the description of the behavior of EM waves in a medium by means of three constitutive parameters, that in general are tensor quantities, but under the assumption of isotropy and homogeneity can be considered scalars: the electric permittivity, ε, the electric conductivity, σ, and the magnetic permeability, μ.
A useful approximation, in the case of a homogeneous isotropic medium, is represented by the damped plane wave solution of the scalar wave equation. In this case each component of the electric (E) and magnetic (H) field at a distance z and time t is related to the corresponding fields at z=0 and t=0 (E0 and H0) by the expressions:
(2.1.11)
(2.1.12)
where
(2.1.13)
(2.1.14)
α is called absorption constant and β is called the phase constant.
The constitutive parameters ε and σ are, in general, complex numbers and have in‐phase (d.c.) components, namely ε’ and σ’, and out of phase (high frequency) components, namely ε” and σ” (Turner and Siggins, 1994) relating with each other by the following:
(2.1.15)
(2.1.16)
At most radar frequencies the out of phase component of the electric conductivity (σ”) is generally negligible, while the out‐of phase component of the electric permittivity (ε”) is not. Moreover, most geological materials, which are best suited for GPR investigations, are low loss (tanδ << 1), non‐magnetic media (μ ≅ μ0). Under these conditions, approximated expressions for the above α and β can be written as:
Figure 2.1.4 Electromagnetic‐wave velocity measurements: (a) the known object depth; (b) the two‐way travel time related to reflection event by known object (e.g. a pipe).
(2.1.17)
(2.1.18)
Where:
ω = 2 π f, is the radian frequency
μ = μ0μr = (4π) 10−7 Henry/m (μr = 1), is the magnetic permittivity
ε = ε0 εr = 8.85 10−2 εr = εr /(36 π 109) F/m, is the dielectric constant
c = 1/(ε0μ0)1/2= 3x108m/s, is the electromagnetic velocity in free space
Z0 = (μ0/ε0)1/2=376,8 ohm, is the intrinsic impedance in the free space
K′=ε′/ε0 is the real part of the relative permittivity (or dielectric constant) of the medium.
From the above equations results that for the materials with electric conductivities less than 50 mSiemens/m the electromagnetic wave velocity of propagation depends exclusively on the real part of the dielectric constant and is not frequency‐dependent (Figure 2.1.6):
(2.1.19)
And the medium attenuation can be approximated by:
(2.1.20)
The dielectric constant varies from its “free space” value of 1 to a maximum of 80 for water, whose presence, therefore, strongly influences the dielectric constant of rock‐ (or soil‐) water mixtures. It is also clear that GPR is a method suited for sounding dielectric low‐loss materials. Attenuation increases as the conductivity of the ground increases. Materials having high conductivities, as water‐saturated clay or saltwater, rapidly dissipate the radar energy and restrict the investigation depths. The amplitude of radar waves is further reduced by spherical spreading losses, reflection, and transmission at discontinuities as well as by small scale heterogeneity scattering which, in turn, increases with increasing frequencies.
Figure 2.1.5 Electromagnetic wave velocity analysis with the hyperbola adaptation method using a commercial software.
Figure 2.1.6 Relation between EM wave velocity and frequency (a) and between attenuation and frequency (b) at different values of electric conductivities
(Modified from Davis and Annan, 1989).
For these reasons, the penetration capability of GPR decreases as the center frequency of the antenna increases. When a wave arrives at a boundary separating two media with different EM characteristics, energy is partially reflected and partially transmitted. For normal incidence and in the case of non‐magnetic low‐loss materials, the amplitude reflection coefficient, R can be expressed either in terms of the radar wave velocity in the two layers (v1 and v2):
(2.1.21)
Davis and Annan (1989) published a table that summarizes the values of relative dielectric constant, electromagnetic wave velocity, conductivity, and electromagnetic wave attenuation related to several soil materials (Table 2.1.1).
It can be seen that the dielectric constant of water is 80, while the dielectric constant of many dry geological materials is in the range of 4–8: this great difference explains why the electromagnetic wave velocity is strongly dependent on the water content in the traversed materials.
Very important in GPR surveys is the choice of the antenna
