Читать книгу Electromagnetic Methods in Geophysics - Fabio Giannino - Страница 17

As already mentioned above, GPR method is based on the propagation of Electromagnetic (EM) waves in the ground. And the Maxwell’s equations provide the starting point to understand how electromagnetic fields can be used in Georadar exploration to obtain information about the electric and magnetic properties of the soil which is an electrically neutral medium (ρ = 0 where ρ indicates the charge density). This is a set of the four Maxwell’s equations:

(2.1.6)

(2.1.7)

(2.1.8)

(2.1.9)

In the above, E is the electric field vector, B is the magnetic induction vector, D is the electric displacement vector, H is the magnetic field intensity vector, and J is the conduction current density. The EM field relates to these quantities by means of empirical relationships known as constitutive equations (Keller, 1987; Ward and Hohmann, 1987):

(2.1.10)

where σ, ε, and μ are respectively the electrical conductivity (Siemens/m), the electrical permittivity (Farad/m), and the magnetic permeability (Henry/m).

These relations allow 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 to use to obtain the best result: the ability to resolve buried objects and the depth to be reached are, in fact, mainly determined by the frequency and therefore by the length of the transmitted wave.

Other factors that must be considered in the study of the electromagnetic wave propagation are the penetration depth and the resolution. The penetration depth decreases as the frequency increases, while radar resolution increases with higher frequencies. The resolution is a crucial point both in defining the acquisition geometry and interpreting georadar data. Resolution relates to how close two points can be, yet still, be distinguished.

On this regard, two “types” of resolution are illustrated and discussed in order to derive their implication in terms of targets detectability, namely the “vertical resolution” and the “horizontal resolution”.

The vertical resolution relates to the (minimum) depth separation between two boundaries to give separate reflection events; it is determined by the bandwidth that is considered about equal to the center (or dominant) frequency. Reflections from two boundaries, separated by a distance Δz, are separated for high center frequency pulses and are merged for low center frequency pulses. The acceptable threshold for vertical resolution generally is a quarter of the dominant wavelength (Sheriff, 1994), although this criterion is subjective and depends on the noise level in the data.

The above criterion implies that the minimum depth separation (Δz) is:

(2.1.22)

On this purpose, the following experimental Table 2.1.2 is given illustrating the relationship between wavelength and frequency of EM emitted waves (Leucci, 2015):

As for the horizontal resolution, instead it refers to how close two reflecting points can be situated horizontally yet be recognized as two separate points rather than one.

Table 2.1.1 Values of the relative dielectric constant ε_r, electrical conductivity σ, electromagnetic‐wave velocity, and attenuation in some geophysical materials (Davis and Annan, 1989. With permission of John Wiley & Sons).

Material Type	Relative Dielectric Constant εr = ε/ε0	Electrical Conductivityσ (mS/m)	EM Waves Velocity V (m/ns)	EM Waves Attenuation α (dB/m)
Air	1	0	0.30	0
Distilled water	80	0.01	0.033	2*10−3
Fresh water	80	0.5	0.033	0.1
Salt water	80	3*104	0.01	103
Dry sands	3‐5	0.01	0.15	0.01
Saturated sands	20‐30	0.1‐1	0.06	0.03‐0.3
Limestone	4‐8	0.5‐2	0.12	0.4‐1
Shale	5‐15	1‐100	0.09	1‐100
Silt	5‐30	1‐100	0.07	1‐100
Clay	5‐40	2‐1000	0.06	1‐300
Granite	4‐6	0.01‐1	0.13	0.01‐1
Dry salt	5‐6	0.01‐1	0.13	0.01‐1

Table 2.1.2 Wavelength values λ as a function of the frequency at several electromagnetic‐wave velocities of propagation (From Leucci, 2015).

Freq. (MHz)	P(ns)	λ (m) @ v= c	λ (m) @ v= (1/3) c	λ (m) @ v= (1/6) c
1	1000	300	100	50
10	100	30	10	5
30	33	10	3.3	1.65
100	10	3	1	0.5
300	3.3	10	3.3	1.65
500	2	0.6	0.2	0.1
1000	1	0.3	0.1	0.05
2000	0.5	0.15	0.05	0.025
3000	0.33	0.1	0.03	0.015

The electromagnetic waves transmitted by a standard antenna are irradiated through the ground in a generally elongated elliptical cone. The radiation lobe is generated by a horizontal dipole antenna, to which some protection elements are added (often metallic foils) which reduce the emitted radiation upwards (shielding). When a dipole antenna is placed in the air, the path of the radiation is approximately perpendicular to the antenna axis. When instead it is placed in contact or near the ground and/or the surface of the investigated materials, there is a change in the shape of the radiation lobes due to the coupling with the ground.

Variation of both the shape and the lobe directivity also occurs at the variation of h/λ, where h is the height from the ground of the antenna and λ is the wavelength of the pulse in the first medium (air).

The radiation cone (related to the first Fresnel zone) that intercepts a horizontal flat surface illuminates an ellipse‐shaped area with the major axis parallel to the antenna’s trailing direction (Annan et al., 1991). The radiation lobe in the subsoil enables “looking” not only directly under the antenna but also in front, back, and sides as the antenna travels along the ground. This is known as horizontal resolution (Leucci, 2019). Two reflecting points separated by a distance less than the first Fresnel zone radius ® are considered indistinguishable as observed from the earth’s surface. The first Fresnel zone radius is given by the following:

(2.1.23)

and, in addition to velocity and frequency, is also depth dependent. Since the Fresnel zone generally increases with depth, the spatial resolution also deteriorates with depth.

Figure 2.1.7 Elliptical cone of GPR penetration into the ground.

In a simple way the angle of the cone is defined by the relative dielectric constant of the material traversed by the electromagnetic waves and by the frequency of the transmitter antenna. An equation that can be used to estimate the width of the transmission beam at various depths (the footprint) is (Conyers and Goodman, 1997):

(2.1.24)

where A is the approximate dimensions of the radius of the footprint, λ is the wavelength of the electromagnetic impulse, D is the depth at which the reflecting object is located, and ε_r is the relative dielectric constant of the crossed medium (Figure 2.1.7).

Among other constraints (Conyers and Goodman, 1997), in a GPR survey, the central frequency of the antenna is chosen to obtain a viable compromise between the desired penetration depth and vertical resolution. Moreover, the lateral resolution is important in planning the acquisition geometry and in particular, the spatial sampling along the survey line (inline spacing) and the distance between consecutive lines (crossline spacing). The latter requirement is seldom fulfilled due to time and positioning problems.