Three-Dimensional Localization Method of Underground Target Based on Miniaturized Single-Frequency Acoustically Actuated Antenna Array

Abstract

The acoustically actuated antenna technology enables a significant reduction in antenna dimension, facilitating miniaturization of ground-penetrating radar systems in the very high-frequency (VHF) band. However, the current acoustically actuated antennas suffer from narrow bandwidth and low range resolution. To address this issue, this paper proposed a three-dimensional (3D) localization method for underground targets, which combined two-dimensional (2D) array direction-of-arrival (DOA) estimation with continuous spatial sampling without relying on range resolution. By leveraging the small dimension of acoustically actuated antennas, a 2D uniform linear array was formed to obtain the target’s angle using DOA estimation. Based on the variation pattern of 2D angles in continuous spatial sampling, the genetic algorithm was employed to estimate the 3D coordinates of underground targets. The numerical simulation results indicated that the root mean square error (RMSE) of the proposed 3D localization method is 1.68 cm, which outperforms conventional methods that utilize wideband frequency-modulated pulse signals with hyperbolic vertex detection in theoretical localization accuracy, while also demonstrating good robustness. The gprMax electromagnetic simulation results further confirmed that this method can effectively localize multiple targets in ideal homogeneous underground media.

1. Introduction

As a non-invasive underground detection method, ground-penetrating radar (GPR) has been widely applied in various fields, including mineral prospecting [1,2], pavement layer analysis [3,4], archaeology [5], glacier research [6], and landmine detection [7,8], due to its unique capabilities in high resolution, operational efficiency, and cost-effectiveness. GPR usually operates in lossy underground media, where electromagnetic wave attenuation increases with frequencies. Consequently, lower operating frequencies offer better penetration capability in such lossy media.

Conventional antennas rely on electromagnetic resonance to transmit and receive signals, requiring their physical dimensions to match the wavelength. Very high frequency (VHF, 30 MHz–300 MHz) electromagnetic waves are particularly suitable for underground detection. However, conventional VHF antennas currently suffer from large dimensions (as shown in

Table 1. Common VHF band GPR antenna dimension.

Radar Manufacturer	Center Frequency	Antenna Dimension
Sensors & Software Pluse EKKO	50 MHz	2 m (length)
	100 MHz	1 m (length)
	200 MHz	0.5 m (length)
GSSI	100 MHz	2.031 m × 0.965 m × 0.305 m
	200 MHz	0.6 m × 0.6 m × 0.3 m

), which fundamentally limit further miniaturization of low-frequency antennas and consequently hamper technological advancements in underground detection.

In 2017, the Defense Advanced Research Projects Agency (DARPA) proposed a mechanical antenna concept for antenna miniaturization, which has attracted global research attention and has become a research hotspot in the field of miniaturized antennas [9,10,11,12]. As a category of mechanical antennas, acoustically actuated antennas utilize acoustic resonance in magnetoelectric heterostructures—composed of piezoelectric and magnetostrictive materials—to replace conventional electromagnetic resonance. The operational principle specifies that these antennas require dimensional matching with acoustic wavelengths instead of electromagnetic wavelengths [13,14,15]. Given that acoustic wave propagation velocity in air is lower by orders of magnitude than electromagnetic wave velocity, acoustic wavelengths at the same frequency are substantially shorter than electromagnetic wavelengths. This fundamental distinction theoretically enables acoustically actuated antennas that operate at lower frequencies to achieve significant dimensional miniaturization.

Current acoustically actuated antenna technology integrated with semiconductor fabrication processes enables dimensional reduction in antennas to within 1% of the operating electromagnetic wavelength [16,17]. In this study, acoustically actuated antennas with a center frequency of 60 MHz were utilized, measuring 10 mm × 3 mm. Multiple antennas were integrated on a single wafer (5 × 3 array, measuring 30 mm × 20 mm, as shown in Figure 1), forming a transmitting or receiving element. This enables the miniaturization of VHF-band radar systems, allowing integration into size-constrained platforms such as small unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs).

GPR systems typically employ wide bandwidths to achieve high-range resolution, utilizing signal types such as impulse signals, frequency-modulated continuous waves (FMCW), and linear frequency-modulated (LFM) pulses. Underground target detection and localization generally use the B-scan method, which involves spatial sampling along a linear path to generate a two-dimensional (2D) cross-sectional image. Target localization is accomplished through hyperbolic vertex detection. For three-dimensional (3D) localization or imaging methods, multiple parallel B-scans need to be constructed to obtain information from multiple parallel cross-sectional images, which is also known as a C-scan. In recent years, numerous novel methods have been proposed to enhance 3D localization efficiency. Reference [18] demonstrated a uniform linear array (ULA) configuration where the B-scan direction orthogonal to the array provides an equivalent C-scan effect (shown in Figure 2a). Alternatively, Reference [19] proposed a circular array system deployed in vertical borehole movement, enabling 3D target localization via direction and range measurements at different depths (shown in Figure 2b).

Acoustically actuated antenna technology is rapidly developing, but it still faces significant bandwidth limitations that prevent the use of conventional broadband signal schemes. This constraint results in low-range resolution, and the aforementioned 3D localization methods, which rely on high-range resolution, are not applicable. Regarding the signal type, single-frequency continuous waves (CW) are the most suitable for acoustically actuated antennas. However, conventional CW radar inherently lacks the ability to measure range. By leveraging the miniaturization advantage of acoustically actuated antennas, the directional estimation performance can be improved by composing an array, thereby achieving indirect localization of subsurface targets.

At present, there have been some studies using direction of arrival (DOA) estimation in underground exploration. For example, circular DOA estimation is used in drilling exploration in references [19,20]. However, in locating the underground target, the measured distance is still used in addition to the angle estimated by DOA. Reference [21] proposes a narrowband signal-based imaging method that employs ULAs (shown in Figure 2c). This method uses the same spatial sampling direction as the array direction, acquires angular information via DOA estimation, and implements imaging through back-projection algorithms, addressing underground localization challenges in narrowband systems. However, this method only resolves 2D localization in a single B-scan operation, requiring multiple scanning iterations for complete 3D coordinate determination.

During the investigation, we revealed the variation patterns of subsurface targets’ 2D angular parameters in continuous spatial sampling: specifically, the functional curves relating the targets’ angular parameters and the array center’s displacement coordinates along the movement direction are uniquely determined by the targets’ 3D spatial coordinates. Consequently, estimating the 3D coordinates of subsurface targets constitutes a nonlinear optimization problem. Genetic algorithm, as a global optimization method that has been successfully applied in subsurface localization scenarios [22,23], provides an effective solution framework for this challenge.

This study aims to address the application challenges of acoustic-wave-excited antenna technology in underground target detection, achieve miniaturization of VHF-band GPR systems, and realize high-efficiency 3D localization of subsurface targets. This paper proposes a method that combines 2D DOA estimation with continuous spatial sampling based on acoustically actuated antennas (shown in Figure 2d). This method does not rely on range resolution and utilizes acoustically actuated antennas to form a 2D ULA, which is employed to obtain the target’s angle using DOA techniques. Based on the variation pattern of the target’s angle in continuous spatial sampling, a genetic algorithm is employed to estimate the 3D coordinates of underground targets, enabling high-precision localization.

2. Methods

2.1. Signal Model

The receiving antenna array is configured as an L-shape with 2N − 1 elements, where each orthogonal axis comprises N elements with one shared element at the intersection. A single transmitting antenna element is deployed, and the spatial arrangement of the antenna array is illustrated in Figure 3.

To facilitate the distinction of information from two orthogonal directions, the array parallel to the spatial sampling direction is designated as the parallel array, measuring target angle $\varphi$, while the array orthogonal to the spatial sampling direction is defined as the orthogonal array, determining target angle $\theta$, as shown in Figure 4.

This paper adopts a two-dimensional DOA estimation method that decouples parallel array DOA estimation and orthogonal array DOA estimation. This approach not only satisfies the localization requirements of the proposed method but also enables compatibility with a ULA under limited system resources to achieve 3D localization through two spatial sampling operations (one array parallel to the sampling direction and another array orthogonal to the sampling direction).

This paper establishes the following assumptions for the detection scenario:

• Signal sources are zero-mean and non-Gaussian;
• The array elements receive ideal Gaussian white noise that is statistically independent of the signal sources;
• The number of signal sources is smaller than the number of array elements;
• All elements in the receiving array maintain identical reception characteristics across all directions without mutual interference or coupling effects.

As shown in Figure 5, taking the orthogonal array as an example, the receiving antenna array is a ULA composed of N elements numbered sequentially as 0 to N − 1, with inter-element spacing d. Assuming there are K signal sources, where the direction of the source k is ${\theta }_k$. Based on geometric relationships, the time delay between signal arrivals at adjacent array elements can be derived as

$${\tau }_k={\displaystyle \frac{d\sin ({\theta }_k)}{c}}$$

(1)

where $c$ denotes the speed of light in a vacuum.

Assuming the signal received by element 0 from the source k is $s_k\left(t\right)$, then the signal received by element n from the source k can be expressed as

$$s_k(t-n\tau )=s_k(t)\exp (-jn{\omega }_0{\tau }_k)=s_k(t)\exp (-j{\displaystyle \frac{2n\mathsf{\pi }d\sin ({\theta }_k)}{\lambda }})$$

(2)

where ${\omega }_0$ denotes the angular velocity of the signal, ${\omega }_0=2\pi f=2\pi c/\lambda$.

Considering K targets, the composite signal received by element n can be expressed as

$$x_n(t)={\displaystyle \sum _{k=1}^K{s}_k(t)\exp (-jn{\omega }_0{\tau }_k)}+n_n(t)$$

(3)

Therefore, the signal received by the entire array can be expressed in vector form as

$$X(t)=A(\theta )S(t)+N(t)$$

(4)

where $\mathit{A}\left(\theta \right)$ denotes the array manifold, $\mathit{X}\left(t\right)$ denotes the received data vector, $\mathit{S}\left(t\right)$ denotes the source signal vector, and $\mathit{N}\left(t\right)$ denotes the noise vector.

$$A(\theta )=\left[\begin{array}{cccc}a({\theta }_1)& a({\theta }_2)& \cdots & a({\theta }_K)\end{array}\right]$$

(5)

$$X(t)={\left[\begin{array}{cccc}{x}_0(t)& x_1(t)& \cdots & x_{N-1}(t)\end{array}\right]}^{{\rm T}}$$

(6)

$$S(t)={\left[\begin{array}{cccc}{s}_1(t)& s_2(t)& \cdots & s_K(t)\end{array}\right]}^{{\rm T}}$$

(7)

$$N(t)={\left[\begin{array}{cccc}{N}_0(t)& N_1(t)& \cdots & N_{N-1}(t)\end{array}\right]}^{{\rm T}}$$

(8)

In Equation (5), $\mathit{a}\left({\theta }_k\right),k=1,2,\dots ,K$ denotes the steering vector.

$$a({\theta }_k)=\left[\begin{array}{c}1\\ \begin{array}{l}\exp (-j{\omega }_0{\tau }_k)\\ \exp (-j2{\omega }_0{\tau }_k)\end{array}\\ ⋮\\ \exp (-j(N-1){\omega }_0{\tau }_k)\end{array}\right]$$

(9)

The preceding analysis assumes electromagnetic wave propagation in free space. When propagating through underground lossy media, electromagnetic waves experience both amplitude attenuation and velocity variation. Specifically, the change in propagation velocity affects the phase differences of received signals between array elements. The relationship between the phase coefficient in underground media and the medium’s electrical properties is given by

$$\beta ={\omega }_0\sqrt{\mu \epsilon }\sqrt{{\displaystyle \frac{1}{2}}\left[\sqrt{1+{\left({\displaystyle \frac{\sigma }{{\omega }_0\epsilon }}\right)}^2}+1\right]}$$

(10)

In Equation (10),

$\mu$ denotes the magnetic permeability of the underground medium;

$\epsilon$ denotes the permittivity of the underground medium;

$\sigma$ denotes the electrical conductivity of the underground medium.

Typically, the electrical parameters of underground media satisfy the following relationship:

$${\displaystyle \frac{\sigma }{\omega \epsilon }}\ll 1$$

(11)

Therefore, the phase coefficient in Equation (10) can be simplified as

$$\beta =\omega \sqrt{\mu \epsilon }$$

(12)

Therefore, the propagation velocity of electromagnetic waves in underground media can be approximated as

$$v={\displaystyle \frac{\omega }{\beta }}={\displaystyle \frac{c}{\sqrt{{\epsilon }_r}}}$$

(13)

where $\sqrt{{\epsilon }_r}$ denotes the relative permittivity of the medium.

This proposed method specifically targets ground-coupled detection of GPR. This approach assumes that the entire propagation path can be reasonably approximated as occurring within underground media, in which geometric relationship alterations caused by surface refraction can be neglected. Considering the velocity variations within underground media, the steering vector in the received signal model requires modification. The modified steering vector formulation becomes

$$a({\theta }_k)=\left[\begin{array}{c}1\\ \begin{array}{l}\exp (-j\sqrt{{\epsilon }_r}{\omega }_0{\tau }_k)\\ \exp (-j2\sqrt{{\epsilon }_r}{\omega }_0{\tau }_k)\end{array}\\ ⋮\\ \exp (-j(N-1)\sqrt{{\epsilon }_r}{\omega }_0{\tau }_k)\end{array}\right]$$

(14)

2.2. DOA Estimation of Underground Targets

In radar target detection applications, the signals received by antenna arrays are the reflected waves generated by the interaction between transmitted signals and targets. Echo signals from different targets exhibit coherence with fixed phase differences. However, the signal type transmitted by the acoustically actuated antenna in this method is a single-frequency CW, which limits the discrimination of multiple coherent target echoes using conventional time-domain or frequency-domain processing techniques.

To address the coherent multi-target echo problem, the Forward-Backward Spatial Smoothing Multiple Signal Classification (FBSS-MUSIC) algorithm is applied in this method. Given that DOA estimation can be performed independently in orthogonal and parallel array directions, the following analysis will focus on the orthogonal array as a representative example.

From Equation (4), the covariance matrix of the received array signals can be derived as

$$R_X=\mathrm{E}[X\left(t\right)X^{\mathrm{H}}(t\left)\right]=A(\theta \left)R_S{A}^{\mathrm{H}}\right(\theta )+{\sigma }_{\mathrm{noise}}^{2}I$$

(15)

where the following is true:

${\mathit{R}}_s$ denotes source signal covariance matrix;

${\sigma }_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}^2$ denotes noise power;

$\mathit{I}$ denotes the identity matrix.

By performing eigenvalue decomposition on the covariance matrix ${\mathit{R}}_x$, when the array receives K signals, there are K larger eigenvalues determined by the signals, while the remaining smaller eigenvalues are determined by noise, with their values equal to ${\sigma }^2$.

$$R_X={\displaystyle \sum _{i=1}^N{\xi }_i}{u}_i{u}_i^H=U_S{\Sigma }_S{U}_S^{\mathrm{H}}+U_N{\Sigma }_N{U}_N^{\mathrm{H}}$$

(16)

where the following is true:

${\xi }_1\ge {\xi }_2\ge \dots \ge {\xi }_K\gg {\xi }_{K+1}\ge \dots {\xi }_N$: eigenvalues sorted in descending order;

${\mathit{U}}_{\mathit{S}}=[{\mathit{u}}_1,{\mathit{u}}_2,\dots ,{\mathit{u}}_K]$: signal subspace eigenvectors;

${\mathit{U}}_{\mathit{N}}=[{\mathit{u}}_{K+1},{\mathit{u}}_{K+2},\dots ,{\mathit{u}}_N]$: noise subspace eigenvectors.

However, when signal sources are coherent, the rank reduction in the covariance matrix ${\mathit{R}}_X$ causes the signal subspace dimension to be smaller than the number of sources [24]. This leads to some steering vectors of coherent sources being unable to achieve full orthogonality with the noise subspace. Consequently, the directional estimation accuracy degrades severely.

The spatial smoothing method effectively restores the rank of the covariance matrix, with its principle shown in Figure 6.

The ULA is partitioned into P overlapping subarrays, each containing L array elements, with the quantitative relationship expressed as: $N=P+L-1$.

For subarray $p$,

$$X_p(t)=[\begin{array}{cccc}{x}_p& x_{p+1}& \dots & x_{p+L-1}\end{array}]=A_{\mathrm{L}}{D}^{p-1}S(t)+N_p(t)$$

(17)

where

$$D=\left[\begin{array}{cccc}\exp (j{\displaystyle \frac{2\pi d\sin ({\theta }_1)}{\lambda }})& 0& \dots & 0\\ 0& \exp (j{\displaystyle \frac{2\pi d\sin ({\theta }_1)}{\lambda }})& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & \exp (j{\displaystyle \frac{2\pi d\sin ({\theta }_K)}{\lambda }})\end{array}\right]$$

(18)

The covariance matrix of this subarray ${\mathit{R}}_p$ is expressed as

$$R_p=A{D}^{(p-1)}{R}_s{D}^{(p-1)H}{A}^H$$

(19)

The covariance matrix of each submatrix in forward smoothing and backward smoothing is ${\mathit{R}}_p^f$ and ${\mathit{R}}_p^b$.

Rank restoration is achieved by averaging the covariance matrices of each subarray. The processed covariance matrix after forward smoothing becomes

$$R_f={\displaystyle \frac{1}{P}}{\displaystyle \sum _{p=1}^P{R}_p^f}$$

(20)

Reference [25] demonstrates that when $L\ge K$ and $P\ge K$, the matrix ${\mathit{R}}_f$ preserves full rank. Building upon this foundation, by considering reversed-order subarrays, the backward smoothed covariance matrix can be expressed as

$$R_b={\displaystyle \frac{1}{P}}{\displaystyle \sum _{p=1}^P{R}_p^b}$$

(21)

The forward-backward smoothed covariance matrix ${\mathit{R}}_{fb}$ is the average of both, specifically defined as

$$R_{fb}={\displaystyle \frac{1}{2}}(R_f+R_b)$$

(22)

Reference [26] demonstrates that when $L\ge K$ and $2P\ge K$, the covariance matrix ${\mathit{R}}_{fb}$ achieves full rank under optimal weighting conditions. Compared with pure forward or backward smoothing techniques, this approach achieves lower aperture loss.

The original covariance matrix ${\mathit{R}}_{\mathit{X}}$ is replaced by ${\mathit{R}}_{fb}$, and new signal subspace, noise subspace, and their eigenvector are obtained by eigenvalue decomposition. Under ideal conditions, the signal subspace and noise subspace preserve orthogonality. Consequently, the steering vectors of the signal subspace are orthogonal to the noise subspace.

$$a^{\mathrm{H}}(\theta )U_N=0$$

(23)

In practical computations, the data array has a finite length, and the maximum likelihood estimate of the data covariance matrix is given by

$${\widehat{R}}_X={\displaystyle \frac{1}{L}}{\displaystyle \sum _{i=1}^{L}X(t)X{(t)}^H}$$

(24)

where $L$ is the number of snapshots.

As the snapshot count attains statistical sufficiency, the estimated covariance matrix asymptotically converges to its theoretical counterpart. In practice, imperfect orthogonality between the signal subspace’s steering vectors and the noise subspace manifests as residual inner products with $|⟨\mathit{a}(\theta ),{\mathit{U}}_{\mathrm{n}}⟩|<\mathsf{ϵ}$. Because of this non-ideal situation, DOA estimation is typically a numerical optimization process:

$$\widehat{\theta }=\arg \underset{\theta }{\min }{a}^H(\theta )U_n{U}_n^{H}a(\theta )$$

(25)

Therefore, the spatial spectrum is

$$P(\theta )={\displaystyle \frac{1}{a^H(\theta )U_n{U}_n^{H}a(\theta )}}$$

(26)

The peaks in the spatial spectrum correspond to coordinates containing target angle information. Following DOA estimation completion in both array directions, a set of angles $\phi$ and $\theta$ are obtained, which will be utilized for subsequent localization of underground targets.

2.3. Target Position Estimation Based on Genetic Algorithm

When the antenna array conducts spatial sampling along the specified direction, the angle of underground targets relative to the array varies with sampling positions. Taking the positive y-axis direction as the movement direction for spatial sampling, the angular variation relative to the orthogonal array is shown in Figure 7, while the corresponding angular variation relative to the parallel array is shown in Figure 8.

Assume that the center coordinates of the antenna array are $(x,y,z)$, and the coordinates of the underground target are $(x_0,y_0,z_0)$. The angular variation of the target concerning the antenna array movement can be expressed as

$$\tan (\theta )={\displaystyle \frac{x-x_0}{\sqrt{{\left(y-y_0\right)}^2+{(z-z_0)}^2}}}$$

(27)

$$\tan (\phi )={\displaystyle \frac{y-y_0}{\sqrt{{(x-x_0)}^2+{(z-z_0)}^2}}}$$

(28)

To complete localization, the 3D coordinates $\left(x_0,y_0,z_0\right)$ need to be estimated. During spatial sampling, the target direction $(\theta ,\varphi )$ of each position is recorded after 2D DOA estimation. By equipping the GPR system with a Position and Orientation System (POS), the position $\left(x,y,z\right)$ and timing of the antenna array in each spatial sampling point can be achieved. With known antenna center coordinates $\left(x,y,z\right)$ and target directions $(\theta ,\varphi )$ from multiple spatial sampling points, estimating the target’s 3D coordinates $\left(x_0,y_0,z_0\right)$ constitutes a nonlinear curve-fitting problem.

This paper employs the genetic algorithm to perform nonlinear curve fitting for underground target localization. The genetic algorithm is an optimization method based on natural selection and genetic mechanisms, simulating biological evolution through iterative operations to progressively approach the optimal solution. The workflow of the genetic algorithm implemented in this paper is shown in Figure 9.

The steps of the genetic algorithm in this method are as follows:

• Population initialization: set the population size to M, and randomly generate an initial population containing M coordinates within the system’s detectable range:

  $$P_0=\left\{S_1=\left(x_1,y_1,z_1\right),S_2=(x_2,y_2,z_2),\dots ,S_M=(x_M,y_M,z_M)\right\}$$

  (29)

2.

Fitness evaluation: Substitute individual parameters and antenna positions into the target model to calculate the modeled angle. For the individual m in the population at the sampling point i, the modeled angle ${\widehat{\theta }}_{m,i}$ and ${\widehat{\varphi }}_{m,i}$ are

$${\widehat{\theta }}_{m,i}=\mathrm{actan}\left({\displaystyle \frac{x_m-x_i}{\sqrt{{(y_m-y_i)}^2+{(z_m-z_i)}^2}}}\right)$$

(30)

$${\widehat{\varphi }}_{m,i}=\mathrm{actan}\left({\displaystyle \frac{y_m-y_i}{\sqrt{{(x_m-x_i)}^2+{(z_m-z_i)}^2}}}\right)$$

(31)

The squared error between the modeled angles $({\widehat{\theta }}_{m,i},{\widehat{\varphi }}_{m,i})$ and the DOA results $({\theta }_i,{\phi }_i)$ serves as the cost function. When the squared error is smaller, the coordinates represented by the individual are closer to the true position. The cost function can be expressed as:

$$J(S_m)={\displaystyle \sum _{i=1}^I[{({\varphi }_i-{\widehat{\varphi }}_{m,i})}^2}+{({\theta }_i-{\widehat{\theta }}_{m,i})}^2]$$

(32)

where $S_m$ is the individual m. The value of the cost function is always positive, and the reciprocal of the cost function is used to measure the fitness of the individual. Therefore, individuals with higher fitness are more likely to be passed on to the next generation:

3.

Selection operation: Based on the fitness of individuals, those with higher fitness in the current population are selected and copied to the next generation. To ensure the robustness of the algorithm, this paper adopts the elitism strategy, in which the individual with the highest fitness is directly transferred to the next generation without participating in subsequent operations. The remaining individuals are selected using the roulette wheel method, where the wheel is divided into sectors proportional to each individual’s fitness. The pointer randomly stops at one of the sectors, and the selected individual is passed to the next generation. The probability of selection is determined by the central angle of each sector, which is based on the relative fitness of the individuals, as shown in Equation (37).

$$P(S_m)={\displaystyle \frac{F(S_m)}{{\displaystyle \sum _{\mathrm{m}=1}^{M}F(S_m)}}}$$

(33)

4.

Crossover operation: This operation simulates the genetic recombination process by combining the traits of parent individuals to generate new offspring. In this paper, a single-point crossover is used, where a position is selected in the parent individuals’ genes, and the segment starting from that position is swapped, producing two new offspring. Let the gene representations of parent individuals a and b be as follows:

$$S_{\mathrm{a}}=(x_{\mathrm{a}},y_{\mathrm{a}},z_{\mathrm{a}}),S_{\mathrm{b}}=(x_{\mathrm{b}},y_{\mathrm{b}},z_{\mathrm{b}})$$

(34)

Assuming the crossover point is randomly selected as the y coordinate, the offspring is generated.

$$S_{\mathrm{a}}^{\prime }=(x_{\mathrm{a}},y_{\mathrm{b}},z_{\mathrm{a}}),S_{\mathrm{b}}^{\prime }=(x_{\mathrm{b}},y_{\mathrm{a}},z_{\mathrm{b}})$$

(35)

5.

Mutation operation: This operation introduces random changes to an individual’s genes, increasing the diversity of the population and preventing it from getting trapped in local optima. In this paper, a small perturbation is added to a randomly selected dimension of an individual’s parameters, causing a slight shift in the current coordinate position in a random direction. Taking x-coordinate mutation as an example, this operation can be formulated as

$$(x_m+\Delta x,y_m,z_m)$$

(36)

where $\mathsf{\Delta }x$ is a small perturbation that follows a normal distribution.

6.

Iteration: The offspring generated through the aforementioned steps form a new population generation. This population iteratively undergoes fitness evaluation, selection, crossover, and mutation to produce the next generation.

The stopping criteria are defined as either reaching the maximum generation limit or when the best fitness improvement falls below a preset threshold. Upon termination, the highest-fitness individual’s coordinates are identified as the optimal estimate for the underground target’s location, finalizing the 3D coordinate estimation process.

2.4. Algorithm Flow

The operational process of the underground target 3D localization mainly includes performing continuous spatial sampling with the radar system and signal processing (shown in Figure 10). To achieve high-precision underground target localization, the process requires acquiring precise coordinates and angular deviations of the radar system corresponding to each spatial sampling point through integration with a POS. The POS requires initial calibration after system startup and temporal interpolation prior to data utilization to ensure spatiotemporal synchronization with the GPR system’s spatial sampling points.

The signal processing involves five steps: direct wave removal, 2D DOA estimation, angle calibration, data association, and target position estimation (shown in Figure 11). The 2D DOA estimation and target position estimation are discussed in Section 2.2 and Section 2.3, respectively:

• Direct wave removal:

In the raw signals, the direct wave signal exhibits substantially higher energy compared with target echoes, which can overwhelm the echoes. Methods such as the averaging method, wave-number domain notch filtering, adaptive filtering, singular value decomposition, and wavelet transform can be used to remove these interference components.

• Angle calibration:

The POS data are aligned with the sampling points through interpolation, obtaining the angular deviation of each sampling point relative to the initial state. These deviations are used to compensate for and correct the DOA estimation results.

• Data association:

The DOA estimation results are paired with a known target trajectory. For each sampling point, the system may detect multiple targets simultaneously. The angle information from each sampling point needs to be clustered into distinct subsets associated with the underground targets, with each subset corresponding to a single target source with maximum probability. Data association can be implemented through methods such as global nearest-neighbor data association, joint probability data association, and multiple hypothesis tracking [24].

3. Experiments

3.1. Numerical Simulation

3.1.1. Effect of Array Dimension on DOA Estimation Accuracy

To ensure the proposed method maintains a dimensional advantage over conventional GPR, the inter-element spacing in the array was designed significantly smaller than the typical spacing $1/2\lambda$.

In the simulation setup, the parameter settings for the underground medium were referenced to soil with a volumetric water content of 10%, with the relative permittivity configured as 6. The receiving array was set up as a ULA of six acoustically actuated antennas with lengths ranging from 0.3 m to 2 m. A single transmitting antenna was set up, and the transmitting signal is a CW with a center frequency of 60 MHz. The noise model adopted additive white Gaussian noise, with signal-to-noise ratio (SNR) levels spanning 0–20 dB in 5 dB increments.

The root mean square error (RMSE) of DOA estimation is defined as

$${\mathrm{RMSE}}_{\theta }={\displaystyle \frac{1}{K}}{\displaystyle \sum _{k=1}^K\sqrt{\frac{1}{\mathrm{Monte}}{\displaystyle \sum _{n=1}^{\mathrm{Monte}}{({\widehat{\theta }}_{kn}-{\theta }_k)}^2}}}$$

(37)

where the following is true:

$\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{e}$ denotes the number of Monte Carlo experiments;

${\theta }_k$ denotes the true angle of the target k relative to the array;

${\widehat{\theta }}_{kn}$ denotes the estimated angle of target k relative to the array in the n-th Monte Carlo experiment.

The simulation results are shown in Figure 12.

Doubling the array length induced approximately 50% RMSE reduction, corresponding to a twofold enhancement in estimation precision. Referring to Table 1 for conventional antenna dimensions, the proposed configuration maintained dimensional advantage when the antenna length remained below 1 m. Under SNR ≥ 10 dB with aperture loss induced by forward–backward spatial smoothing, the system sustained $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}\le 1°$.

3.1.2. Effect of Number of Array Elements on DOA Estimation Accuracy

For a practical GPR system, increasing the number of receiving antennas proportionally increased both receiver channels and analog-to-digital converters (ADCs), leading to linear growth in system weight and cost. Therefore, a trade-off between the number of antennas and the system’s weight and cost was necessary to select an appropriate number of receiving antennas. The array length was fixed at 1 m, and the number of antennas was set to range from three to eleven, with other simulation conditions remaining the same as in Section 3.1.1.

As shown in Figure 13, with a fixed array length, the improvement in DOA estimation accuracy was gradual. For the FBSS-MUSIC method used in this paper, as the number of array elements increased, the aperture loss during multi-target estimation decreased, and the ability to resolve more coherent targets improved. Therefore, the selection of the number of array elements should primarily depend on the number of underground targets.

3.1.3. Effect of Coherent Signal Discrimination

A receiving ULA (6 elements, length 1 m) and two targets were arranged in 3D space with their positional relationship shown in Figure 14. SNR was 10 dB, and other simulation parameters remained the same as in Section 3.1.1. In this configuration, the echo signals from the two targets were coherent. The spatial spectra obtained using the FBSS-MUSIC algorithm and conventional MUSIC algorithm were shown in Figure 15, where the two red vertical lines indicated the true angles of the targets. The FBSS-MUSIC algorithm successfully estimated both target angles, whereas the conventional MUSIC algorithm failed to resolve the coherent signals and produced an erroneous single-angle estimation.

3.1.4. Underground Target Localization Performance

In a 3D Cartesian coordinate system, the underground medium was defined as the spatial region with z-axis coordinates less than −0.5 m, where the z = −0.5 m plane parallel to the xoy-plane represented the ground surface. The antenna array was configured in an L-shape, with the intersection point of its two-directional arrays initially positioned at (1 m, 1 m, −0.5 m). The array elements extend along the positive x-axis and positive y-axis directions, respectively, each directional array containing six elements with a length of 1 m. A single transmitting antenna was positioned at (2 m, 2 m, −0.5 m). The radar system moved along the positive x-axis direction, performing spatial sampling at 0.1-meter intervals. The SNR was set to 10 dB, while other simulation parameters remained the same as in Section 3.1.1.

Two targets were configured with their true coordinates and the estimated coordinates using the proposed method listed in

Table 2. True and estimated coordinates of the underground targets in simulations.

	True Coordinate (x, y, z)/m	Estimated Coordinate (x, y, z)/m	Coordinate Estimation Error/m
Target 1	(11, 3.5, −3.5)	(11.0003, 3.4937, −3.5082)	0.0104
Target 2	(17, 0.5, −2.5)	(17.0024, 0.5074, −2.5175)	0.0192

. The DOA estimation angles for targets and the genetic algorithm fitting results are shown in Figure 16, where red markers indicate the DOA estimated angles and the blue curve represents the genetic algorithm fitting curves.

The coordinate fitting results of 100 Monte Carlo experiments were calculated. Figure 17 shows the coordinate estimation values across all dimensions for the 100 Monte Carlo experiments. It can be observed that the algorithm demonstrates good robustness, as no abnormal estimation results were found during the 100 experiments.

The RMSE can be calculated to measure the accuracy of the coordinate estimation. The RMSE of each dimension coordinate was defined as follows:

$${\mathrm{RMSE}}_x=\sqrt{{\displaystyle \frac{1}{\mathrm{Monte}}}{\displaystyle \sum _{n=1}^{\mathrm{Monte}}{({\widehat{x}}_n-x_0)}^2}}$$

(38)

$${\mathrm{RMSE}}_y=\sqrt{{\displaystyle \frac{1}{\mathrm{Monte}}}{\displaystyle \sum _{n=1}^{\mathrm{Monte}}{({\widehat{y}}_n-y_0)}^2}}$$

(39)

$${\mathrm{RMSE}}_z=\sqrt{{\displaystyle \frac{1}{\mathrm{Monte}}}{\displaystyle \sum _{n=1}^{\mathrm{Monte}}{({\widehat{z}}_n-z_0)}^2}}$$

(40)

The RMSE of 3D coordinate synthesis of underground targets was defined as follows:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{\mathrm{Monte}}}{\displaystyle \sum _{n=1}^{\mathrm{Monte}}\left[{({\widehat{x}}_n-x_0)}^2+{({\widehat{y}}_n-y_0)}^2+{({\widehat{z}}_n-z_0)}^2\right]}}$$

(41)

where $(x_0,y_0,z_0)$ were the true 3D coordinates of the underground target, and $({\widehat{x}}_n,{\widehat{y}}_n,{\widehat{z}}_n)$ were the genetic algorithm estimation results for the n-th Monte Carlo experiment.

Based on the above formula, the RMSEs were calculated (as shown in

Table 3. RMSE for each dimension and overall RMSE of each target.

	True Coordinate (x, y, z)/m	${\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}}_{\mathbf{x}}$/m	${\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}}_{\mathbf{y}}$/m	${\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}}_{\mathbf{z}}$/m	RMSE/m
Target 1	(11, 3.5, −3.5)	0.0025	0.0042	0.0157	0.0165
Target 2	(17, 0.5, −2.5)	0.0017	0.0073	0.0153	0.0170

).

The overall RMSE from three Monte Carlo experiments was 0.0168 m.

In conventional GPR systems using wideband LFM pulse signals for 3D target localization in underground media, spatial sampling is required from at least two orthogonal directions, and localization is performed based on the vertex positions of hyperbolas in the B-scan image. Under the same conditions, we simulate the conventional GPR systems. Here, an LFM pulse radar with the same center frequency was used, and hyperbola detection was performed on the results after pulse compression. The relationship between the localization accuracy and bandwidth is shown in Figure 18. It can be seen that the theoretical positioning accuracy of the proposed method is significantly higher than that of conventional methods.

3.2. Electromagnetic Simulation of the Underground Scene

In this paper, gprMax (version 3.1.7) was used to simulate the underground target detection [27]. It solves Maxwell’s equations in 3D using the Finite-Difference Time-Domain (FDTD) method. It is a commonly used electromagnetic simulation software in the field of GPR.

3.2.1. Echo Signals in Different Scenarios

In practical scenarios, the interference from direct wave signals manifests significant detrimental effects. During the 2D DOA estimation process, direct wave signals tend to generate spurious targets while simultaneously obscuring the phase information embedded in target echoes. However, the efficacy of existing direct wave suppression methodologies exhibits substantial variations across different operational scenarios.

The simulation space was configured as a 30 m × 4 m planar area, with initial consideration given to a homogeneous underground medium. Target types included metallic spheres and cavities, both with radii of 0.1 m. Other simulation parameters were configured as

Table 4. Parameters of gprMax simulation.

Parameter	Value
relative permittivity	6
conductivity	0.001 S/m
relative permeability	1
type of waveform	continuous sine
center frequency	60 MHz

:

Direct wave suppression was applied to the simulation results, with the corresponding effectiveness illustrated in Figure 19.

Comparative analysis reveals that within homogeneous media environments, both metallic targets and cavity structures exhibit effective suppression of direct wave components. The investigation subsequently transitions to metallic spherical targets embedded within inhomogeneous media configurations, as systematically characterized in the following experimental framework.

Figure 20 demonstrates that within inhomogeneous media environments, direct wave suppression demonstrates inadequate performance. Consequently, constrained by the inherent limitations of current direct wave cancellation algorithms, the proposed methodology cannot effectively process and precisely localize targets embedded in such complex media configurations.

3.2.2. Simulation of Uniform Medium Metal Ball Target Scenario

A space with dimensions of 30 m × 4 m × 4 m was set, where the medium extended to a depth greater than 0.5 m. The parameters for the simulation scenario were the same as in Table 4.

The targets were two metallic spheres with a 0.1 m radius. The positions of targets and antenna array parameters were the same as in Section 3.1.4. The simulation model is shown in Figure 21.

As shown in Figure 22a, the raw data generated by gprMax electromagnetic simulation primarily consists of direct waves. After removing the direct waves, the target echo signals are shown in Figure 22b.

DOA estimation, position estimation based on genetic algorithm, and other operations were carried out on the signal after the removal of the direct wave, and the results are shown in Figure 23.

Using this method, the final localization results are shown in

Table 5. Comparison of electromagnetic simulation localization results with true positions.

	True Coordinate (x, y, z)/m	Estimated Coordinate (x, y, z)/m	Coordinate Estimation Error/m
Target 1	(11, 2.5, −3.5)	(10.9588, 2.4881, −3.5116)	0.0444
Target 2	(17, −0.5, −2.5)	(16.9983, −0.54129, −2.5269)	0.0498

:

4. Discussion

Numerical simulation results indicate that the proposed method achieves a comprehensive RMSE of 1.68 cm, theoretically demonstrating its high-precision capability for underground target localization. Compared with conventional GPR using broadband LFM pulse signals, this approach exhibits superior accuracy. Analysis from Table 3 reveals distinct dimensional characteristics: the x-coordinate shows the smallest RMSE, followed by the y-coordinate, while the z-coordinate’s RMSE is significantly higher.

Analysis of the sources of dimensional information reveals that, in the numerical simulations, the spatial sampling direction was parallel to the x-axis. The x-coordinate information manifests in angular variation curves, serving as the symmetry axis in the $\theta$ direction and the central symmetry point in the $\varphi$ direction, exhibiting distinct characteristics. In the $\varphi$ direction, the y- and z-coordinates are fully coupled, while in the $\theta$ direction, the y-coordinate serves as a scaling factor for the entire curve, demonstrating a more pronounced characteristic representation. Therefore, the three dimensions coordinate exhibit different levels of accuracy.

The localization results of gprMax electromagnetic simulation show that this method can locate underground targets, but the positioning error is higher than that of numerical simulation. This occurs because electromagnetic simulation data require direct wave removal operations. However, in the process of removing direct waves, current methods such as the averaging method and wave-number domain notch filtering also remove part of the echo. This affects the phase information in the echo, causing errors in the DOA estimation results and subsequently affecting the localization accuracy. From the perspective of parallel and orthogonal directions, respectively, the signal loss of each element of the parallel direction array was relatively consistent, while the orthogonal direction experienced varying signal loss due to phase differences with the direct wave, resulting in a greater impact.

In the current processing workflow, the removal of abnormal values in DOA estimation helps reduce certain impacts (as shown in Figure 21). However, fundamentally solving this problem and enhancing localization accuracy necessitates further investigation into optimized direct wave suppression methods.

The methodology’s applicability is primarily constrained in two aspects: (1) The requirement for direct wave suppression as a prerequisite processing step, coupled with the technical limitations of current direct wave suppression algorithms, results in unsatisfactory performance in inhomogeneous media. For homogeneous media of different types, while theoretically processable, practical implementation faces radar system constraints—specifically, excessive attenuation in subsurface media leads to undetectable radar echoes. (2) The proposed methodology does not directly utilize target echoes to estimate position but instead employs DOA estimation in the middle processing. Since DOA estimation specifically targets small point-like objects to derive angular parameters, this approach becomes inapplicable to volumetric targets. Furthermore, the system’s target discrimination capability is fundamentally constrained by the angular resolution of DOA estimation, rendering two targets with angular proximity indistinguishable.

5. Conclusions

This paper introduces the application of acoustically actuated antenna technology in underground target detection, achieving 3D localization for a miniaturized VHF-band GPR. To address the issue of the narrow bandwidth of current acoustically actuated antennas, this method employs a 2D ULA configuration and utilizes coherent DOA estimation to acquire target angular information. The approach is independent of antenna bandwidth and demonstrates significant dimensional superiority compared with conventional antennas. Based on the variation of the angles of underground targets relative to the antenna array center during spatial sampling, genetic algorithms are employed to estimate the 3D coordinates of underground targets. This allows for high-precision target localization with a single continuous spatial sampling, achieving higher operational efficiency compared with conventional GPR systems. Numerical simulation results demonstrate that the RMSE of the 3D localization method proposed in this paper is 1.68 cm, which outperforms conventional methods that utilize wideband LFM pulse signals with hyperbolic vertex detection in theoretical localization accuracy, while also demonstrating good robustness. The gprMax electromagnetic simulation results also verify that this method can effectively localize multiple targets in ideal homogeneous underground media.