are known as Maxwell’s equations. This set of equations are designed to draw a relation between the above described vectors and their source, and also with the electric charge density q (C/m3), and with the density of electric current I (A/m2) (P.V. Sharma, 1997).
Further to the above, vectors E, D, B, H, are linked to ε dielectric constant (in F/m), μ magnetic permittivity (H/m), and σ electric conductivity (S/m), by the following relations:
(2.2.5)
(2.2.6)
(2.2.7)
The ε dielectric constant (in F/m), μ magnetic permittivity (H/m), and σ electric conductivity (S/m), are physical properties that can be associated to natural soils or man‐made objects buried in the ground, for subsoil modeling and interpretation.
In order to obtain a set of two equations describing the propagation of the electric and the magnetic field in a homogeneous and isotropic mean whose physical properties are given by ε, μ, σ, Maxwell’s equation can be further reduced to:
Where,
In (2.2.8) and (2.2.9), the term Iωμσ relates with the so‐called conduction currents, whereas the term εμω2 relates with the displacements current. It can be observed that, at frequency lower than 10 KHz, conduction currents (hence the term Iωμσ) prevail with respect to the displacements current (εμω2); this condition is known as inductive regime. On the other hand, when the frequency of the EM signal is higher than 10 MHz, displacement current (Iωμσ), prevails with respect to the conductive current, and thus we have the so‐called radar regime.
In FDEM methods, EM fields propagates according to the inductive regime, whereas the Georadar (GPR) method follows the radar regime.
It may also be useful recalling here, the meaning of ε (dielectric constant) and μ (magnetic permittivity): dielectric constant is defined as the physical property of a material of holding an electric charge when an electric field is applied. The higher the electrical conductivity σ, the higher ε. ε is measured in Farad (F).
Magnetic permittivity μ represents the physical property of a material of holding the magnetization when a magnetic field is applied. μ is measured in Henry (H). The greatest part of geological material has a very low value of μ which is about (4π*10−7 H). However, it dramatically increases in those substances having magnetic properties.
2.2.2. The Relation Between the Primary and Secondary EM Field
As mentioned above, when an alternating current of frequency f and intensity Ip flows within a transmitting coil, it generates a magnetic field that is called primary magnetic field. This magnetic field, spreads in all the directions, including the subsoil. Also, if a receiving coil is located at a distance s from the transmitter, part of the magnetic field directly arriving at the receiver from the transmitter coil, holds the same phase and amplitude of the EM signal spread out form the transmitter (Figure 2.2.3).
At the same time, the portion of magnetic field (primary) reaching the subsoil, will induce the subsoil itself (according to the Faraday’s law) to generate an electric current (eddy current) having the same frequency of the primary field, but with a 90° (π/2) phase difference.
These electric currents (the eddy currents, the so‐called secondary), in turn, will flow into the subsoil according to the Ohm’s law (I = ∆V/R, where ΔV is the voltage and R is the electric resistance of the medium), and shows, according to the Ampere’s law, a magnetic field associated (secondary) that, once detected by the receiver coil, shall have a phase difference with respect to the primary, equals to φ.
The phase difference φ depends upon the conductors (medium) physical properties, and it is represented by the following mathematical equation:
(2.2.10)
In 2.2.10, L represents the inductance (being this the property of an electric conductor or circuit that causes an electromotive force to be generated by a change in the current flowing) and R the electrical resistance.
Thus, the total phase difference between the primary and the secondary magnetic field, shall be:
(2.2.11)
Figure 2.2.3 Sketch of the propagation of an EM filed generated by a transmitter. Ip is the electric current generated at the transmitter (primary, dashed line), Is is the induced electric current (secondary, dashed‐point line). Dashed‐point lines indicates the secondary magnetic field, whereas dashed lines indicate the primary magnetic field. S is the coils spacing
(modified from F. Giannino, 2014).
A sketch of the phase relation between the primary and the secondary, is illustrated in Figure 2.2.4.
It can be observed that, when EM measurements are carried out at high‐conductive areas, R tends to 0 and φ tend to π/2 (90°), and the phase of S is about ‐180° with respect to R.
On the other hand, when high‐resistive targets are present in the subsoil, R tends to ∞ and φ tends to 0, with the phase of S at ‐90° with respect to R (D.S. Parasnis, 1979).
Furthermore, it should be noticed that S may be factorized into its components S sin φ and S cos φ: S sin φ is the so‐called Real component (Re, in phase) and S cos φ is the imaginary component
