2
cos u q y sin u
1 1
ž ž
´ ´
1 1
For details on the derivation of these equa- tions, please refer to any textbook on EM theory
Ž . Fig. 3. Schematic diagram of perpendicular and parallel polarized EM waves. a shows a perpendicular polarized EM wave
where the E field points out of the page perpendicular to the plane of incidence and the H field is parallel to the plane of
Ž .
incidence. b shows a parallel polarized EM wave with the E field parallel to the plane of incidence and the H field perpendicular to the plane of incidence.
Fig. 4. Magnitude and phase portion of reflection coefficients for two different velocity contrasts. Parallel and perpendicular polarizations are shown when the EM wave goes from low velocity to high velocity and from high velocity to low velocity.
such as Electromagnetic Waves and Radiating Ž
. Systems Jordan and Balmain, 1968 . Plots of
the reflection coefficients vs. angle of incidence for both the parallel and perpendicular polarized
waves are shown in Fig. 4. This figure shows the magnitude and phase portion of the reflec-
tion coefficients.
For parallel polarized waves, there exists an angle where no reflected wave exists. This an-
gle is referred to as the Brewster angle and Ž .
given by Eq. 7 . The Brewster angle occurs Ž .
when the numerator in Eq. 6 goes to zero: ´
2
tan u s .
7
Ž .
1
´
1
The Brewster angle is more easily understood when one looks at the magnitude and phase
portions of Fig. 4. The Brewster angle occurs in Fig. 4b and d when a step change of 908 in
phase occurs prior to reaching the critical angle where all energies are reflected from the inter-
face. When a step change of 908 in phase occurs prior to all energies being reflected, the re-
flected energy goes to zero. When viewing Fig. 4c and d, other phase changes are evident and
these occur after all energies have been re- flected. It should be noted that these phase
changes are not step changes in phase.
4. Application of theory
