Developing a realistic numerical equivalent of a GPR antenna transducer using global optimizers

Numerical modelling of a ground-penetrating radar (GPR) has been widely used for predicting and assessing its performance. As the transmitter and the receiver are the most essential components of a GPR system, an accurate representation of them should be included in a model. Simulating a real system is particularly challenging, especially when it comes to commercial GPR systems. A three-dimensional model based on a 2000 MHz ‘palm’ antenna from Geo-physical Survey Systems, Inc. (GSSI) is presented in this paper. The geometric features of the transducers were modelled via visual inspection, whereas their unknown dielectric properties were estimated using global optimizers in order to minimize the differences between real and synthetic measurements. In particular, the antenna was calibrated in free space and on top of a metal plate. Subsequently, the resulting model was successfully tested in various case studies to assess its performance. Models of two units of the same transducer were developed, showing that units of the same system in general are not identical. The results support the premise that global optimizers can be used to provide information on key aspects of the dielectric structure of the transducer and allow us to accurately model its behaviour in various environments.


INTRODUCTION
Forward modelling of ground-penetrating radar (GPR) using the finite-difference time-domain (FDTD) method is frequently used to assist the interpretation of GPR data (Cassidy, 2007;Daniels, 2004).A typical GPR case study consists of the host medium, the targets and the antenna system itself.Therefore, realistic representations of these elements should be included in the simulations.In order to simulate a specific transducer, knowledge of the geometry and the dielectric properties of its elements is essential (Warren & Giannopoulos, 2011).Although the antenna system plays an important part, nonetheless, it is a common practice amongst GPR practitioners to use a theoretical source instead This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.© 2023 The Authors.Near Surface Geophysics published by John Wiley & Sons Ltd on behalf of European Association of Geoscientists and Engineers.
of an accurate numerical equivalent of the actual GPR antenna system (Giannakis et al., 2019;Stadler & Igel, 2018;Warren & Giannopoulos, 2011).The use of simple theoretical sources, such as an infinitesimal dipole, is helpful for simulations that reproduce overall patterns in GPR scans, but the resulting A-Scans using simplistic sources can differ substantially in a number of key aspects from what is observed in real measurements (Diamanti et al., 2014).This is more important for high resolution, near-interface GPR applications where the targets are usually located in the near-field of the antenna (Giannakis et al., 2016).
Many researchers have developed and numerically modelled their own custom antennas.In Bourgeois and Smith (1996), a three-dimensional FDTD representation of an experimental scale-model of a bowtie GPR antenna has been described.Accurate implementations of resistive loaded bowtie antennas have been presented in Lestari et al. (2004Lestari et al. ( , 2010)), Uduwawala et al. (2005), and Shlager et al. (1994).A model of a spiral antenna GPR system is introduced in McFadden and Scott (2009).The accuracy of the previous models was validated by comparing simulated and observed responses, showing good agreement in all cases.Additional numerical models of antenna systems for designing purposes have been described in Holliger et al. (2003), Uduwawala and Norgren (2006), Liu et al. (2008), and Uduwawala et al. (2004).
Creating a model of a commercial GPR antenna structure is more challenging than simulating a custom-built antenna system.Although key aspects of the geometry can be obtained by observation, the excitation pulse and most of the values of the dielectric properties of materials used in building the final product are unknown, most often due to commercial sensitivity issues that is often compounded by genuine lack of detailed information on the wide-band dielectric properties for some materials (e.g.microwave absorbers).Furthermore, a fully functioning commercially available GPR transducer incorporates many other elements that might not play a major part in the GPR signal radiation.It is important to note that this is not something that is known a priori or always easily to deduce by inspection.A few studies have tried to simulate widely used GPR systems.In Klysz et al. (2006), a 3D FDTD model of a Geophysical Survey Systems, Inc (GSSI) 1.5 GHz antenna is presented.The authors of Warren and Giannopoulos (2011) developed models on the basis of a 1.5 GHz antenna from GSSI and on a 1.2 GHz antenna from MALÅ Geoscience and important values of key properties were estimated using Taguchi optimization, and a Gaussian-shaped pulse is used as excitation.The model presented in Warren and Giannopoulos (2011) was further improved in Giannakis et al. (2019) using a linear/non-linear full-waveform inversion scheme which simultaneously updates the dielectric properties and derives an optimized excitation waveform, overcoming the limitation of the Gaussian-shaped pulse in Warren and Giannopoulos (2011).In Stadler and Igel (2018), a commercial GSSI 400 MHz antenna was also modelled using Taguchi optimization to tune the dielectric properties of the antenna structure subject to real measurements, whereas a model of the same GPR system was presented in Stadler and Igel (2022), where both material properties and the feed pulse were optimized by fitting modelled data to a real response of a distant reflection from a metal plate.Note that each GPR system has its own unique characteristics (geometry, components, pulse, frequency content and radiation pattern) and therefore its replication can be considered a different optimization problem that requires different techniques for its solution.A complex and more advanced system with a large number of components might require a complex optimizer in order to acquire an accurate digital twin, in contrast to a simpler system.
Global optimizers, such as genetic algorithms (GA) and particle swarm optimization (PSO), have been extensively used for the numerical modelling of real systems in a broad spectrum of applications (Rahmat-Samii, 2003).Their success is based on their ability to handle the non-linearity and non-convexity of many problems that require a global search approach.So far, several authors have applied these methods to design new antennas by optimizing different sets of parameters.Specific radiation characteristics were obtained in Pantoja et al. (2007), by optimizing several parameters of a log-periodic dipole array through a PSO scheme.In Jin and Rahmat-Samii (2005), a parallel PSO/FDTD algorithm was applied for developing multiband and wideband patch antennas.GAs were employed in Altshuler and Linden (1997) to design four different antennas, whereas in Haupt (1994), global optimizers are implemented for thinning planar and linear arrays of antennas.In van Coevorden et al. (2006), the geometry and the resistive loading of a GPR thin-wire bow-tie antenna were chosen using GAs.Each of these schemes is used to optimize certain properties in order to achieve different radiation requirements.This indicates the applicability of global optimizers to a variety of problems related to antenna designing.
In this paper, we present a 3D numerical model as a digital equivalent of the 2000 MHz GSSI 'palm' antenna.The geometry is considered known as it can be observed by inspection.Preliminary development of our model and some early results were presented in Patsia et al. (2021).Estimates of unknown modelling parameters are obtained using a GA and a PSO scheme.Although global schemes were implemented to design new antennas, attempts to acquire estimates of the properties of an existing commercial antenna system had not been made in the literature, prior to Patsia et al. (2021).The outputs of the final model are validated by conducting a series of experiments over sand and over a reinforced concrete slab using the real transducer, replicating the same scenarios in the simulations, and comparing the collected A-scans with their corresponding simulated responses.A comparison of the output values of the schemes is also made, showing how the two different approaches converge, and the radiation patterns of the final model are presented.In addition, a model of a second unit of the same GPR system is also constructed, to support the general case that units representing the same transducers are not an exact replica of each other.Finally, the main reason for developing such a model is to be able to use its predictive power in developing and improving automated quantitative GPR data interpretation which depends on the availability of faithful and realistic forward modelling.

GEOMETRY OF THE ANTENNA
The numerical method chosen is a second-order accurate, in both space and time, FDTD scheme originally introduced in Yee (1966).In particular, we have used gprMax, an open source FDTD solver tuned for GPR applications (Warren et al., 2016(Warren et al., , 2019)), and we intent to make this model available in gprMax to be used by other researchers in the same way to the ones developed by Giannakis et al. (2019), Stadler and Igel (2018), and Warren and Giannopoulos (2011).A fine 1-mm grid is employed to construct the model based on a grid independence analysis.Inspecting the geometry of the real antenna structure, the most important components are chosen to be included in the model.The key design hypothesis in building such a model is to include as much as possible the parts that affect and shape directly the electromagnetic (EM) radiation and reception and avoid the detailed modelling of circuit components.Also, due to the small size of such components, an extremely fine grid that would greatly increase the computational cost would be needed.
Figures 1 and 2 illustrate the geometry of the simplified numerical equivalent of the investigated antenna system.The model includes two planar surface bowties, for the transmitting and the receiving antennas, with additional rectangular extensions at the ends of both bowties, reducing the antenna dimensions and also increasing the wideband characteristics.The bowties are etched from copper onto printed circuit boards (PCBs).Underneath the PCBs, two different types of EM absorber foams are inserted in order to reduce the back-cavity radiation.In addition, electromagnetic interference (EMI) shielding gaskets are placed on the bottom of the second absorber in the vicinity of each bowtie's midpoint.The bowties are placed in the same case, which is modelled as a square metal box.In the middle of the box, as shown in the side view of the model in Figure 2, there is a plate that separates the transmitter from the receiver, which are centred in each section of the box.The box is open on the side where the bowties lie and is further enclosed in a plastic case.The skid plate is also included in the model, resulting in an overall size of 86 × 86 × 68 mm.
For the excitation, a voltage source is used in a singlecell gap between the two arms of the transmitter bowtie (feed point).The shape of the real pulse is unknown.Via trial and error, it was derived that a Gaussian-shaped pulse produces the best results when exciting the model.The centre frequency was determined by the optimization process, resulting in a value close to the frequency specified for the real system.Using the approach presented in Giannakis et al. (2019), considering the shape of the pulse as an unknown, did not affect the resulting pulse, supporting the premise that a Gaussian pulse is a good approximation for the actual pulse.A single-  cell gap between the arms of the receiver bowtie with an edge of unknown conductivity is also used to model the receiver circuitry.The conductivity of this edge as well as the dielectric properties of the components of the antenna structure are to be estimated using global optimizers subject to real measurements.

OPTIMIZATION OF THE ANTENNA
GA (Goldberg, 1989) and PSO (Kennedy & Eberhart, 1995) belong to a family of global optimization schemes that search for global solutions to problems that hold multiple maxima or minima (Rahmat-Samii, 2003).In particular, they belong to the stochastic optimization methods, meaning that they introduce random variables in the formulation of the problem.Global approaches are utilized to find solutions to multi-parametric optimization problems when common traditional optimization techniques cannot.Thus, the multi-parametric nature of a GPR transducer makes it a suitable candidate for these schemes.
The aim of both methods is to reach a certain value of an objective function minimizing or maximizing it.Thus, they are searching for the set of parameters that achieve the best possible fit for a given problem.These parameters form an n-dimensional vector: which is the output of the optimization.Due to the non-uniqueness of the problem, many combinations of parameters, which might not be realizable, can lead to an optimal solution for a given problem.In order to acquire values that are accurate and realistic, boxbounded constraints that set the upper and lower limits for the optimization space were implemented penalizing this way the parameters and constraining the range of values they can obtain.
The objective function can take many different forms depending on the problem.One of the most common functions that was chosen to be minimized is the residual sum of the square differences (RSS) between the synthetic and real data: where X i and Y i are vectors of the modelled and real data for the ith scenario, respectively, x i,j and y i,j are the jth time sample belonging to the ith scenario, N is the total number of the investigated scenarios, and M is the number of time samples in a scenario.Each vector, X i or Y i , represents amplitude values versus time and the whole time series collected are used as part of the optimization.
If N = 1, the model of the antenna structure will produce valid results only for that specific setting and it will fail to respond reliably to different cases in the future (i.e., overfitting).To handle this problem, two data sets are used for the optimization: (a) free-space cross-coupling; and (b) the response over a perfect electric conductor (PEC).These scenarios are selected due to their simplicity and ease of implementation, both in practice and in the simulations.Thus, the algorithms search for the model parameters that will imitate the behaviour of the real GPR system for these two scenarios as close as possible.
As the true excitation pulse of the real GPR system is unknown, the real free-space and PEC responses are both normalized to the maximum amplitude value of the free-space response.The corresponding modelled free-space and PEC responses are normalized to the equivalent maximum amplitude value of the simulated free-space response.In order for the modelled and the real data to be comparable, the synthetic data are downsampled to 512 samples, which is the number of samples collected for each real A-scan over a time window of 8 ns, the real and synthetic responses for each scenario.Finally, the real and synthetic A-scans are aligned by their negative peak, for example for the freespace scenario, the synthetic response is shifted by a certain number of time steps until its negative peak is aligned with the negative peak of the real response.
GA is an iterative method that starts by creating an initial population of candidate solutions, based on specified constrains.All solutions together form the population matrix.At each iteration, the population forms a generation, and each solution from the population matrix is evaluated based on the selected objective function and the solutions with the best fit, called 'parents' , are chosen.The number of parents chosen, E, called the elite count, automatically survive to the next generation, whereas the rest N s − E solutions for the next iteration, where N s is the number of total solutions, are created by combining and/or mutating the 'parents' to form the 'children' .The two most common processes are crossover and mutation.The crossover and mutation fractions specify how many of the N s − E solutions will be produced by crossover and mutation, respectively.From this new generation, 'parents' are again selected and the process described before continues until the algorithm converges to a single optimal solution.PSO, which is also an iterative process, has a population, called swarm, of candidate solutions, called particles.The algorithm starts by creating particles with initial locations and velocities.The velocity decides towards which direction and what distance each particle will travel in the optimization space.Similar to the GA, at each iteration, the objective function is calculated for each particle in the swarm and the particle with the best fit and its corresponding position are determined.All the particles update their velocities based on their current velocities, their best known positions and the position of the particle with the best fit of the swarm.The effect of each factor is controlled by the three weighting factors; the inertia weight constant, w, the cognitive coefficient, c 1 , and the social coefficient, c 2 , that affect the optimization process.Afterwards, these velocities are used to find the new particle positions.This way the entire swarm moves towards the best solution at every iteration and eventually reaches an optimal solution.
Both optimization schemes were formed as single objective problems subject to inequality constraints.Before running the optimization process, it is necessary to tune the solvers and test different initial conditions via a grid search in order to determine the optimal hyperparameters for the problem and based on the number of variables to be optimized and their variability.From this process, a population size of 80 was chosen for the GA, resulting in a population matrix of 80 × 15 with a crossover fraction of 0.6, a mutation fraction of 0.4, and an elite count of 6.The initial population was chosen randomly following a uniform distribution over the interval [0,1] adjusted to the constraints.
For the PSO, a swarm size of 40 was chosen as optimal through the grid search.The three weighting factors, which were also tuned, were set to w = 0.7 and c 1 = c 2 = 1.6.Increasing the number of possible solutions did not seem to improve the final output for neither of the two methods.The maximum number of iterations for both schemes was set to 40, whereas the stopping criteria are to obtain a relative RSS value of 1%, which translates to an RSS value of 0.18 for the specific data, or to reach the maximum number of iterations.

Optimization domain
Different sets of parameters were considered to be optimized.After examining the options, the electrical properties of the materials that affect the radiation characteristics of the transducer the most are chosen to be optimized.These form a 15-dimensional vector that consists of the following parameters: (a) Absorber 1 relative permittivity (ε a1 ) (b) Absorber 2 relative permittivity (ε a2 ) (c) PCB relative permittivity (ε pcb ) (d) Skid relative permittivity (ε s ) (e) Plastic case relative permittivity (ε p ) (f) Shield relative permittivity (ε sh ) (g) Absorber 1 conductivity (σ a1 ) (h) Absorber 2 conductivity (σ a2 ) (i) PCB conductivity (σ pcb ) (j) Skid conductivity (σ s ) (k) Plastic case conductivity (σ p ) (l) Shield conductivity (σ sh ) (m) Source impedance (R s ) The above dielectric properties of the materials were assumed to be frequency independent.Remaining variables of the components of the antenna structure and the geometry of the model were kept fixed.The metallic components of the antenna system are modelled as PECs.All the elements of the model are considered nonmagnetic, having relative permeability μ r = 1 and zero magnetic loss σ μ = 0.The size of the domain used is 250 × 200 × 170 mm with a step size of 1 mm for both simulations, whereas the time step is set to Δt = 1.92 ps as calculated by the Courant-Friedrichs-Lewy condition (Taflove, 2000).
The GA optimization required an overall runtime of ∼3 days on an Intel(R) Xeon(R) CPU E5-2640 v4@2.40GHzCPU, whereas the PSO scheme required an execution time of ∼1.9 days on the same system.Although the optimization algorithms were executed on CPU, each A-scan was executed on an NVIDIA TITAN X 12 GB GPU to speed up the process, where each A-scan required ∼40 s to be generated.The execution times are presented for the optimization solvers executed using the final settings that resulted after tuning.

Optimization results
The two optimization schemes terminated after reaching the maximum number of iterations, obtaining the smallest possible RSS value they could converge to.For the GA algorithm, the RSS value at the first iteration was 15.4, which corresponds to a relative RSS of ∼85%, whereas the optimization ended with an RSS = 1.15 or equivalently a relative RSS of 6.5%.The PSO algorithm started with an RSS of 16.3 and reached a final value of 1.31% or 7.3% as a relative RSS. Figure 3 shows the results of the GA and the PSO techniques compared to the real data from the free-space and the PEC cases, respectively.Both the observed and simulated responses have been normalized to the maximum absolute amplitude of the free-space responses and aligned by their negative peak in order to be comparable.The normalized amplitudes represent the normalized induced voltages at the antenna terminals.The only filters applied to the real data were stacking and a 10 MHz vertical high-pass infinite impulse response filter.In both figures, the real and synthetic data are in very good agreement.Small differences occur in the amplitudes between the responses, whereas the phase has been captured correctly with negligible differences.
The unknown parameters of the antenna system that were optimized and their ranges are presented in Table 1 along with their resultant values obtained from the GA and the PSO, where it can be seen that both methods  schemes converged to similar values.The greatest differences exist in the impedance of the source, with the GA value being 100 Ω larger than the one obtained from the PSO, and in the conductivities of the absorbers.These values demonstrate how the two approaches arrive in different combinations of parameters as realistic solutions to the same problem.The GA model resulted in a lower error compared to the PSO, and therefore, the GA values are chosen for the final model.Note that the resultant values are only estimates of the true properties of the materials that were used to construct the GPR system and might be different from the true ones.The aim was to acquire a combination of realistic values of these properties, which will mimic the behaviour of the real antenna system, regardless if they are the true ones or not.

RADIATION PATTERNS
Electric and magnetic field radiation patterns can provide valuable insight into the performance of an antenna.
For an ultra-wideband GPR antenna, it is more useful to create a radiation pattern of the total emitted energy in a given angular direction, instead of presenting a pattern at a single frequency.A measure of the total emitted energy ℰ tot given by Diamanti and Annan ( 2013) and adapted from Warren and Giannopoulos ( 2017) is: where E is the electric field, r and θ are the observation radius and angle, respectively, and T is the time window.
A summation over time is performed and the pattern is presented at a specific radial distance from the centre of the transmitting antenna.The ℰ tot represents the energy included in the E-field for the whole spectrum of the radiated pulse at a given angular distance.
As the GSSI 2000 MHz system is mostly used for concrete scanning, the radiation patterns over two concrete slab with different moisture content were studied.The first scenario represents a dry concrete slab, which was modelled using a single Debye pole with a relative permittivity at infinite frequency ϵ ∞ = 4.5, zero-frequency relative permittivity ϵ p = 4.82, relaxation time of t p = 0.83 × 10 −9 s and a conductivity of σ = 0.0006 S/m.The second case represents a concrete with higher water content, where concrete was simulated with a relative permittivity at infinite frequency ϵ ∞ = 7.3, zero-frequency relative permittivity ϵ p = 12.2, relaxation time of t p = 0.62 × 10 −9 s and a conductivity of σ = 0.05 S/m.
Figure 4 demonstrates the radiation patterns, for the 2000 MHz modelled antenna system, of the E-plane and the H-plane.The E-plane is the plane that contains the E-field and the direction of maximum radiation, whereas the H-plane contains the H-field and the direction of maximum radiation (Balanis, 2012).The plots are presented in a logarithmic scale.The radiation patterns are displayed for observation distances between 0.11 and 0.35 m of the centre of the antenna structure with a 0.015m step.All patterns show a main lobe with most of the energy going into the subsurface.The maximum energy of the E-plane occurs at 180 • , perpendicular to the antenna, whereas for the H-plane occurs at 160 • due to the transmitter and receiver being offset from each other.The existing backlobes are energy going into the air and are smaller due to the shielding of the antenna as well as their ground coupling.As expected, as the relative permittivity of the subsurface increases, the main lobe becomes narrower and more directive.The energy escaping in the air is minimized with increasing permittivity as more energy gets drawn into the subsurface.The greater attenuation of the concrete model with higher water content can clearly be seen by comparing the radiation patterns of the two cases.

VALIDATION OF THE ANTENNA MODEL
In order to demonstrate the accuracy of the model, real data were acquired from two experiments and the scenarios were reproduced in the simulations and compared with the real responses.
The first test was conducted on a wooden box filled with dry sand having a metal plate buried at a depth of 19 cm.The sand was modelled as a non-dispersive homogeneous material with ε r = 3 and a loss of σ = 0.01 S/m, whereas the metal plate was modelled as PEC.Both the sand box and the metallic plate are shown in Figure 5, whereas the modelled geometry of this scenario is illustrated in Figure 6.The second scenario was from a reinforced concrete slab, where responses from two rebars, annotated in Figure 7, were investigated.The first rebar is buried at a depth of d = 6.3 cm and the second rebar at d = 10 cm, whereas both rebars have a radius of r = 1.25 cm.Both rebars were simulated as cylindrical PEC targets.The polarization of the antenna was parallel to the main axis of the rebar.Concrete is a dispersive material, and therefore its dielectric behaviour can be modelled using an extended Debye model (Bourdi et al., 2012).Considering the values of the properties of the extended Debye model that were obtained in Bourdi et al. (2012) and the condition of the concrete, the slab was modelled using a single Debye pole, having a relative permittivity at infinite frequency ε ∞ = 7.3, zero-frequency relative permittivity ε p = 12.2, relaxation time of t p = 0.62 × 10 −9 s and a conductivity of σ = 0.05 S/m.To replicate accurately the real concrete slab, the third rebar in the vicinity of the two investigated  Figure 9 shows the real versus the modelled responses from the two scenarios.It is clear that the simulated data follow closely the pattern of the real    responses, predicting accurately the shape and time of both the direct wave and the received wavelets from the targets, with minor differences in the amplitude.It should be noted that in both models, the background media were assumed homogeneous and of infinite extent (due to computational constrains not allowing us to fully model the investigated scenarios), which is obviously not the case for both the finite depth concrete slab and the sandbox.In addition, unwanted responses from the bottom of the concrete slab, along with other sources of noise, are visible in the real responses, which are not present in the modelled data.
To demonstrate the importance of including a numerical model of a real GPR transducer in the simulations, which was discussed by Diamanti et al. (2014), and also the need for optimization of unknown parameters, responses from the free-space and PEC scenario were generated using a Hertzian dipole with a 2 GHz Ricker waveform in the simulations.Figure 10 shows the real versus the synthetic responses using a Hertzian dipole for both cases.Observing the results using a theoretical source, it is obvious that the modelled data have significant differences with the real responses, and in many cases cannot be used to assist the interpretation of real data, supporting the need for including realistic transducer models in the simulations.

COMPARISON BETWEEN TRANSDUCER UNITS
For a commercial GPR system, many transducer units of the same design are constructed.These might produce slightly different responses when tested in the same scenarios.One reason for testing for small discrepancies in performance is that it is extremely difficult to create the required materials that are needed to construct a specific transducer, with the exact same dielectric properties, especially when it comes to EM absorber foams.Thus, for each transducer unit, materials with slightly different properties might be used, which affect the radiation of each antenna and lead to variations in the responses.Another reason is that due to degradation in the antenna elements with time, the antenna performance could be degraded over time.To demonstrate this, data from the same scenarios and using the same system settings were collected using two different units of the GSSI 2000 MHz 'palm' antenna.Figure 11 shows the normalized free-space direct coupling and the  metal plate responses of both units compared, where No-1 represents the real antenna system that was used to build the model described above and No-2 represents the second unit.It is obvious that the responses from the two units are not identical, resulting from differences in the dielectric properties of their components.Small variations could be also due to slight differences in the field measurements (lines slightly offset, external interference noise etc.).
In order to test which components' dielectric properties differ the most and cause these dissimilarities in the responses, optimization was utilized and a model of the second unit was built, as well.The modelled geometry of the second unit was the same as with the first, whereas the GA scheme was used to optimize the same antenna parameters.For both units, the same initial random state was used in the optimization, which would have resulted in the same values for the parameters if the two systems were identical.
The GA optimization of the second unit started with a relative RSS of 76% and reached a final value of 6.8%, which is similar to the errors obtained from the optimizations for the first unit, having a 0.3% and a 0.5% difference from the GA and the PSO resultant RSS values, respectively.Table 2 shows the resultant parameter values for the model of the second unit along with the ones from the first.For most variables, small differences are observed between the two models, relative to the ranges of values each variable can obtain, whereas the permittivities and conductivities of the absorbers had the largest contrasts.Note that neither of the resultant values of each model corresponds to the true dielectric properties of the absorbers, as mentioned before.
The optimization, resulting in larger contrasts for the absorbers compared to the rest of the parameters, simply supports the premise that the absorbers are the key components that cause most of the differences between the responses from each unit due to the variation in their material properties.

CONCLUSIONS
A realistic 3D model of the 2000 MHz 'palm' antenna from GSSI was created.GA and PSO were successfully used to acquire estimates of the unknown parameters of the antenna structure, while considering the geometry known.Two distinct data sets were used for this purpose, placing the antenna on top of a metal plate and in free space.The responses from the model were very similar to the real responses.The validity of the model was further demonstrated using comparisons with real A-scans from a reinforced concrete slab and a metal plate buried in dry sand.It was shown that the synthetic data were in a very good agreement with the real responses both in phase and amplitude, demonstrating the effectiveness of the model and its ability to capture the behaviour of the real GPR system in different scenarios.The resultant model was compared to a model of another unit of the same real antenna system, demonstrating that the two systems are not identical.The above results show that global optimizers can be applied to estimate the properties of essential parts of the models of real GPR antenna systems.The accuracy of the model provides the ability to be used in supporting the development of data-driven interpretations such as machine learning and artificial intelligence which we are currently focusing on with very promising preliminary results.

FIGURE 1 FIGURE 2
FIGURE 1 Modelled geometry of the GSSI 2000 MHz 'palm' antenna with the skid removed.

FIGURE 3
FIGURE 3 Real and modelled responses from free-space (a, c) and perfect electric conductor (PEC) (b, d) using the optimized values from the GA (a, b) and the PSO (c, d) scheme.

FIGURE 4
FIGURE 4 Radiation patterns of the 2 GHz modelled antenna for observation distances between 0.11 and 0.35 m from the centre of the antenna.For a dry (left) and a wet (right) concrete slab.

FIGURE 5
FIGURE 5 Real set-up of scenario 1.The metal plate on the right was buried in the dry sand on the left.

FIGURE 6
FIGURE 6 Modelled geometry of validation scenario 1. Sandbox with metal plate.

FIGURE 7 FIGURE 8
FIGURE 7Real set-up of scenario 2. Reinforced concrete slab.

FIGURE 9
FIGURE 9 Real versus modelled data from (a) metal plate buried in sand, (b) reinforced concrete slab for rebar 1 and (c) reinforced concrete slab for rebar 2.

FIGURE 10
FIGURE 10 Real versus modelled responses using a hertzian dipole in the simulations for (a) free-space and (b) PEC responses.

FIGURE 11
FIGURE 11Comparison between the real responses of the two transducer units for the free space (top) and the metal plate (bottom).No-1 represents the real transducer unit that was used to build the model above, whereas No-2 represents the second unit.

TABLE 1
Optimized antenna parameters.

TABLE 2
Optimized antenna parameters for the two transducer units.