Ground-penetrating radar (GPR) systems are powerful tools for nondestructive, noninvasive subsurface detection.^1,2 These systems are suitable for measuring the geometric parameters of buried objects,³ locating bars of armor, detecting cavities or breaks inside structures,⁴ and groundwater detection.⁵ GPR has widespread application in civil engineering for such tasks as investigating reinforced concrete structures⁶ and localizing bars of reinforcement in concrete. GPR systems can also be used to determine defects, internal morphology, and lack of homogeneity in materials in civil engineering and preventive characterization of soils and contextual mapping of subsoils for structural safety analysis.⁷ In the current study, the use of chirp transmission and reception in a frequency-modulated continuous-wave (FMCW) GPR will be examined for defining the mass density of an illuminated ground layer.

Subsurface radar systems have been built using a variety of modulation types depending on the particular requirements of penetration depth, resolution, known target characteristics, software available for data processing, reduction of electromagnetic interference, system size, and cost. The four GPR architectures are pulse wave radar,^8,9 continuous-wave (CW) radar, ¹⁰ modulated CW radar,^{11, 12} and synthetic pulse radar. Ground mass density is an important parameter in civil engineering, and the FMCW GPR is a useful tool for evaluating this parameter. This type of radar provides less penetration than a pulsed type radar, but with better resolution. Efforts have been made to improve the detection and evaluation capabilities of GPR systems by developing wideband antennas,^5,13 imaging and signal processing algorithms,^{14, 15} and by improvements in transmitter and receiver circuitry. Figure 1 presents the transmitted waveforms of different radars in one measurement cycle.

Most GPR systems are based on discrete pulse transceivers, although there can be some problems in applying signal processing with this approach. It may be necessary to determine the amplitude-time dependencies of electromagnetic (EM) field components due to wave scattering at a material interface.16 In some cases, the time needed for successful detection may be long and impractical.¹⁷

An FMCW GPR system can overcome these difficulties, working in the frequency domain (Fig. 2).

The frequency-domain behavior of an FMCW GPR is shown in Fig. 3. During each period, T, the transmitted frequency is linearly increased to achieve the desired bandwidth, B. During the process of transmitting and receiving, a delay, τ, will occur. This delay is the difference in frequency between transmitted and received signals for the duration of the chirp. It is called the beat frequency and is represented by f_B. The transmitter and receiver frequencies, f_T and f_R, are determined by Eqs. 1 and 2, respectively:

f_T = f₀ + at (1)
f_R = f₀ +a (t –t ) (2)
where
a = B/T,
B = the bandwidth of the chirp,
and
T = the period of the chirp.

Equation 3 shows that the beat frequency is proportional to the permittivity of the transmission medium,
where
c = the speed of light,
e_r = the relative permittivity of the medium, and
R = the range (the distance between the transmitter and the receiver) of the GPR.

For civil engineering, one of the most important properties of a medium is its density. This parameter, which is the mass per unit volume of a substance, can be used to determine the compressive strength of road pavement and other materials.¹⁸ A material can be thought of as a system of molecules that will react to an incident electric field. The electric polarization P (average electric dipole moment per unit volume) is given by Eq. 4¹⁹:

P = ?ae₀E_loc (4)
where
? = the density of the material’s molecular structure,
a = the mean molecular polarizability of the material,
e₀ = the permittivity of a vacuum, and
E_loc = the local electric field.

The electric polarization can also be written as
P = (e_r – 1)e₀E (5)
where
e_r = the dielectric constant (relative permittivity) and
E = the applied electric field.

The use of a Lorenzian transformation with Eqs. 4 and 5 yields Eq. 6:
E_loc = E + (P/3e₀) = (1/3 )(e_r + 2)E (6)

The Clausius–Mossotti relation can be applied to show that the density of a medium, ?, is proportional to relative permittivity, er:
e_r – 1)/(e_r + 2 ) = a?/3 (7)

If the mean molecular polarizability and relative permittivity of a medium are known, its density can be found by the Clausius–Mossotti relation. By combining Eqs. 3 and 7, it can be shown that the beat frequency at the receiver of an FMCW GPR system is related to the density of the illuminated medium, as in Eq. 8:
f_B = 2RB/CT = [(3 + 2?a )/(3 – ?a)]^0.5 (8)

The permittivity of a material layer, e_r is proportional to the density of the layer, ?; the EM velocity though the medium depends mainly on the medium’s dielectric property, and can be calculated by Eq. 9²⁰:
BM = C/( e_r )^0.5 (9)

Changes in the EM propagation velocity result in changes in the time delay, τ, for a signal to travel to the target and back. The density of a material is proportional to the time delay, so that a certain time delay, τ, represents a certain density, ?.

For road and pavement measurements, GPRs can operate over short ranges but must deliver high resolution. An FMCW GPR can provide such performance, and can be improved by means of basic modifications. One of the most important GPR parameters for impacting accuracy is chirp-signal duration. This signal is produced by a voltage-controlled oscillator (VCO) with duration related to the saw-tooth (ramp) duration. For analysis, simulated saw-tooth and chirp signals are shown in Figs. 4(a) and 4(b).The bandwidth of the chirp signal is 900 MHz. When the saw-tooth signal changes from 0 to 10 V, the chirp frequency changes from 1 to 1.9 GHz and is then mixed with a 3.6-GHz local oscillator (LO). The output of the mixer is filtered to achieve a signal that changes from 1.7 to 2.6 GHz.

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