Electromagnetic Methods in Geophysics. Fabio Giannino
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the zero‐offset section of the seismic reflection), where the two‐way time is plotted vertically below the midpoint position, between the Tx and the Rx, even if the actual ray path is slanted, as for the reflection from dipping interfaces or from small‐size targets (diffraction).
Typically, the vast majority of commercial GPR, are built according to a monostatic architecture. However, GPR data can be acquired using other modes depending on the relative geometry of transmitter(s) and receiver(s). These acquisition modes are known as: The Common Mid‐Point (CMP) or Common Depth Point (CDP), the Wide‐Angle Reflection and Refraction (WARR), and the transillumination (Figure 2.1.3) The first two are mainly used for the electromagnetic (EM) wave velocity determination whereas the last is used in tomographic studies.
In general, any GPR is built to measures EM waves reflection events at a given time. This means that, once the EM signal is emitted by the Tx, it travels in the ground and when the wave encounters a reflector it is scattered back and recorded by the receiver. The time spent by the EM wave to travel from the Tx to the reflector and back to the Rx is known as two‐way travel time. Hence, the electromagnetic wave propagation velocity plays an important role in the GPR data analysis, because it allows the conversion of the two‐way travel time window into depth. The EM wave, propagates at a different velocity in different mediums, depending on their physical (dielectric) properties.
Beside CMP and WARR methods to estimate EM waves velocity, other methods can be used. They are (i) the location of objects at known depth, and (ii) the reflection from a source point. In the first method, two‐way travel time is the time that an electromagnetic wave takes to travel through the ground, from the transmitting antenna to the object and back to the receiving antenna (Figure 2.1.4).
Denoting the depth of the known object with a zknown and the velocity of the electromagnetic wave with v, the two‐way travel time for a monostatic configuration of the antenna is given by:
Since the depth of the object is known, it can be taken the double travel time from a radar section and express the velocity of the electromagnetic wave using Eq. 2.1.1 (Figure 2.1.4 a).
Figure 2.1.2 Schematic illustration of data acquisition in the reflection profiling mode (a), corresponding radar time section (b) and the waves characterization (c).
The second method is based on the phenomenon that a small object, for example, the cross section of a pipe, reflects radar waves in almost all directions (Figure 2.1.4 b).
Denoting the depth of the object with z and the lateral distance of the monostatic antenna from the object with x, the length w of the wave path can be simply expressed by:
(2.1.2)
and therefore, the function of the two‐way travel time with:
(2.1.3)
Denoting with t0 the two‐way travel time, on the vertical to the object, one has:
(2.1.4)
Therefore:
which is the formula for the so‐called “diffraction hyperbola” method. Many commercially available GPR data processing software, allows for the computation of the EM velocity propagation, automatically, based on this method.
Since from the radar section for each x position the corresponding two‐way travel time t (x) is known, the velocity can be calculated by inverting Eq. (2.1.5). The shape of the hyperbola is governed by the velocity of the wave through the ground and by the geometry of the buried object (Fruhwirth et al., 1996) (Figure 2.1.5).
Figure 2.1.3 Schematic illustration of data acquisition in the a) CMP, b) transillumination, and c) WARR, (Tx: transmitter, Rx: receiver).
2.1.3. Electromagnetic wave propagation
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)