I. INTRODUCTION
Nondestructive testing (NDT) techniques have emerged with the intention of examining any object, material, or system without affecting its future utility. A wellaccepted NDT technique is ground penetrating radar (GPR), which has been widely explored and applied in many areas, such as archaeology, geophysics, and engineering [^{1}]–[^{3}]. The GPR technique represents a faster alternative to assess the interior of different structures and also contributes to their preservation.
Several structures, such as railways [^{4}], pavements [^{5}], bedrock and retaining walls [^{1}] can be modeled as multilayer media and analyzed using GPR. In broad terms, GPR detects the reflections originating from electromagnetic discontinuities on the structure. In this way, the information obtained by GPR can be used to determine the characteristics of each layer, e.g. thickness and relative electric permittivity. This process represents an inverse electromagnetic problem that can be solved using two broad approaches: analysis techniques in the frequency and time domains [^{6}]. Nevertheless, choosing a solution technique for the inverse problem has a direct impact on the GPR operating frequency, and, consequently, the antenna to be used.
Frequencydomain analysis techniques usually utilize optimization tools to solve the electromagnetic inverse problem [^{7}], [^{8}]. However, defining the optimization problem based on the reflected electricfield phasor is a challenge, because the uniqueness of the solution must be guaranteed. This means that the objective function must be ensured to have only a unique minimum or maximum point. In a comprehensive study of this problem, Oliveira et al. [^{8}] presented a methodology that ensures smoothness in the error function of the inverse problem to facilitate its solution. In contrast, the proposed methodology imposes restrictions on the GPR operating frequency, which will limit the minimum size of the antenna. Furthermore, the findings in [^{8}] suggest working with narrowband antennas, which have an analytic closedform that facilitates the design.
In contrast, timedomain analysis techniques generally use the travel time of the electromagnetic wave between the layers and the reflected peak amplitude to define the layers’ dielectric properties and thicknesses [^{9}], [^{10}]. This analysis technique uses GPR measurements, easy calculations and two assumptions to solve the inverse problem: propagation in (a) lowloss dielectric, i.e., conductivity σ = ωε, where ω is the angular frequency and ε is the permittivity; and (b) homogeneous layers, which means that the dielectric constant is assumed to be equal throughout the layer. This time domain analysis approach, based on travel time through the layers and presented in detail in [^{9}], will be considered for this study. Nevertheless, this method may become inaccurate if inadequate reflection detections (due to overlapping) are performed. To avoid this drawback, a good vertical resolution is necessary. Consequently, there is a need to design antennas with the ability to transmit and receive short pulses with low distortion.
It is believed that the most significant advances in a GPR system can be achieved with the antenna design (transmitter–receiver). In this regard, a complete review of GPR antenna characteristics is presented in [^{11}]. Several studies with specific applications for GPR have been developed to seek antenna systems with optimal designs and low cost [^{12}], [^{13}]. On this matter, patch antennas have reached increasing prominence by achieving versatile solutions in multiple applications where a radiant element is required [^{14}]–[^{16}].
For instance, authors in [^{17}] proposed using multiobjective optimization to design 3 antennas with applications in 3 different areas, including GPR. In that research, the solution of contradictory objectives related to directivity, impedance matching, cross polarization, and frequency range using stochastic and deterministic algorithms in the solution was carried out. To find improvements over traditional approaches that only use spectral analysis to optimize antennas, a multicriteria optimization approach was proposed directly in the time domain [^{18}] for a bowtie antenna. The results demonstrated a good balance between radiation and admittance matching in the analyzed frequency range. In addition, another interesting analysis was suggested in [^{19}] to design an optimized antenna that adapts to specific input signals. They ensured a high correlation between the input signal in the time domain and the output signal in the receiving antenna. Further, the authors in [^{19}] achieved not only a minimum voltage standing wave ratio (VSWR), but also low temporal dispersion.
Although previous studies have proposed antennas for general GPR applications, none of them are based on the specific problem to establish the antennadesign characteristics, such as the central frequency, bandwidth, and temporal response. Therefore, we propose a practical methodology to obtain the specific antennadesign requirements that allow problems modeled as multilayer media to be solved with timedomain solution techniques while avoiding pulse overlapping. In other words, we suggest correlating, in an unequivocal (analyticdirect) way, some apriori physical knowledge of the multilayer problem with the antenna characteristics to be designed. This apriori knowledge means that some typical estimates are known for the relative electric permittivity and thickness of layers, which are generally obtained through other techniques. In this sense, the methodology presented in this paper represents a practical guide for GPR antenna design, enabling the production of a specific antenna for any analyzed multilayer problem. Consequently, this guide can contribute to obtaining dedicated GPR equipment.
The paper is organized as follows: Section II presents the step by step formulation to obtain the spectral and temporal antenna design requirements. Then, Section III contains a description of a simple multilayer medium used to test the methodology proposed in Section II. Later, Section IV explains the ultrawideband (UWB) antenna selection and parametric analysis to fulfill the requirements obtained for the test problem in Section III. Finally, the results are presented and discussed in Section V, and Section VI lists the conclusions of the work
II. PROPOSED METHODOLOGY
As was pointed out in the introduction, the antenna design is dependent on the GPR operating frequency, which in turn depends on the type of analyzed multilayer structure and the inquiring signal. From this correlation, the following methodology was developed. The methodology starts with the analysis of the inquiring GPR signal. Then, some information from the timedomain inverse problem solution is extracted, and finally, a signaloverlap criterion is established according to [^{20}]. Following these methodology steps, the requirements for antenna design can be obtained for a specific problem. Thus, if an antenna meets those requirements, the overlapping reflected pulses would be avoided. In this way, the proposed methodology serves as a practical guide in the design of GPR antennas.
A. Ricker wavelet
The Ricker wavelet is the pulse selected for use in the proposed methodology. It is frequently used in GPR applications [^{21}], [^{22}] because it solves the problem of lowfrequency components (spectrum near DC) presented in the Gaussian pulse.
The equation that describes Ricker wavelet is as follows:
where A is the amplitude, t_{0} is the time shift, and τ is the time constant.
The Fourier transform amplitude X_{r}(f) of the Ricker wavelet is given in Eq. (2).
The peak emission frequency f_{p} can be found by deriving Eq. (2) and equating it to zero. As a consequence, the peak emission frequency should satisfy
As shown, it is possible to define, through Ricker wavelet analysis, the peak emission frequency f_{p}. This frequency represents the antenna’s central frequency, which is an important parameter in the antenna design. As can be seen in Eq. (3), the pulse width t_{p} is still an unknown variable which will be obtained in the next methodology step.
B. Impacts of selecting the inverse problem solution approach
Selecting an approach (either in the time or frequency domain) to solve the inverse problem imposes a set of restrictions leading to different GPR operating frequencies. The timedomain analysis approach uses a short pulse to feed the GPR antenna, which results in a spectrum rich in frequency components. The pulse width in this technique is determined by the analyzed multilayer structure.
In the timedomain analysis approach, an adequate identification of each reflected pulse is necessary because the layers’ characteristics, such as thickness and relative electric permittivity, are obtained from the time between multiple pulse reflections and their amplitudes [^{9}]. In this way, the total time of the incident pulse for each interface should be less than the time for the transmitted wave portion to travel through the next layer, reflect on the next interface, and return.
The intersection of subsequent pulses will be null if
where t_{p} is the Ricker wavelet width,
where ε_{i} and d_{i} are the relative electric permittivity and thickness of layer i, respectively, and C is the speed of light. Thus, by substituting Eq. (5) in Eq. (4), the general equation for any multilayer problem is obtained
It is evident from (6) that both the thinnest layer thickness and the lowest permittivity determine the selection of t_{p}. In this way, the pulse is defined in terms of the relative electric permittivity and thickness of each pavement layer using some apriori knowledge about the analyzed structure, as will be explained in the next section.
Now, the appropriate requirements for designing the antenna from the analyzed multilayer structure can be completely defined. First, the peak emission frequency f_{p} can be calculated using Eq. (3) where t_{p} is the pulse width satisfying Eq. (6). Then, the cutoff frequencies and bandwidth can be defined for f_{p}. These frequencies are commonly selected depending on the application. For instance, ANSI STD C63.102013 [^{23}] and IEEE STD 6862017 [^{24}] define for UWB and radar devices respectively that the cutoff frequencies f_{L} and f_{H} are the lowest and highest frequencies where the signal is at least 10 dB below the peak power level. On the other hand, circuit theory commonly defines these frequencies when the peak power falls at half (3dB) [^{25}]. For this work, the halfwidth cutoff frequencies at −6 dB were used because it allows a pulse reconstruction closer to the Ricker wavelet. Thus, the approximations for the frequencydomain parameters of the Ricker wavelet as a function of f_{p}, as determined by Wang in [^{26}], are as follows:
where f_{c} is the central frequency, f_{L} and f_{H} are the lower and upper cutoff frequencies, respectively and BW is the bandwidth. These specifications guarantee that there will not be any pulse overlap which contributes to the accuracy of the solution technique of a GPR inverse multilayer problem. In summary, the antenna to be used in GPR equipment must be capable of transmitting a Ricker wavelet respecting the characteristics described by Eqs. (3) and (7)–(10).
C. Pulse overlap criterion
The spectral and temporal antenna characteristics were already obtained in Subsections II.A and II.B. Nevertheless, antennas are inherently dispersive elements and, therefore, can cause undesired reflections. It is important to know which is the antenna’s permissible dispersion (
Fig. 1 shows the extremes for two Ricker wavelets, in which the pulse is characterized by the width at half energy W defined by the time difference between the extremes of the pulse when they achieve half of the maximum value of energy. Fig. 1(a) depicts pulses that are clearly separable without any overlap. In turn, Fig. 1(b) presents pulses that overlap but are still considered distinguishable (separable). According to [^{20}], two Gaussian pulses are distinguishable if they are separated by at least the width at half amplitude. However, for the Ricker wavelet, the width at half energy is used instead of that at half amplitude, as shown in Fig. 1.
Therefore, if certain pulses are allowed to overlap by the width at half energy
where
By substituting Eq. (12) into (11) and using (5), the allowed maximum pulse width is given by
As a result, the dispersive characteristic of the antenna is defined by the separation threshold between two pulses.
Thus, given an estimate of the multilayer problem to be assessed, the design antenna requirements can be defined using equations (7)–(10) and (13), as presented in Section III.
III. TEST PROBLEM
A multilayer structure was selected in this work to show how the methodology proposed in Section II can be used. GPR has been used commonly as an indirect measurement for construction and maintenance of highways [^{27}], where pavement can be modeled by multiple overlapping layers. This topic has been widely explored and much research has been carried out, such as [^{28}], [^{29}].
The threehomogeneouslayer model presented in Fig. 2 was used as a test pavement problem. In Fig. 2, Tx and Rx represent the transmitter and receiver antennas, respectively, d_{0} is the distance between the pavement surface and the antennas. d_{1} and d_{2} are the unknown layer thicknesses; ε_{1}, ε_{2} and ε_{3} are the unknown relative electric permittivities related to each layer (1 for asphalt, 2 for the base and 3 for subgrade layer).
To apply the proposed methodology described in the previous section, the first step is to calculate the Ricker wavelet width by using Eq. (6). This equation is dependent on
Layer 

d_{i} (cm)  t_{p} (ns) 

1. Asphalt  3–12  6–10  0.69 
2. Base  5–10  20–30  2.98 
3. Subgrade  10–25     
The most restrictive case was selected from Table I to generate a worstcase analysis. This means that the smallest values of both permittivity and thickness of each layer in Table I were used to calculate t_{p}, and, from these t_{p} values, the shortest travel time was selected. In this way, the nonoverlap in the GPR reflected pulses will be ensured, regardless of the real value of the pavement parameters. Thereby, the pulse width of Ricker wavelet was calculated for each layer using Eq. (6), and the obtained values are listed in Table I.
It can be defined from the data in Table I that if the pulse width is at most 0.69 ns, the overlap between layers reflections for the pavement problem would be avoided. Thus, it is demonstrated that the pulse was actually defined based on apriori pavement physical knowledge assuming a worstcase scenario.
The next step is to calculate the peak emission frequency (f_{p}) using Eq. (3) where t_{p} is the one obtained for the thinnest layer in Table I (t_{p} = 0.69 ns). Then, the antenna design values were calculated using Eqs. (7)–(10). Thus, all the antenna requirements were estimated; they are presented in Table II.
Finally, the maximum antenna permissible dispersion (
IV. UWB ANTENNA
Once the spectral and temporal requirements are obtained for the test problem using the proposed methodology in Section III, an antenna topology should be selected. In general, an antenna for GPR applications needs to consider the following characteristics: wide frequency bandwidth to obtain high resolution, good gain, radiation efficiency, good impedance matching, low dispersion, small size and low weight [^{11}]. Many GPRUWB antennas have been studied in the literature that could meet the obtained requirements: bowtie antennas [^{32}], [^{33}], spiral slot antennas [^{34}], circular antennas [^{35}], Vivaldi antennas [^{15}], resistive linear antennas [^{36}], among others [^{11}].
For this study, the selected microstrip patch antenna geometry was inspired from [^{37}], as it presents a simple geometry for construction, ease of parametrization, and low implementation cost. Therefore, it represented a good starting point for a parametric study. The antenna in [^{37}] has a gain that varies from 3 to 6 dB and the bandwidth from 3.6 to 12 GHz, which means that it is not in the desired frequency band. Besides, it has an omnidirectional radiation pattern. To solve this, a reflector plane was added behind the antenna and a parametric study was carried out, as shown in Fig. 3.
Fifteen parameters were used to describe the antenna geometry: the width W_{d} and length L_{d} of the dielectric substrate, the width W_{f} and length L_{f} of the feed line, the width W_{p} and length L_{p} of the patch, the width W_{g} and length L_{g} of the ground plane, the position on the x and y axes of one of the arms (x_{a}, y_{a}), and the width W_{xa} and length L_{ya} of the arm. To reduce the search space, the dimensions of the other arm were obtained assuming that the antenna is symmetrical. Also, the width W_{R} and length L_{R} of the reflector plane, and the distance D_{R} between the reflector plane and the antenna were considered. The dielectric substrate thickness was defined as a constant (h = 1.6mm).
V. RESULTS AND DISCUSSION
A. Parametric analysis of the antenna
A parametric analysis of the proposed antenna that aims not to distort the Ricker wavelet was performed. This procedure resulted in the antenna parameters listed in Table III.
B. Simulation of antenna performance
The new antenna geometry was simulated in HighFrequency Structure Simulator (HFSS) where the microstrip line, patch, arms, ground, and reflector plane were modeled as perfect electric conductors (PECs), and a port was connected to the microstrip line as the feed. The material selected for the substrate was FR4 (tanδ = 0.02, ε_{r} = 4.4). The designed antenna with the reflector plane was simulated and analyzed (see Fig. 4).
The 3D radiation pattern simulated in the central operating frequency (3GHz) for the antenna is presented in Fig. 4(a). It can be observed from Fig. 4(a) and Fig. 4(c) that the inclusion of the reflector plane made the radiation pattern directive (as expected) not only at central frequency but in the complete bandwidth. On the other hand, the VSWR presented an increase at low frequencies of the analyzed band compared with the original reference (see Fig. 4(b)). However, this does not represent a drawback, because the most important antenna parameter to be guaranteed in the GPR inverse problem solution technique is the pulse fidelity rather than the power transmission.
C. Antenna measurements
To perform the measurements, a Rohde & Schwarz vector network analyzer (VNA) with coaxial testing cables was used in a nonanechoic environment. The coaxial cables were properly calibrated using the thru–open–short–match (TOSM) calibration and the ZVZ135 female calibration kit from Rohde and Schwarz. Finally, the antenna was connected to the coaxial cables using a female SMA–50Ω connector. The antenna was built on a 47.9 × 78.5 × 1.6 mm^{3} FR4 substrate and it is shown in Fig. 5. The transmitter–receiver link measurements were performed following the methodology presented in [^{38}], placing two equal antennas separated by a distance considered to be farfield (5λ of the operating central frequency). At the same time, this link was modeled in HFSS for antenna validation, which comprises comparing the measured results of gain, and transmission losses with the simulated ones.
The Sparameters from the antenna measurement were exported and compared with the simulation ones (see Fig. 6). It is observed from Fig. 6 that the results present a good correlation at frequencies higher than 3 GHz. It can also be seen that the reflection coefficient (s_{11}) is better in measurements within the whole bandwidth. This indicates that the reflection losses are negligible and less than 10% of the voltage energy returns to the source. Differences in the measured and simulated values presented in Fig. 6 may be due to the nonanechoic environment and uncertainties in the FR4 electromagnetic properties.
Next, the Friis transmission equation was used as shown in Eq. (14) to calculate the antenna realized gain, where r is the distance between two equivalently designed antennas, λ is the freespace wavelength, and s_{21} is the transmission coefficient.
Fig. 7 shows the gains calculated using Eq. (14) from the HFSS simulated link (dashed line) and the measured link (solid line), comparing them with the HFSS gain for a single antenna (dotted line). This comparison was performed as a validation of the Friis approximation, where the distance between the antennas must be farfield. Thereby, Fig. 7 illustrates that the defined Friis approximation using the distance of 50 cm provides good results comparing the Friis simulated gain (dashed line) and the HFSS antenna gain (dotted line).
In contrast, gain from the measurements (solid line: Friis measured) was satisfactory at frequencies higher than 3GHz. However, the results differ from simulated ones at lower frequencies. This result was expected because the Friis equation (Eq. (14)) depends on the s_{21} parameter, which has a similar behavior between the simulated and measured results (see Fig. 6).
D. Timedomain analysis
A pulse temporal analysis was performed using the designed antenna to visualize the signal spread. To achieve this, the Sparameters were exported from HFSS and from measurements to the Advanced Design System (ADS) software, where they are modeled as a twoport black box. Fig. 8 shows the complete simulated circuit of a Ricker wavelet transmission and reception using the designed antenna. The only change between measured and simulated pulses is the black box containing the Sparameters. The Ricker wavelet was imported to ADS from MATLAB with a pulse width t_{p} = 0.69 ns and connected to a 50Ω resistance.
The voltage signal at each node (1Transmitted, 2Coupled, 3Received in Fig. 8) was simulated and the results are presented in Fig. 9. In Fig. 9, small reverberations in the signal coupled to the antennas (dashed line) can be seen. These are due to impedance matching, and therefore, existing reflections are added to the input signal (solid line). The dotted line in Fig. 9 represents the output signal, which was multiplied by a factor (of 50) for visualization purposes.
To perform the spread (dispersion) analysis, the power at the system output was analyzed. A shorter spread time indicates that there will be no overlap of the reflected signals. The simulated and measured power graphs from the transmitted and received signals are shown in Fig. 10. Table IV contains the total pulse duration at the output
Signal  Pulse width (


Transmitted  0.69 ns 
Received (simulated link)  0.8 ns 
Received (measured link)  1.39 ns 
As demonstrated in Table IV, the pulse width obtained from measurements was 1.39 ns. Therefore, the condition where the pulse width with dispersion
VI. CONCLUSIONS
This paper has highlighted the importance of an adequate detection of the reflected peaks measured by a GPR to improve the GPR inverse problem solution technique in multilayer problems. Consequently, we have devised a new practical step by step methodology to obtain specific antenna requirements (its operating frequency and temporal response) through the analysis of homogeneous and lowloss dielectric multilayer problems. In this way, the proposed methodology serves as a guide for GPR design to generate efficient solutions.
A test pavement problem was defined to estimate the antenna requirements using the proposed methodology. Finally, a lowcost antenna was proposed to fulfill the requirements. The validation results of the antenna showed low dispersive behavior, which indicates that there will be no overlap of the GPR reflected signals.