An Energy Focusing-Based Scanning and Localization Method for Shallow Underground Explosive Sources

Abstract

To address the problem of slow speed and low accuracy for recognizing and locating the explosive source in complex shallow underground blind spaces, this paper proposes an energy-focusing-based scanning and localization method. First, the three-dimensional (3D) energy field formed by the source explosion is reconstructed using the energy-focusing properties of the steered response power (SRP) localization model, and the velocity field is calculated based on a multilayered stochastic medium model by considering the random statistical characteristics of the medium. Then, a power function factor is introduced to quantum particle swarm optimization (QPSO) to search for and solve the above energy field and to approach the real location of the energy focus point. Additionally, the initial population is constructed based on the logistic chaos model to realize global traversal. Finally, extensive simulation results based on the real-world dataset show that compared to the baseline algorithm, the focusing accuracy of the energy field of the proposed scheme is improved by 117.20%, the root mean square error (RMSE) is less than 0.0551 m, the triaxial relative error (RE) is within 0.2595%, and the average time cost is reduced by 98.40%. It has strong advantages in global search capability and fast convergence, as well as robustness and generalization.

1. Introduction

Shallow underground explosive source localization technology is the key to realize shallow underground explosion, microseismic safety monitoring in shallow mining, and health monitoring of composite material structures [1,2], and is also often applied to earthquake disaster assessment and emergency rescue response. At present, the localization method based on vibration signals is the most promising method for detecting the source location. However, compared with deep underground seismic source localization in large areas, shallow underground explosive source localization within small areas has complex geological structure, unknown velocity information, etc., resulting in limited localization accuracy and low efficiency. Therefore, it is difficult to further promote and apply.

The shallow underground explosive source localization has been studied in the fields of natural earthquakes and microearthquakes. The traditional localization method mainly depends on travel time inversion, and the key link is to identify and pick up accurate travel time information. However, because of serious weak interference, an accurate travel time is difficult to pick up. In order to reduce the influence of travel-time pickup error on source localization, improved methods, such as the relative localization method [3,4] and the double-difference localization method [5,6] are developed. However, such methods usually assume that the underground medium is homogeneous and that the propagation speed of the vibration wave is already known. In the case of low signal-to-noise ratio (SNR), travel-time pickup error is large, making localization error and stability restricted [7]. Moreover, only travel time information is used, and the abundant information provided by the vibration signal waveform is not fully utilized.

With the precise deployment of intensive sensor arrays, waveform-based source localization methods have been catalyzed. They can not only avoid pickup errors caused by travel time with low SNR, but also make full use of waveform information to predict the source location by searching for the largest focus point both in space and time. Waveform stacking is the most widely used source localization method based on the signal waveform, including scattering stacking and interference stacking [8,9,10,11,12]. Existing research results show that localization uncertainty exists in both scattering and interference stacking, and this uncertainty is highly dependent on the frequency of vibration signals [13]. In order to improve the spatial resolution, the signal waveform is often processed to form a feature function, which is stacked to achieve source localization and reduce uncertainty. Compared with the travel time-based localization method, waveform stacking is more robust, but it is also more sensitive to imaging conditions and velocity models. Reversal time imaging (RTI) is based on the theory of time reversal and uses the dissimilarity of wave propagation to reverse the waveform in the reversed time direction to realize energy focusing for the source in space [14,15,16,17]. RTI has a perfect theoretical basis, but it requires a dense sensor array deployment and an accurate velocity model. Therefore, its practicability is poorer, and it is often limited to theoretical research [18]. Wavefront tomography also requires a wide range of high-density sensor arrays with a sufficient aperture for tomography inversion to obtain a high-precision velocity model; otherwise, it may lead to source localization failure [19,20]. The full waveform inversion (FWI) localization method is used to acquire the source location by minimizing the residual between the observed waveform and the synthesized waveform to complete waveform matching [21,22]. However, FWI requires continuous forward modeling and velocity updates, which have a high computational cost and require a more accurate initial velocity model. Otherwise, the algorithm may not converge, so it is not suitable for real-time and routine applications [23]. Given the above information, waveform-based localization methods all obtain the localization result through different imaging methods, and naturally pose a great challenge for imaging resolution. In order to obtain high-quality imaging results, extensive calculations are often required, which increases the time cost of the algorithm to a certain extent and also causes localization errors.

With the rapid development of machine learning, underground source localization methods based on machine learning have been developed [24,25,26], but they are still immature and not universal. The main reasons are: (1) there are not enough prior samples for network model training, and a large number of training samples obtained through forward simulation are not universal; (2) when underground medium changes (such as continuous hydraulic fracturing, continuous mine exploitation, etc.), the trained localization model will fail to locate the source because of a lack of generalization.

Additionally, since the source location is coupled with the propagation velocity of the vibration wave within the underground medium, no matter what kind of localization method is used, the accuracy of the velocity model directly determines the source localization accuracy. Although some excellent works have been conducted on the joint inversion of source location and velocity [27,28,29,30,31], medium randomness is usually neglected. To counter the problems above, based on the velocity model of the multilayered stochastic medium, an energy-focusing-based scanning and localization method for shallow underground explosive sources is proposed by jointly optimizing quantum particle swarm optimization (QPSO) and steered response power (SRP).

The main contributions of this paper are summarized as follows:

• Different from the most common velocity field model that only relies on the average velocity value (a fixed value) or one-dimensional (1D) velocity parameter, this paper considers the multilayered and inhomogeneous structure characteristics of shallow underground medium and proposes a velocity field modeling method based on multilayered stochastic medium. Through the prior medium model with statistical properties, the vibration wave is more consistent with the actual propagation scenario in the shallow underground medium.
• The signals obtained by the vibration sensor array are used to reconstruct the three-dimensional (3D) energy field in underground space through reverse time migration (RTM), and the source localization model based on SRP energy information is improved to enhance the focusing accuracy of the energy field. Additionally, in order to reduce the time cost, a lookup table is introduced into the model-building steps.
• Aiming at the problem that QPSO is easy to fall into local convergence, the power function is introduced to quickly scan and search for the source, and the particle search range is optimized and expanded by introducing the logistic chaos model. From this, convergence to global optimum is ensured while achieving accurate calculations for the above energy field model.
• A Python-based simulator is developed to implement the proposed algorithm and other baseline algorithms. Extensive simulation results based on the dataset from outfield static explosion experiments are provided to show the excellent performance of the proposed scheme.

The rest of the sections of this paper are arranged as follows: Section 2 mainly explains the localization principle of the proposed scheme. Section 3 elaborates on the detailed contents used in the design and analysis of the improved QPSO algorithm. The outfield experiments and data analysis are performed in Section 4. At last, the whole work is concluded in Section 5.

2. The Principle of Energy Focusing-Based Source Scanning and Localization

2.1. The Overall Design Framework

Figure 1 illustrates the overall design framework. First of all, the vibration sensor array in the monitoring area is utilized to obtain the vibration signals generated by the source explosion. Then, considering the characteristics of the multilayered stochastic medium, the energy field localization model is reconstructed through amplitude cumulative multiplication in the reversed time direction. Finally, an improved QPSO algorithm is designed to globally scan and quickly approach the energy focus point, which is output as the source position. The main notations used in the paper are summarized and listed in

Table 1. Main symbol and variable list.

Notation	Definition	Notation	Definition
m(x,y,z)	The prior information model of the stochastic medium model.	P(e)	The output power of beamforming, i.e., SRP.
m0(x,y,z)	The large-scale heterogeneity of the medium.	Fm(ω)	The Fourier transform of the received signal from sensors.
σm(x,y,z)	The small-scale stochastic distribution characteristics of medium.	${\tau }_{mn}(\mathit{e})$	The signal-time difference between two sensors.
δm(x,y,z)	The standard deviation of the stochastic medium model.	v	The propagation velocity of the vibration wave.
f(x,y,z)	The stochastic interference coefficient.	$R_{mn}\left({\tau }_{mn}\left(\mathit{e}\right)\right)$	The GCC between the received signals of two sensors.
R(ξx,ξy,ξz)	The ellipsoidal autocorrelation function.	$\widehat{\mathit{e}}$	The source location.
ξx, ξy, ξz	The spatial offsets in three spindle orientations of R(ξx,ξy,ξz).	pi,j	The local attractor of particles.
a, b, c	The triaxial autocorrelation lengths, i.e., scaling factors.	x	The particle among the particle swarm.
r	The roughness factor.	f(x)	The fitness function.
Rx, Ry, Rz	The triaxial wavenumbers.	y	The variable transformed by particle x.
φ	The random phase information.	k	The number of iterations.
Vp	The P-wave velocity.	η	The degree of chaotic mapping.
Vb	The background P-wave velocity of the medium.	xi,j	The j-dimensional position of the i-th particle.
V	The velocity field model.	ρ	The contraction-expansion coefficient.
ξ	The mean of σm(x,y,z).	ψi,j	The uniformly distributed value.
χ2	The variance of σm(x,y,z).	pi	The position update of the i-th particle.
M	The number of sensors.	pbest	The local optimal position.
$\mathcal{M}=\left\{1,2,\dots ,m,\dots ,M\right\}$	The collection of all sensors.	gbest	The global optimal position.
xm(t)	The signals received by sensors.	mbest	The average optimal position.
s(t)	The source signal.	ϕ	A random value uniformly distributed on [0,1].
hm(t)	The shock response from the source to sensors.	Q	The size of particle swarm.
nm(t)	The received noise signal.	g(x)	The probability density function.
e	The focus point in underground space.	ϖ, u	The random numbers.
Y(e)	The beamforming of sensor array.	μ	The translation term.
τ(e,m)	The signal delay received by sensors.	λ, ε, β	The constant terms.
Y(ω,e)	The Fourier transform of Y(e).	δ(x)	The Dirac delta function.
Y*(ω,e)	The complex conjugate of Y(ω,e).	κ	An arbitrarily small positive number.

.

2.2. The Principle of Velocity Field Modeling Based on Multilayered Stochastic Medium

Theoretical research and practical experiences show that the shallow underground medium can be divided into different velocity domains in layers [31], and the propagation velocity of vibration waves in underground rock strata is closely related to the structural characteristics of rock strata, such as rock composition and origin, medium density, etc. Therefore, the propagation velocity will vary with the change in wave impedance of different mediums. The values of P-wave velocity and wave impedance in the common shallow underground medium are shown in

Table 2. Comparison table of P-wave velocity and wave impedance in the common shallow underground medium.

Medium Name	P-Wave Velocity (m·s−1)	Wave Impedance (Ω)
Weathered zone	100~800	1.2~1.4
Gravel, clastic rock, dry sand	200~800	2.8~16
Sandy clay	300~900	3.1~18
Green sand	600~800	3.8~19
Clay	1200~2500	18~55
Loose rock	1500~2500	14~16
Dense rock	1800~4000	27~60

.

Stochastic medium modeling uses statistical methods to take the physical parameters of the medium as random variables in space and reflect the heterogeneity of geological structure by means of stochastic perturbation. In 3D space, the stochastic medium model is generally considered to be composed of a large-scale model and some small-scale stochastic (relative) perturbation [32], and its prior information model can be expressed as [33]:

$$m(x,y,z)=m_0(x,y,z)+{\sigma }_m(x,y,z),$$

(1)

where m₀(x,y,z) represents the large-scale heterogeneity of the underground medium, i.e., the geological model in the conventional sense. σ_m(x,y,z) is the small-scale stochastic distribution characteristic, which is usually represented by a random process with a zero mean. Under the third-order stationary assumption, σ_m(x,y,z) can be denoted as:

$${\sigma }_m(x,y,z)={\delta }_m(x,y,z)\cdot f(x,y,z),$$

(2)

where δ_m(x,y,z) is the standard deviation of the stochastic medium model. f(x,y,z) is a random variable (also known as the stochastic interference coefficient) with a mean value of 0 and a standard deviation of 1, and its distribution follows a certain autocorrelation function R(ξ_x,ξ_y,ξ_z). Usually, the ellipsoidal autocorrelation function is used to describe the spatial correlation degree of the underground medium.

$$R({\xi }_x,{\xi }_y,{\xi }_z)=\exp [-{(\frac{{\left({\xi }_x\right)}^2}{a^2}+\frac{{\left({\xi }_y\right)}^2}{b^2}+\frac{{\left({\xi }_z\right)}^2}{c^2})}^{1/\left(1+r\right)}],$$

(3)

where ξ_x, ξ_y, and ξ_z are spatial offsets in the three spindle orientations of the ellipsoid of the autocorrelation function, respectively. a, b, and c are the autocorrelation lengths corresponding to three axes and are also known as scaling factors. r is the roughness factor. In general, when r = 0, R(ξ_x,ξ_y,ξ_z) is a Gaussian ellipsoidal autocorrelation function. When r = 1, it is an exponential ellipsoidal autocorrelation function. When 0 < r < 1, it is an intermixed ellipsoidal autocorrelation function (e.g., Von-Karman type). Von-Karman autocorrelation function can provide a spatial description for the medium on a large scale; therefore, it is usually adopted to characterize the geological structures of underground medium [34].

According to the Wiener–Hinchin theorem, the power spectral density function of the spatial random variable f(x,y,z) is the Fourier transform of the autocorrelation function R(ξ_x,ξ_y,ξ_z). Additionally, the stochastic interference coefficient is the inverse Fourier transform of the random spectrum and can be calculated by autocorrelation function and random phase.

$$\mathcal{F}[R({\xi }_x,{\xi }_y,{\xi }_z)]={\displaystyle \iiint {}_{-\infty }^{+\infty }}R({\xi }_x,{\xi }_y,{\xi }_z)\exp \left[-j2\pi \left(R_{x}x+R_{y}y+R_{z}z\right)\right]dxdydz,$$

(4)

$$f(x,y,z)={\mathcal{F}}^{-1}\left(\sqrt{\mathcal{F}\left(R\left({\xi }_x,{\xi }_y,{\xi }_z\right)\right)}\cdot e^{i\varphi \left(R_x,R_y,R_z\right)}\right),$$

(5)

where R_x, R_y, and R_z represent wavenumbers in the triaxial direction, respectively. φ is the random phase information with a uniform distribution of [0,2π].

The stochastic medium model can be transformed into the velocity field model by using the relationship between the stochastic medium and the corresponding background velocity. According to Brich’s principle, the relative perturbation of P-wave velocity V_p can be utilized to describe the small-scale heterogeneity of underground medium, namely:

$$\sigma V_p=m(x,y,z)\cdot V_b,$$

(6)

where V_b is the background P-wave velocity of the medium. Thus, the velocity field model can be shown as:

$$V=V_b+\sigma V_p=V_b(1+m).$$

(7)

The modeling steps of the velocity field model based on the multilayered stochastic medium model are shown in Figure 2.

Step 1: The autocorrelation function R(ξ_x,ξ_y,ξ_z) is constructed by selecting appropriate triaxial autocorrelation lengths a, b, and c. Additionally, its Fourier transform $\mathcal{F}[R({\xi }_x,{\xi }_y,{\xi }_z)]$ is calculated according to Equation (4).

Step 2: The initial random phase φ is generated within the interval range of [0,2π].

Step 3: According to the stochastic process calculation method, the power spectrum of the stochastic medium is obtained by using the product of the power spectrum of the autocorrelation function and the 3D random phase.

$$\mathcal{F}\left({\xi }_x,{\xi }_y,{\xi }_z\right)=\sqrt{\mathcal{F}\left[R\left({\xi }_x,{\xi }_y,{\xi }_z\right)\right]}{e}^{i\varphi \left(R_x,R_y,R_z\right)}.$$

(8)

Step 4: The inverse fast Fourier transform (IFFT) is applied to the power spectrum calculated in step 3 to acquire the stochastic perturbation model.

$$\sigma (x,y,z)={\displaystyle \iiint {}_{-\infty }^{+\infty }}\mathcal{F}\left[R({\xi }_x,{\xi }_y,{\xi }_z)\right]\exp \left[j2\pi \left(R_{x}x+R_{y}y+R_{z}z\right)\right]d{R}_{x}d{R}_{y}d{R}_z.$$

(9)

Step 5: The stochastic perturbation model σ(x,y,z) is normalized to meet the requirements with mean ξ and variance χ², that is, σ_m(x,y,z).

Step 6: A large-scale medium model m₀(x,y,z) is constructed to conform to the regularity of the underground medium.

Step 7: The underground stochastic medium model m(x,y,z) is obtained on the basis of Step 6 by adding the stochastic perturbation model generated in Step 5.

Step 8: The multilayered stochastic medium model is constructed according to the characteristics of different rock strata shown in Table 2.

Step 9: The 3D shallow underground velocity field model is constructed according to Equations (6) and (7).

Compared with the homogeneous velocity model and the 1D velocity model, which cannot characterize the randomness of the underground medium, the velocity field model based on the multilayered stochastic medium is more in line with the actual propagation of vibration waves.

2.3. The Principle of Energy Field Localization Model Based on Improved SRP

The essence of SRP is the output power obtained through beamforming. The SRP-based localization model forms a beam by summing preprocessed vibration signals with time delay compensation and then guides the beam in the actual search process for the source. The point with the maximum output power is just the source location.

Suppose that the test sensor array has M sensors, denoted by a collection of $\mathcal{M}=\left\{1,2,\dots ,m,\dots ,M\right\}$. The signal received by the m-th sensor can be expressed as:

$$x_m(t)=h_m(t)\ast s(t)+n_m(t),$$

(10)

where s(t) is the source signal. h_m(t) is the shock response from the source to the m-th sensor. n_m(t) is the received noise signal. “*” represents the convolution operation.

For any focus point e in underground space, the beamforming of the sensor array is defined as:

$$Y(\mathit{e})={\displaystyle \sum _{m=1}^M{x}_m(t-\tau (\mathit{e},m))},$$

(11)

where e = [e_x,e_y,e_z] represents the 3D spatial coordinates of any focusing point. τ(e,m) is the signal delay received by the m-th sensor. Thus, the output power of beamforming, i.e., SRP, is defined as:

$$P(\mathit{e})={\displaystyle {\int }_{-\infty }^{+\infty }Y(\omega ,\mathit{e})Y^{*}(\omega ,\mathit{e})d\omega },$$

(12)

where $Y(\omega ,\mathit{e})$ is the Fourier transform of $Y(\mathit{e})$. $Y^{*}(\omega ,\mathit{e})$ is the complex conjugate of $Y(\omega ,\mathit{e})$. In this way, SRP can be written as:

$$\begin{array}{l}P(\mathit{e})={\displaystyle {\int }_{-\infty }^{+\infty }Y(\omega ,\mathit{e})Y^{*}(\omega ,\mathit{e})}d\omega \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\displaystyle {\int }_{-\infty }^{\infty }\left({\displaystyle \sum _{m=1}^M{F}_m(\omega )e^{-j\omega \tau (\mathit{e},m)}}\right)} \left({\displaystyle \sum _{n=1}^M{F}_n{}^{*}(\omega )e^{j\omega \tau (\mathit{e},n)}}\right)d\omega \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^M{F}_m(\omega )F_n{}^{*}(\omega )e^{-j\omega \left(\tau (\mathit{e},m)-\tau (\mathit{e},n)\right)}}}}d\omega \end{array},$$

(13)

where F_m(ω) denotes the Fourier transform of the received signal of the m-th sensor.

In practical application, the energy of the received sensor signal is limited; therefore, the integration of Equation (13) is convergent, and the operation order of summation and integration can be interchanged, namely:

$$P(\mathit{e})={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^M{\displaystyle {\int }_{-\infty }^{\infty }{F}_m(\omega )F_n^{*}(\omega )e^{-j\omega \left(\tau \left(\mathit{e},m\right)-\tau \left(\mathit{e},n\right)\right)}}d\omega }}.$$

(14)

Since the signal time difference between the m-th sensor and the n-th sensor at a certain location e is:

$${\tau }_{mn}(\mathit{e})=\tau (\mathit{e},m)-\tau (\mathit{e},n)=\frac{\left|\mathit{e}-{\mathit{e}}_m\right|-\left|\mathit{e}-{\mathit{e}}_n\right|}{v},$$

(15)

where v is the propagation velocity of the vibration wave, which is provided by the velocity field model based on the multilayered stochastic medium in the previous subsection. Then, Equation (14) can be rewritten as:

$$P(\mathit{e})={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^M{\displaystyle {\int }_{-\infty }^{\infty }{F}_m(\omega )F_n^{*}(\omega )e^{-j\omega \left({\tau }_{mn}\left(\mathit{e}\right)\right)}}d\omega }}.$$

(16)

Additionally, the generalized cross-correlation (GCC) between the received signals of the m-th sensor and the n-th sensor can be represented as:

$$R_{mn}\left({\tau }_{mn}\left(\mathit{e}\right)\right)=\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }{F}_m(\omega )F_n^{*}(\omega )e^{-j\omega \left({\tau }_{mn}\left(\mathit{e}\right)\right)}}d\omega .$$

(17)

Therefore, SRP can be shown as the sum of the GCC of all sensor pairs, i.e.,

$$P(\mathit{e})={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^M{R}_{mn}}}({\tau }_{mn}(\mathit{e})).$$

(18)

Removing the autocorrelation value in Equation (18), the following is obtained:

$$P\left(\mathit{e}\right)={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=m+1}^M{R}_{mn}}}\left({\tau }_{mn}\left(\mathit{e}\right)\right).$$

(19)

Finally, the source location is the maximum value of SRP in the monitoring area [35], that is:

$$\widehat{\mathit{e}}=\underset{\mathit{e}}{\arg \max }P(\mathit{e}).$$

(20)

In this way, the source localization can be realized by scanning spatial grids for SRP-based localization models and outputting the maximum energy value. The SRP localization principle is shown in Figure 3.

The source localization model based on SRP can be acquired by using Equations (19) and (20). However, in this process, the energy field image is prone to the phenomenon of blurred focus regions and poor imaging accuracy. Additionally, the calculation amount will also exponentially increase with the increase in grid number, increasing the time cost of the algorithm. Accordingly, in order to make the SRP-based source localization model obtain a better focusing effect, this paper jointly optimizes two tasks. One is to establish a lookup table and optimize the traditional grid scanning process of performing $C_M^2$ groups of cross-correlations at each step into the process of finding cross-correlation values. The other is to optimize the accumulation of cross-correlation values (shown in Equation (19)) into cumulative multiplication (shown in Equation (21)) to further increase the correlation degree between cross-correlation values:

$$P(\mathit{e})=\prod _{m=1}^M\prod _{n=m+1}^M{R}_{mn}({\tau }_{mn}(\mathit{e})).$$

(21)

The calculation method of the waveform cross-correlation function shown in Equation (21) not only improves detection ability for effective signals but also suppresses random noise in vibration data. Although the calculated cross-correlation coefficients have positive and negative association rules, they will not wipe one another out by multiplication. On the contrary, due to the higher correlation degree of waveforms at the source location, large values after multiplication will be much larger than those of small values, which makes the energy value of the source location more “prominent” and more focused. Additionally, it can suppress the false image formed by noise in the spatial imaging domain. Thus, imaging resolution and localization accuracy of the source are both improved.

3. The Source Scanning Algorithm Based on Improved QPSO

QPSO is a population-based optimization algorithm in continuous space. Since the position and velocity of the particle cannot be simultaneously determined in quantum space, the particle position is signified by a wave function and is solved by means of the Monte Carlo method. To ensure global convergence, particles should be clustered towards their local attractor (defined as p_i,j) with a certain probability distribution. In the aggregation process, the farther the particle from p_i,j is, the smaller its occurrence probability. However, it cannot be infinitely far from the target. At the same time, the narrower the probability density function, the higher the probability that the particle appears near p_i,j, and the faster the convergence speed of the algorithm. It is prone to premature convergence and low convergence accuracy in later iterations [36]. To solve these problems, this paper reconstructs the probability density function, exploits logistic chaos initialization to improve the quality of particles, and utilizes chaos perturbation to avoid the search process falling into the local extremum.

3.1. The Fitness Function

The fitness function determines the location selection of the local optimum and global optimum of the population, which is the key to particle update and directly affects the accuracy of the source localization result. Therefore, in the process of solving the localization model, the fitness function is used to measure the localization result of the algorithm. Based on the improved SRP-based source localization model, the fitness function is constructed as:

$$f(x)=\prod _{m=1}^M\prod _{n=m+1}^M{R}_{mn}({\tau }_{mn}\left(x\right)),$$

(22)

where $R_{mn}({\tau }_{mn}(x))$ refers to the GCC function of the signal received by the m-th sensor and the n-th sensor. ${\tau }_{mn}(x)$ denotes the time difference from the particle position x to the m-th sensor and the n-th sensor.

3.2. The Particle Initialization

In the QPSO algorithm, the initialization for the population and each update iteration of particles follow the general random process, and the particles are usually distributed as evenly as possible in the region. However, randomly generated particles will inevitably have a concentrated coverage, resulting in poorer quality and an ineffective initialization effect, and the subsequent calculation needs more iterations. In order to deal with this problem, the logistic chaos model is introduced to improve the particle initialization process.

$$y(k+1)=\eta \ast y(k)\ast (1-y(k)),$$

(23)

$$s.t. y=\{\begin{array}{cc}2x,\hfill & 0\le x\le 0.5\hfill \\ 2x-1,& 0.5<x\le 1\hfill \end{array},$$

(24)

where y is the variable into which particle x is transformed according to the mapping relationship shown in Equation (23). k denotes the number of iterations and η stands for the degree of chaotic mapping.

3.3. The Evolution Formula Based on Improved Probability Density Function

The particle position of QPSO is updated as:

$$x_{i,j}\left(t+1\right)=p_{i,j}\left(t\right)\pm \rho \left|m_{best\_j}\left(t\right)-x_{i,j}\left(t\right)\right|\ln \left[1/{\psi }_{i,j}\left(t\right)\right],$$

(25)

where x_i,_j signifies the j-dimensional position of the i-th particle. $\rho \le 1$ is the contraction-expansion coefficient. ${\psi }_{i,j}\left(t\right)~\left(0,1\right)$ is a uniformly distributed value, and the probabilities of taking “+” and “−” are all equal to 0.5. The local attractor p_i,j of each particle is jointly determined by the local optimal position and the global optimal position, which are defined as:

$$p_i=\varphi \cdot p_{best\_i}+\left(1-\varphi \right)\cdot g_{best},$$

(26)

$$m_{best}=\frac{1}{Q}{\displaystyle {\sum }_{i=1}^Q{p}_{best\_i}},$$

(27)

where p_i is used for the position update of the i-th particle. p_{best_i} is the optimal position experienced by the i-th particle in the current iteration, i.e., the local optimal position. g_best is the current global optimal particle. ϕ is a random value uniformly distributed on [0,1]. Q is the size of the particle swarm. m_best is the average of p_{best_i} and represents the average optimal position of particles in the past.

In order to further improve the probability of the particles appearing near the local attractor, suppose that the probability density function that particles obey is g(x), then the function should satisfy the following properties [36]:

• The function g(x) is symmetrically distributed for the target, which ensures that the probability of particles appearing at the same distance from the target is the same.
• The function value at the target is the largest. The function value becomes smaller when particles are farther away from the target, which ensures that the probability of particle appearance is at its maximum when the target is used as the local attractor.
• $\underset{x\to +\infty }{\lim }g\left(x\right)=\underset{x\to -\infty }{\lim }g\left(x\right)=0$, which ensures the global performance of particles appearing in the whole solution space.

Based on this, the normal function operator and the power function operator are introduced to construct new particle evolution formulas [36].

3.3.1. The Probability Density Function with a Normal Distribution

Supposing that $\epsilon =1/\sqrt{2\lambda }$, then the probability density function of the random variable x and the local attractor p_i,j is:

$$g_1(x)=\frac{1}{\sqrt{2\pi }\epsilon }{e}^{-\frac{x^2}{2{\epsilon }^2}}.$$

(28)

If the current particle position is supposed to be x_i, then (x_i−p_i,j) follows the normal distribution of N(0, ε²). By generating a random number ϖ that satisfies the standard normal distribution, the normal distribution-based QPSO evolution formula, i.e., NQPSO, can be obtained as:

$$x_{i,j}(t+1)=p_{i,j}\pm \varpi /\sqrt{2{\lambda }_{i,j}},$$

(29)

where λ_i,j > 0 is defined as:

$${\lambda }_{i,j}=\frac{1}{\rho (m_{best}-x_{i,j}(t))}.$$

(30)

3.3.2. The Probability Density Function with the Power Function Operator

Assuming that the probability density function of the random variable x is:

$$g_2(x)=\frac{\lambda (\beta -1)}{2{\mu }^{\beta -1}{(\lambda |x|+\mu )}^{\beta }},$$

(31)

where λ > 0, μ > 0, and β > 1. It should be pointed out that the parameter μ acts as a translation, ensuring that the function has an extreme value at zero. To simplify the operation, μ is set to 1 in this paper. That is:

$$g_2(x)=\frac{\lambda (\beta -1)}{2{(\lambda |x|+1)}^{\beta }}.$$

(32)

The distribution function G(x) is solved with a random number u, which obeys the uniform distribution:

$$G(x)={\displaystyle {\int }_{-\infty }^x\frac{\lambda (\beta -1)}{2{(\lambda |t|+1)}^{\beta }}}dt=u.$$

(33)

Then:

$$x=\pm \frac{1}{\lambda }\left(u^{-1/\left(\beta -1\right)}-1\right).$$

(34)

Thereby, the power function based QPSO evolution formula, i.e., PQPSO, can be obtained as:

$$x_{i,j}(t+1)=p_{i,j}\pm \frac{1}{{\lambda }_{i,j}}(u^{-1/\left(\beta -1\right)}-1),$$

(35)

where u is a random number in the interval [0,1]. To guarantee convergence, the constant term β > 1, and β should be chosen as small as possible. In this paper, the value of β is set to 3 to achieve the best results of the algorithm [36].

In order to ensure the convergence of the improved QPSO algorithm, when the random variable (x_i−p_i,j) follows the probability density function of Equation (29) or (35), then when the number of iterations tends to infinity, if $\lambda \to +\infty$, then the convergence probability of the particle x_i equals 1, that is:

$$\underset{t\to +\infty }{\lim }P\left(\left|x_i-p_{i,j}\right|<\kappa \right)=\underset{t\to +\infty }{\lim }{\displaystyle {\int }_{-\kappa }^{\kappa }g\left(x\right)}dx={\displaystyle {\int }_{-\kappa }^{\kappa }\underset{t\to +\infty }{\lim }g\left(x\right)}dx={\displaystyle {\int }_{-\kappa }^{\kappa }\delta \left(x\right)}dx={\displaystyle {\int }_{-\infty }^{+\infty }\delta \left(x\right)}dx=1,$$

(36)

where δ(x) is the Dirac delta function and κ is an arbitrarily small positive number. Therefore, as long as λ_i,j is set to an infinite function with the increase in iterations, then at the beginning of the iteration, λ_i,j is small, and at this time, the particle will appear at a position far away from the target with a larger probability to realize a global search for the target. With the increase in iterations, λ_i,j also increases accordingly, and finally converges to the target (i.e., the source location) to ensure absolute convergence.

The specific steps of the improved QPSO algorithm are presented:

Step 1: particle swarm initialization.

The particle dimension of the population is set to 3, the total number of iterations is set to T. The logistic chaos model is used to generate Q particles in the test area, which are regarded as the initial first generation of particles.

Step 2: particle filtering.

According to the fitness function shown in Equation (22), the fitness values of particles in the initial population are calculated. The particle with the largest fitness value is selected as the global optimal g_best by comparison, and the average optimal position m_best of the current initial population is calculated.

Step 3: iteration repetition.

When the maximum iterations make it to the end, the particle position is continuously updated according to Equation (29) or (35), and the fitness value is recalculated based on the current position. If the current fitness value is greater than the original value, then the local optimal p_best and the corresponding fitness value are updated, otherwise, the original particle position and the fitness value are retained. Meanwhile, the average optimal position m_best in the current iteration is constantly updated. The position with the largest fitness value among the current population is taken as the current global optimal g_best.

Step 4: output the result.

After completing the iterations, the global optimal g_best is output as the final source coordinates.

The improved QPSO algorithm is thoroughly described in Algorithm 1.

Algorithm 1: The improved QPSO algorithm
1	begin
2	Establish the logistic chaos model.
3	Initialize the particle swarm and iterations.
4	Calculate fitness values of particles in the initial population according to Equation (22).
5	for each iteration do
6	Calculate gbest and pbest;
7	Calculate mbest of the population;
8	Update the particle position according to Equation (29) or (35);
9	Recalculate the fitness value based on the current position;
10	Update pbest, mbest, gbest, and the fitness value in the current iteration;
11	Update the particle swarm.
12	end
13	Output the localization result, i.e., the coordinate of the final gbest.
14	end

4. Experiment Verification

In this section, outfield static explosion experiments with shallow underground explosive sources were carried out at the 247 firing range of the China Ordnance Industry. Additionally, extensive simulations based on the experimental dataset were conducted to verify the feasibility and effectiveness of the proposed scheme.

4.1. Experiment Configuration

The distributed underground explosive source localization system, which was researched and developed independently by the Shanxi Key Laboratory of Information Detection and Processing of the North University of China, was adopted to collect vibration signals and obtain relevant localization parameters. The localization system included 38 sets of vibration sensors, 10 sets of multichannel wireless data acquisition modules, a wireless transmission module, a wireless bridge, etc., as shown in Figure 4.

The size of the shallow underground monitoring area was 100 m × 100 m × 50 m, i.e., x[−50 m, 50 m], y[−50 m, 50 m], z[−50 m, 0 m], and was divided into uniform 3D grids with the same size as 1 m × 1 m × 1 m. Thirty-eight sets of vibration sensors were deployed on and near the surface to form a 3D star-shaped sensor array. Three explosion experiments were conducted with 5 kg, 10 kg, and 15 kg TNT as the explosive sources, and the coordinates were (10, 40, −20), (40, −20, −35), and (−30, 20, −45), respectively. Three explosion experiments were performed using stratified continuous initiation. The sampling rate of the wireless data acquisition module was set to 1 MHz, and the sampling time was 5 s. The sensor data was returned through a wireless transmission module. The layout scheme of the sensor array and explosive sources, as well as part of the received signal, are shown in Figure 5.

4.2. Velocity Field Model Based on Multilayered Stochastic Medium

In shallow underground explosive source localization, the source location is relatively shallow, generally no more than 100 m [30]. Therefore, this paper focuses on common shallow structures, such as weathered zone, dry sand, sandy clay, and green sand. The thickness of each layer was set to 10 m, 15 m, 10 m, and 15 m, respectively. The stochastic medium model was set to obey the Von-Karman ellipsoidal autocorrelation function. Combined with the characteristics of each medium layer, a large number of simulation tests were conducted to select the appropriate triaxial autocorrelation lengths that meet the conditions, as shown in

Table 3. The parameter setting of autocorrelation lengths.

Medium Type	a (m)	b (m)	c (m)
Weathered zone	1.3	2.2	1.4
Dry sand	2.6	2.2	2.1
Sandy clay	3.4	4.3	3.5
Green sand	5.2	6.5	7.6

.

The multilayered stochastic medium model is drawn by following the modeling steps described in Section 2.2. In order to facilitate the observation of medium distribution, each layer was sliced along the x-o-y plane at −5 m, −20 m, −30 m, and −40 m along the z-axis, respectively, and the corresponding two-dimensional (2D) slices were shown in Figure 6, where the color bar on the right stands for the normalized value of the current soil medium.

Since the surface rocks in the weathered zone are eroded by sunlight, rain, etc., the later results are lower in rock density and higher in fine sand content, as shown in Figure 6a. By contrast, gravel, clastic rock, and dry sand in Figure 6b are less eroded than the rocks in the weathered zone, so the integrity of the rock mass is comparatively high. But the rock volume is relatively small due to crustal movements. In Figure 6c, the water content of sandy clay gradually increases, and the soil appears to agglomerate. The water content of the green sand in Figure 6d is relatively high, and the agglomeration phenomenon is glaringly obvious. The multilayered medium shown in Figure 7a is composed of the above four typical stochastic mediums. The medium layers are parallel to each other and have different permeabilities. Therefore, it can be concluded that the stochastic medium modeling method can better reconstruct the distribution of underground medium quickly, which is basically consistent with the actual medium characteristics, and achieves the expected effect. It can be used to simulate medium distribution in shallow underground areas under actual working conditions.

Based on the modeling of the multilayered stochastic medium, the P-wave velocity of the above four typical mediums were utilized as the background P-wave velocity of each layer, and Equation (6) as well as Equation (7) were used to transform the multilayered stochastic medium model into the velocity field model, as shown in Figure 7b, where the color bar represents the velocity range at different locations.

As can be seen from Figure 7b, from near the surface down, the velocity range of each layer of the velocity field is 200 m/s~350 m/s in the weathered zone, 320 m/s~400 m/s in the gravel, clastic rock, and dry sand layer, 340 m/s~430 m/s in the sandy clay layer, and 410 m/s~500 m/s in the green sand layer, respectively. In other words, the velocity field model under the condition of the multilayer complex medium obtained through the stochastic medium modeling method basically conforms to the theoretical velocity range of the corresponding medium shown in Table 2. The velocity difference between layers is small, and shows a continuous variation tendency, which improves the modeling accuracy of the velocity field model and can be used as the calculation basis for the subsequent source localization model. The important thing to note here is the propagation path. This paper assumes that the vibration waves propagate along a straight line in the velocity field, and does not consider reflection and refraction of waves between layers, nor does it consider the situation that vibration waves propagate along broken lines or curves.

4.3. Energy Field Model Based on Improved SRP

To compare and analyze the focusing accuracy of energy fields based on SRP and energy fields based on improved SRP, combined with the velocity field model generated in Figure 7, the corresponding 3D energy field images were drawn, and the slices were shown at the preset source locations for revealing visually, as shown in Figure 8.

It can be observed from Figure 8 that the energy-focusing effect of the energy field localization model based on SRP is not great, and there are blurred focus areas and imprecise image effects, which are not conducive to the calculation and recognition of the source location, and affect the localization accuracy. By contrast, the energy field image based on improved SRP is apparently more concentrated in the focus area, and the fuzzy area is greatly reduced. The reason is that the grid scanning process is optimized by establishing a lookup table, and the correlation degree is strengthened by cumulative multiplication on cross-correlation values. Therefore, the subsequent search calculation for the source location can be reduced significantly.

In reality, however, the energy does not converge to a single point, but is distributed in a very small area, and usually shows an ellipsoid shape. Therefore, the half-axis length of the ellipsoid and the commonly used peak signal-to-noise ratio (PSNR) are adopted to quantitatively evaluate the focusing accuracy of the energy field model. The modeling errors of the energy field are presented in

Table 4. Evaluation of the focusing accuracy of the energy field.

Name of Energy Field	Preset Source Locations	Ellipsoid Error			PSNR (dB)
		x-Axis (m)	y-Axis (m)	z-Axis (m)
Energy field based on SRP	(10, 40, −20)	3.231	2.983	3.001	18.483
	(40, −20, −35)	2.593	2.346	3.547	18.092
	(−30, 20, −45)	2.325	2.473	2.947	19.138
	Average value	2.716	2.601	3.165	18.571
Energy field based on improved SRP	(10, 40, −20)	0.113	0.527	0.367	41.874
	(40, −20, −35)	0.027	0.013	0.352	39.103
	(−30, 20, −45)	0.167	0.027	0.053	40.031
	Average value	0.102	0.189	0.257	40.336
	Percentage increase (%)	96.23	92.73	91.87	117.20

.

In Table 4, the ellipsoid error of the energy field model based on SRP is large, the average errors of the x-axis, y-axis, and z-axis are 2.716 m, 2.601 m, and 3.165 m, respectively, and the average PSNR is smaller, i.e., 18.571 dB. By comparison, the ellipsoid error of the energy field localization model based on improved SRP is dramatically reduced; the average errors of the x-axis, y-axis, and z-axis are 0.102 m, 0.189 m, and 0.257 m, respectively, and the average PSNR is 40.336 dB. Compared with the SRP-based baseline method, the average ellipsoid errors are increased by 96.23%, 92.73%, and 91.87%, respectively, and the average PSRN is increased considerably by 117.20%. By reason of the foregoing, it can be concluded that the energy field localization model based on improved SRP has the features of higher imaging accuracy and smaller focusing error.

The advantage of the improved SRP-based energy field is evaluated by focusing accuracy from the spatial dimension. In order to further measure the efficiency of the proposed algorithm from the time dimension, the advantage of improved SRP is further explained from the perspective of running time. The hardware environment was a CPU-based server that had 8GB DDR4, i7-8750H, and an independent display card of GTX1060 6G. The algorithm was run 10 times to calculate its average running time, and the results are shown in

Table 5. The comparison table of the average running time.

Preset Source Locations	SRP (s)	Improved SRP (s)	Difference Value (s)	Percentage Increase
(10, 40, −20)	762.764	12.530	750.234	98.36%
(40, −20, −35)	741.655	12.016	729.639	98.38%
(−30, 20, −45)	771.955	11.924	760.031	98.46%
Average running time	758.791	12.157	746.635	98.40%

.

In Table 5, the average running time of the SRP-based algorithm is 758.791 s. In the process of grid filling, each grid needs to do $C_M^2$ groups of cross-correlations, so the denser the grid partition, the more times the cross-correlation needs to be performed. This process is not only time-consuming, but also occupies a lot of computing resources, which increases the amount of calculation and reduces the efficiency of practical engineering applications. Clearly, the average running time of the improved SRP algorithm is reduced to 12.157 s. This is because the lookup table method not only greatly reduces the number of cross-correlations and calculation amounts, but also reduces the computing resource consumption of the machine, reducing the time cost by 98.40%.

4.4. Localization Results Based on the Improved QPSO Algorithm

In order to analyze the localization effect of the proposed algorithm, it was compared with the QPSO algorithm (as the benchmark algorithm), the classical genetic algorithm (GA), and the traditional machine learning algorithm support vector regression (SVR). The localization performance was compared by taking the source (−30, 20, −45) as an example. The number of iterations of the improved QPSO algorithms and the original QPSO algorithm were all set at 500. The chaotic mapping degree was set to 4, and the number of iterations of the chaos model was set to 300. Fifty particles were randomly generated as the initial particle swarm, and these particles were distributed in the preset region according to the logistic model. The particle distribution and the localization results were plotted, as shown in Figure 9.

In Figure 9a, it can be seen that uniformly distributed particles (red particles) are relatively concentrated, while the particles initialized through the logistic chaos model (blue particles) are relatively scattered and distributed in the whole space. In the edge regions, uniformly distributed particles are relatively few, while the particles initialized through the logistic chaos model still have a small amount of distribution. On the basis of those facts, we can reach a conclusion that compared with random uniform distribution, the spatial distribution of particles after a chaotic search model is more pervasive, which increases the diversity of particles and reduces local convergence caused by the initial particle distribution.

Figure 9b shows the source search track. It is observed that the PQPSO algorithm and the NQPSO algorithm can both expand the scope of the search track and increase the diversity of particle search. Although NQPSO can realize source localization, there still exists the phenomenon of local convergence in the localization process. By contrast, the PQPSO algorithm has a greater advantage in getting rid of local convergence, and the localization result is closer to the real source location. This is because the power function operator introduced by PQPSO can improve the probability of particles appearing near the local attractor and has a wider search range for the source. Next is SVR. Clearly, there is almost no long-term local convergence in the search process for the source, and the localization result is also more accurate. This is because there are barely any local minimum problems; the searching process has strong independence, and does not depend on the whole data. As for GA, it shows obvious “teleport” many times rather than slowly moving to the optimal solution (i.e., the explosive source). The reason is that the GA is theoretically based on crossover and mutation mechanisms, which cannot use the feedback information of the network in time, resulting in a slow search speed and making it difficult to converge to the local optimum.

In order to further compare and illustrate the localization effects of different algorithms, the fitness curves of search process were plotted, as shown in Figure 9c.

By analyzing Figure 9c, it is easy to see that the fitness curve of the QPSO algorithm begins to converge at approximately 6 iterations, and the second iteration for optimization is carried out after the first escape from local convergence at approximately 13 iterations. However, when the iteration reaches approximately 33, the search traps itself in local convergence, which cannot be escaped. Finally, it enters an imprecise convergence interval, and the fitness value is approximately 3232. Additionally, the particle search range of QPSO is limited. As for NQPSO, it also falls into local convergence like QPSO at approximately 5 iterations, but it falls into local convergence for a long time and gets rid of this convergence interval only after approximately 29 iterations. The final fitness value is relatively high, at approximately 3255. That is to say, NQPSO has limited ability to help particles quickly get rid of local convergence. In contrast, when PQPSO enters the convergence interval for the first and second time at approximately 13 iterations and 18 iterations, respectively, it does not fall into convergence too much but expands the search range and accelerates convergence speed to get rid of these two intervals. The convergence speed gradually declines after approximately 26 iterations. Finally, PQPSO has a higher fitness value of approximately 3277 after 37 iterations. Then, for SVR, it falls into local convergence after approximately 9 iterations, 19 iterations, 27 iterations, and 31 iterations, respectively, and does not fall into too much but quickly gets rid of it and reaches the search terminal point at approximately 35 iterations. The final fitness value is 3251, which is lower than NQPSO but better than QPSO. Finally, when it comes to GA, it traps itself in the local convergence interval after approximately 23 iterations and does not escape from this local convergence interval until approximately 68 iterations. After that, the GA quickly “teleport” to the final convergence interval at approximately 84 iterations. The fitness value is the lowest among the above five algorithms, at 3116. In summary, PQPSO is more robust and has the advantages of fast convergence speed, quickly escaping from local convergence, and higher localization accuracy in the search process for the source.

In order to better verify the superiority of the proposed algorithm in time complexity and space complexity, the running time and memory occupancy of QPSO, NQPSO, PQPSO, SVR, and GA were compared and analyzed, and the results are shown in Figure 10.

As shown in Figure 10, with the increase in population size, the time complexity and space complexity of the above five algorithms all show a gradual growth trend. From Figure 10a, compared with NQPSO, QPSO, SVR, and GA, the time complexity of PQPSO is significantly better than the other four algorithms, and the average running time of PQPSO is reduced by 5.74%, 13.66%, 22.55%, and 30.71%, respectively. However, it can be seen from Figure 10b that the space complexity of PQPSO and NQPSO is worse than QPSO, and the memory is increased by 5.57% and 7.92%, respectively. The possible reason is that the state update formulas of these two algorithms are more complex than those of the original QPSO algorithm. On the other hand, compared with SVR and GA, the memory occupation of PQPSO is reduced by 31.01% and 14.50%, respectively. This is because the SVR needs a lot of samples to conduct more iterations, which takes up more memory in storage and requires more time-consuming calculations. Additionally, although the mutation mechanism of GA provides the population with the ability to jump out of the local extremum, the search efficiency in later iterations is low, making it easy to fall into the local optimum. Meanwhile, the algorithm is more complicated to realize, which takes up more memory space. Therefore, by comprehensive comparison, PQPSO has more advantages in terms of time complexity and space complexity, which can help to better optimize and locate the source.

Based on the above analysis, to further verify the superiority of the proposed algorithms, the aforementioned five algorithms were used to solve the above three source localization models. Without loss of generality, the commonly used root-mean-square error (RMSE) and relative error (RE) are used to evaluate the localization results. The source localization results and error analysis are presented in

Table 6. The localization results of the three explosive sources.

Serial Number		x-Axis (m)	y-Axis (m)	z-Axis (m)	RMSE (m)	RE
						x-Axis	y-Axis	z-Axis
1	Pre-set location	10	40	−20	--	--	--	--
	PQPSO	9.9996	40.0267	−19.989	0.0289	−0.0040%	0.0667%	−0.0550%
	NQPSO	9.9112	40.3260	−20.0571	0.3427	−0.8880%	0.8150%	0.2855%
	QPSO	9.7019	39.6119	−19.8782	0.5043	−2.9810%	−0.9703%	−0.6090%
	SVR	9.8549	39.9255	−19.5324	0.4952	−1.4510%	−0.1863%	−2.3380%
	GA	9.8324	38.1521	−19.0328	2.0924	−1.6760%	−4.6198%	−4.8360%
2	Pre-set location	40	-20	−35	--	--	--	--
	PQPSO	39.9826	−20.0348	−35.0235	0.0455	−0.0435%	0.1740%	0.0671%
	NQPSO	39.4097	−20.5698	−35.2767	0.8658	−1.4758%	2.8490%	0.7906%
	QPSO	39.3695	−20.5778	−35.5968	1.0429	−1.5763%	2.8890%	1.7051%
	SVR	39.6322	−20.7257	−35.4429	0.9263	−0.9195%	3.6285%	1.2654%
	GA	39.6359	−20.9688	−34.1495	1.3396	−0.9103%	4.8440%	−2.4300%
3	Pre-set location	−30	20	−45	--	--	--	--
	PQPSO	−30.0051	20.0519	−45.0179	0.0551	0.0170%	0.2595%	0.0398%
	NQPSO	−30.0112	20.2698	−45.0571	0.2760	0.0373%	1.3490%	0.1269%
	QPSO	−30.8996	20.3210	−44.6694	1.0108	2.9987%	1.6050%	−0.7347%
	SVR	−30.3354	20.4215	−44.5981	0.6721	1.1180%	2.1075%	−0.8931%
	GA	−30.5208	21.6674	−43.9348	2.0460	1.7363%	8.3370%	−2.3671%

.

Table 6 explains that, first of all, all the indicators of GA are poor, mainly due to its problems of premature convergence and worse local search ability, leading to poorer localization accuracy. Additionally, because of the strong directivity and sufficient data mining and computing power, the localization results of the traditional machine learning algorithm SVR are better than GA and QPSO. Secondly, the RMSE obtained through the QPSO algorithm is less than 1.0429m, and the REs of the x-axis, y-axis, and z-axis are within 2.9987%, 2.8890%, and 1.7051%, respectively. This is because it is different from GA; QPSO has memory, the optimal position and optimal direction in the previous iteration process will be retained and used to update the particle swarm, and the global search ability is better. Then, although NQPSO and PQPSO have greatly improved the localization results, it is obvious that PQPSO is more excellent. By contrast, PQPSO can reduce the RMSE to 0.0551m, and the REs of the x-axis, y-axis, and z-axis are reduced to less than 0.0435%, 0.2595%, and 0.0671%, respectively. The principal reason for this is that a power function term is introduced into PQPSO, giving it strong competitive advantage in finding the global optimal solution and superior performance in stability, convergence speed, as well as credibility. Compared with other algorithms, the PQPSO algorithm has better generalization, and can achieve efficient and high-precision localization no matter how the source location changes. From what has been discussed above, we may safely draw the conclusion that, based on the velocity field generated by the multilayered stochastic medium model, the source localization results obtained through PQPSO have smaller REs and higher localization accuracy on the basis of using improved SRP to construct a high-resolution 3D energy field. The proposed scheme can basically meet the localization requirements of shallow underground explosive source, and has a certain engineering application value.

5. Discussion

Velocity field modeling is a key step in the course of data analysis for shallow underground explosive sources. The accuracy of the velocity field model has a great influence on the source localization. In this paper, the velocity field model is built on the basis of the multilayered stochastic medium model, but is not modified or optimized by experimental verification. Therefore, it is unable to form a closed loop. Issues to be aware of include, but are not limited to:

(1)

The coherence of velocity: since the deviation of velocity has less influence on the amplitude superposition than on the time, it is therefore necessary to consider not only the effect of the energy field but also the neighboring velocity when modeling the velocity field.

(2)

The revision and optimization of the velocity: velocity analysis is a process of iterative correction, and the medium randomness leads to the deterministic inversion method not being applicable. Therefore, it is necessary to establish a statistical inversion method that jointly inverts the source location and velocity.

In future research, the propagation characteristics of vibration waves in shallow underground medium will be studied in detail to further improve the accuracy of the velocity field. On the one hand, the unstructured mesh finite element technique is used to perform 3D forward simulation for the stochastic medium model. Through Monte Carlo simulation technology, the wave filed modal composition, waveform characteristics, attenuation characteristics, frequency characteristics generated from different stochastic medium models and different source locations, as well as the influence of transient features on the source localization, are analyzed. Thus, guiding the establishment of an appropriate processing method in the service of source parameter inversion with high precision. On the other hand, the ray paths from all possible positions within a certain area around the source to each sensor on the ground are tracked and determined by the minimum travel time of the 3D ray. The optimal source location and the optimal ray path of the current iteration are determined through the Bayesian inversion model. Finally, the velocity model is updated according to a certain velocity model update criterion, which is used to recursively guide the maximization of posterior probability to complete the source parameter inversion, including the source location and velocity. In this way, the proposed method can be better applied to fields such as underground cave exploration and geological disaster prediction, microseismic localization and safety monitoring, stability prediction of coal mines and engineering rock masses, etc.

6. Conclusions

In this paper, a novel high-efficiency and high-precision shallow underground explosive source localization method was proposed. Firstly, the multilayered stochastic medium model was constructed, and the relationship between the medium and vibration wave velocity was used to transform the above medium model into the velocity field model closely related to the shallow underground test environment. Secondly, the energy field localization model based on improved SRP was constructed, and focusing resolution of energy field was tremendously enhanced. Finally, the logistic chaos model was introduced to construct the initial population and QPSO algorithm was improved to realize global and fast search for the source. The results of outfield experiments show that the proposed scheme can effectively locate shallow underground explosive source. It is capable of searching the entire solution space, and has advantages of higher localization accuracy and smaller localization error.