Automatic Identification of Sound Source Position Coordinates Using a Sound Metric System of Sensors Linked with an Internet Connection

Abstract

In this article, we deal with the problem of increasing the accuracy of the automatic determination of the coordinates of the sound source location. We propose a new algorithm for the identification of the sound source’s position coordinates based on a system of three equations of the second order describing the dynamics of acoustic wavefront propagation. The implementation of the algorithm is carried out by a distributed automated system, which includes autonomous sensor-receivers located in the field and connected to the server of this system via wireless communication channels. Sensor-receivers are placed at the vertices of a flat, symmetrical figure with 4 axes of symmetry of the second order (square). The proposed algorithm takes into account the change in the phase speed of the sound wave when the temperature, air humidity, wind direction and speed change and allows for the determination of the coordinates of the position of the sound source with an error of no more than 1%. The experiment with real input data was carried out in a simulated environment, which was created on the Node.js platform.

1. Introduction

Acoustic signals are widely used in various areas of life to detect and locate objects- in industrial, civil and military terms [1]. Sound source localization techniques can be divided into two main types: active and passive sound source positioning [2]. Active acoustic location involves the creation of sound in order to produce an echo, which is then analyzed to determine the location of the object in question [3]. On the other hand, passive acoustic localization uses the sound emitted by an acoustic source to infer information about its spatial location [4,5]. This article focuses on the passive sound source localization technique. The most commonly used positioning methods include the signal’s time of arrival (ToA), time difference of arrival (TDoA), frequency difference of arrival (FDoA), or doppler shift [6,7]. As emphasized by Wu et al. [8], TDoA and ToA methods, compared to FDoA, can achieve higher positioning accuracy; in addition, they are competitive due to computational simplicity and performance [9]. TOA exploits signals’ arrival times to obtain range estimates and each range estimate produces the radius of a circle or sphere for a three-dimensional case. According to this method, the sound source is then estimated to be positioned at the intersection point of these circles, also known as trilateration [10]. The basic idea behind the TDOA method is to use the time difference of arrival between N pairs of sensor nodes. If the sensor nodes have multiple microphones, each node can perform the TDOA computations separately [11,12]. Thus, the central processing unit only needs to combine the received TDOA measurements to estimate the source position.

For sound localization in 3D space, the most common technologies are the beam-former method [13], acoustic vector sensor (AVS) [14] and multiple microphone array [15,16,17]. In addition, location systems can be divided into three distinct components, namely distance and angle estimation, position computation, and location algorithm [18]. Although there have been many scientific proposals in this field, the location of an acoustic source is still a hot research topic [19].

One of the most important challenges in today’s world is security. The UN’s 2030 agenda states that there can be no sustainable development without a peaceful society free from fear and violence [20]. Gunshot detection and localization systems are being investigated, which can alert relevant authorities to potential gunshots and provide their estimated locations [21,22]. Country’s defense systems are also becoming more and more importance, as evidenced by the ongoing war in Ukraine [23,24]. In this context, an important aspect is an increase in the accuracy of rocket artillery. For an effective guidance system in artillery units, information is needed about the location coordinates of batteries, ground artillery mortars, anti-aircraft artillery, multiple rocket launchers and enemy submarines. One of these artillery weapons’ position coordinates determination methods is acoustic signals registration by their shots or explosions. Sonic artillery intelligence methods are also used in counter-sniper combat, which expands the range of intelligence information users and increases requirements for area of coverage by sound means using less power of acoustic signal sources, also in the context of the country’s defense.

When determining the coordinates of artillery batteries, Baluta and Andrei [25] describe the method of identifying the sound source location based on the relative time of the signal (ToA method). A method of determining sound sources based on the equation of hyperbola, which describes a non-linear relationship between the difference in the time of arrival (TDoA) and the location of the source, is also proposed [26,27]. Techniques include beamforming, spherical beamforming, and acoustic holography. However, the existing algorithms [25,28,29] are based on a piecewise linear approximation of the second-order curve, which ultimately leads to an error in the measurement of the direction angle to the target. The value of this error can significantly exceed the error of the sound–metric complex if the distance to the target is proportional to the length of the acoustic base. Moreover, the study [25,28,29] did not take into account the temperature and humidity data of the air in the area, or the speed and direction of the wind. These data are particularly important outdoors, as the speed of sound depends on the weather conditions [9]. Pourmohammad and Ahadi [30] also showed that the speed of sound varied at different temperatures, so taking this parameter into account in identifying the location of the sound source is very important.

In this article, we deal with calculating the position of the sound source [18]. A new algorithm for automatically determining the coordinates of the position of the sound source with an error of no more than 1% using sensors-receivers that are placed at the vertices of a flat symmetrical figure with 4 axes of symmetry of the second order (square) has been proposed. The basis of the new algorithm is a system of three equations of the second order describing the dynamics of the propagation of the spherical acoustic wavefront, which takes into account temperature, air humidity, wind speed and direction.

2. Materials and Methods

This algorithm implementation is carried out by a distributed automated system that includes autonomous sound sensors set and located in the field and connected to this system server via wireless communication channels. The sensor receives sound waves and sends messages to the server via the Internet, which contains the sensor ID, its geolocation, and data on temperature and humidity in the area where sound receivers are located. The sensor-receiver is a complex device that is a single system. It includes the functions of the following electronic gadgets:

• Electronic stopwatch. Their conditional accuracy is up to 0.001 s.
• Thermometer. An ordinary electronic thermometer has a measurement accuracy of up to 0.1 °C.
• Hygrometer. Digital hygrometers have a measurement accuracy of up to 1%.
• Barometer. Digital barometer with an accuracy of 10 Pa.
• A sound and noise meter (microphone) with a frequency range of 31–8000 Hz. By connecting this device to our sensor, we will obtain a sensor-receiver. After registering a sound wave, the sensor receiver will send a message to the server.

An alternative to the sensor-receiver can be an altimeter, which has all the sensors responsible for measuring the above-mentioned input parameters.

The server receives the message and determines the coordinates of the position of the sound source based on the difference in the time of receiving the sound signal by the sensor-receivers. The program also takes into account the speed of sound, and data on temperature and air humidity received from the sensor. The server processes the received data and determines the coordinates of the target point and transmits the data to the relevant users.

One of the possible operating modes of the sensor may be recording and sending a fully received acoustic signal to the server. This will allow for remote listening in on the situation within the sensor’s range. Complications of recognition and classification algorithms can ensure the detection of shots and also other events accompanied by acoustic signals.

The block diagram of finding acoustic wave sources’ geographical coordinates is shown in Figure 1.

Figure 1 shows the placement of objects in space, namely:

• X—shot point, target point, position of which must be found.
• A, B, C, D—sound receivers that can measure air temperature and humidity and have access to the Internet.
• Server—a program that will determine the position of the target point, which can be connected via the Internet.
• Command post—a place where there is the computer on which the program runs (it can be weapon given automatically) that subscribes to the program that is hosted on the server and uses its data.

We propose the following algorithm for finding the geographical coordinates of the acoustic wave source:

• The target point creates sound.
• Sound propagates from the target point to the sensors.
• The sensor receives sound and sends messages to the server via the Internet.
  The message contains:
  - Sensor ID.
  - Sensor geolocation.
  - Air temperature where the sensor is located.
  - Humidity of the air where the sensor is located.
• The server program receives the message.
  - The program determines the connection delay.
  - The program calculates at what point in time the sensor has received the sound and sent a message (taking into consideration a delay)
  - The program also determines the sound speed using the data about temperature and humidity received by the message from the sensor.
• The program waits for 4 such messages, as mentioned above.
• The program receives the fourth message.
  - The program determines t₁, t₂, and t₃ according to the absolute times defined in paragraph 4.
  - The program determines average speeds v₁, v₂ and v₃ according to time, according to speeds defined in paragraph 4.
  - An array of sensor coordinates from which the message was received is transmitted.
• The program determines the position (x, y) of the target point according to the data specified in paragraph 6.

The program having the geolocation of the first sensor and the target point defined position in paragraph 7 determines the geographical coordinates of the target point and provides an API so that other programs can use this data.

2.1. Geometric Model of Finding the Coordinates of the Sound Source

In this paper, the non-linear relationship between the location of the sound source (coordinates of the position of the sound source) and TDOA is described by a system of equations of dynamic circles with centers at points B, C and D so that the point $\mathrm{O}\left({\mathrm{x}}_0,{\mathrm{y}}_0\right)$ also belongs to these circles, respectively. The radii of these circles change over time and describe the dynamics of the propagation of the spherical acoustic wavefront. The proposed method takes into account meteorological parameters (temperature, air humidity, wind speed and direction).

If the acoustic waves are propagating in all directions at the same speed (in a homogeneous environment), then around the source of acoustic waves at the point O (x₀, y₀) it is possible to draw a circle whose radius is equal to the distance to the nearest sensor (sound receiver) A (0, 0). Next, the line segment connecting point O with the sensor at point B (x_b, y_b) is constructed and point B1 at the intersection of line segment OB with the circle of radius R = OA is marked. The line segment BB₁ can be found as follows: since sensor A has recorded the signal at a certain time t_a, and sensor B records at time t_b, then for a time ${\mathrm{t}}_1={\mathrm{t}}_{\mathrm{b}}-{\mathrm{t}}_{\mathrm{a}}$ the sound will manage to pass the distance $\mathrm{S}=\mathrm{B}{\mathrm{B}}_1={\mathrm{v}}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}\cdot \left({\mathrm{t}}_{\mathrm{b}}-{\mathrm{t}}_{\mathrm{a}}\right)$ (Figure 2).

Then, around points A and B, two circles (blue circle centered at point A; green circle centered at point B) have been drawn, around point A with radius OA, and around point B with radius $\mathrm{O}\mathrm{B}=\mathrm{O}{\mathrm{B}}_1+\mathrm{B}{\mathrm{B}}_1=\mathrm{O}\mathrm{A}+{\mathrm{v}}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}\cdot \left({\mathrm{t}}_{\mathrm{b}}-{\mathrm{t}}_{\mathrm{a}}\right)$. As can be seen, these two circles have two points of intersection (point O; point O₁), one of which is the needed signal source (Figure 3).

To eliminate the false acoustic source at point O₁ another sensor at point C (Figure 4) has been introduced, which is the center of the yellow circle. Now, the only point of intersection of these three circles can be point O–the signal source.

2.2. Calculation of the Source Position Coordinates of the Acoustic Wave

To find the coordinates of the sound wave propagation source, four sensors are enough (A, B, C, D, respectively), and the source of the acoustic wave is point O (Figure 5).

If, at points A (0, 0), $\mathrm{B}\left({\mathrm{x}}_{\mathrm{b}},{\mathrm{y}}_{\mathrm{b}}\right)$, $\mathrm{C}\left({\mathrm{x}}_{\mathrm{c}},{\mathrm{y}}_{\mathrm{c}}\right)$, $\mathrm{D}\left({\mathrm{x}}_{\mathrm{d}},{\mathrm{y}}_{\mathrm{d}}\right)$, there are four sensor receivers that obtain an acoustic signal, the source of which is at the point $\mathrm{O}\left({\mathrm{x}}_0,{\mathrm{y}}_0\right)$, then each sensor records the absolute time t_a, t_b, t_c, t_d of signal reception on the corresponding sensors. These data have been obtained from measuring instruments on the sensors. It is possible to calculate the difference between the absolute times measured by the four sensor receivers using the following formulas: ${\mathrm{t}}_1={\mathrm{t}}_{\mathrm{b}}-{\mathrm{t}}_{\mathrm{a}}$, ${\mathrm{t}}_2={\mathrm{t}}_{\mathrm{c}}-{\mathrm{t}}_{\mathrm{a}}$, ${\mathrm{t}}_3={\mathrm{t}}_{\mathrm{d}}-{\mathrm{t}}_{\mathrm{a}}$.

The equation of a circle with a center at point A(0, 0) is ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{R}}^2$, where R is the radius equal to the distance from point A(0, 0) to point $\mathrm{O}\left({\mathrm{x}}_0,{\mathrm{y}}_0\right)$.

Next, the system of circles equations describing the dynamics of the acoustic wavefront propagation is developed with centers at points B, C and D so that the point $\mathrm{O}\left({\mathrm{x}}_0,{\mathrm{y}}_0\right)$ also belongs to these circles:

$$\{\begin{array}{ccc}{\left(\mathrm{x}-{\mathrm{x}}_{\mathrm{b}}\right)}^2+{\left(\mathrm{y}-{\mathrm{y}}_{\mathrm{b}}\right)}^2={\left(\mathrm{R}+\mathrm{v}\cdot {\mathrm{t}}_1\right)}^2& \begin{array}{ccc}& & \begin{array}{ccc}& & \begin{array}{ccc}& & \end{array}\end{array}\end{array}& \left(1\right)\\ {\left(\mathrm{x}-{\mathrm{x}}_{\mathrm{c}}\right)}^2+{\left(\mathrm{y}-{\mathrm{y}}_{\mathrm{c}}\right)}^2={\left(\mathrm{R}+\mathrm{v}\cdot {\mathrm{t}}_2\right)}^2& \begin{array}{ccc}& & \begin{array}{ccc}& & \begin{array}{ccc}& & \end{array}\end{array}\end{array}& \left(2\right)\\ {\left(\mathrm{x}-{\mathrm{x}}_{\mathrm{d}}\right)}^2+{\left(\mathrm{y}-{\mathrm{y}}_{\mathrm{d}}\right)}^2={\left(\mathrm{R}+\mathrm{v}\cdot {\mathrm{t}}_3\right)}^2& \begin{array}{ccc}& & \begin{array}{ccc}& & \begin{array}{ccc}& & \end{array}\end{array}\end{array}& \left(3\right)\end{array}$$

Using the equation of a circle with a center at point A, it is possible to reduce ${\mathrm{x}}^2+{\mathrm{y}}^2$ on the left side and R² on the right in all three equations of the system. Subsequently, it is needed to multiply Equations (2) and (3) in the system by additional factors:

$$\{\begin{array}{ccc}{\mathrm{x}}_{\mathrm{b}}^2+{\mathrm{y}}_{\mathrm{b}}^2=2\mathrm{R}\mathrm{v}{\mathrm{t}}_1+{\left(\mathrm{v}{\mathrm{t}}_1\right)}^2-2\left(\mathrm{x}\cdot {\mathrm{x}}_{\mathrm{b}}+\mathrm{y}\cdot {\mathrm{y}}_{\mathrm{b}}\right)& \begin{array}{ccc}& & \begin{array}{ccc}& & \end{array}\end{array}& \left(4\right)\\ \left({\mathrm{x}}_{\mathrm{c}}^2+{\mathrm{y}}_{\mathrm{c}}^2\right)\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}=2\mathrm{R}\mathrm{v}{\mathrm{t}}_1+{\mathrm{v}}^2{\mathrm{t}}_1{\mathrm{t}}_2-2\left(\mathrm{x}\cdot {\mathrm{x}}_{\mathrm{c}}+\mathrm{y}\cdot {\mathrm{y}}_{\mathrm{c}}\right)\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}& \begin{array}{ccc}& & \begin{array}{ccc}& & \end{array}\end{array}& \left(5\right)\\ \left({\mathrm{x}}_{\mathrm{d}}^2+{\mathrm{y}}_{\mathrm{d}}^2\right)\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}=2\mathrm{R}\mathrm{v}{\mathrm{t}}_1+{\mathrm{v}}^2{\mathrm{t}}_1{\mathrm{t}}_3-2\left(\mathrm{x}\cdot {\mathrm{x}}_{\mathrm{d}}+\mathrm{y}\cdot {\mathrm{y}}_{\mathrm{d}}\right)\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}& \begin{array}{ccc}& & \begin{array}{ccc}& & \end{array}\end{array}& \left(6\right)\end{array}$$

As you can see, the system of three Equations (4)–(6) has three unknowns R, x, y. To solve this system, we subtract Equation (5) from Equation (4), and Equation (6) from Equation (4). Now, after the subtraction operations, we express the variables x and y. As a result, we obtain the coordinates of the position of the sound source (7), (8):

$$\mathrm{x}=\frac{\frac{{\mathrm{x}}_{\mathrm{b}}^2+{\mathrm{y}}_{\mathrm{b}}^2-{\left(\mathrm{v}{\mathrm{t}}_1\right)}^2}{2}\left(\frac{1}{{\mathrm{y}}_{\mathrm{b}}-{\mathrm{y}}_{\mathrm{d}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}}-\frac{1}{{\mathrm{y}}_{\mathrm{b}}-{\mathrm{y}}_{\mathrm{c}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}}\right)+\frac{{\mathrm{v}}^2{\mathrm{t}}_1{\mathrm{t}}_3-\left({\mathrm{x}}_{\mathrm{d}}^2+{\mathrm{y}}_{\mathrm{d}}^2\right)\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}}{2\left({\mathrm{y}}_{\mathrm{b}}-{\mathrm{y}}_{\mathrm{d}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}\right)}+\frac{{\mathrm{v}}^2{\mathrm{t}}_1{\mathrm{t}}_2-\left({\mathrm{x}}_{\mathrm{c}}^2+{\mathrm{y}}_{\mathrm{c}}^2\right)\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}}{2\left({\mathrm{y}}_{\mathrm{b}}-{\mathrm{y}}_{\mathrm{c}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}\right)}}{\frac{{\mathrm{x}}_{\mathrm{b}}-{\mathrm{x}}_{\mathrm{d}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}}{{\mathrm{y}}_{\mathrm{b}}-{\mathrm{y}}_{\mathrm{d}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}}-\frac{{\mathrm{x}}_{\mathrm{b}}-{\mathrm{x}}_{\mathrm{c}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}}{{\mathrm{y}}_{\mathrm{b}}-{\mathrm{y}}_{\mathrm{c}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}}}$$

(7)

$$\mathrm{y}=\frac{\frac{{\mathrm{x}}_{\mathrm{b}}^2+{\mathrm{y}}_{\mathrm{b}}^2-{\left(\mathrm{v}{\mathrm{t}}_1\right)}^2}{2}\left(\frac{1}{{\mathrm{x}}_{\mathrm{b}}-{\mathrm{x}}_{\mathrm{d}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}}-\frac{1}{{\mathrm{x}}_{\mathrm{b}}-{\mathrm{x}}_{\mathrm{c}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}}\right)+\frac{{\mathrm{v}}^2{\mathrm{t}}_1{\mathrm{t}}_3-\left({\mathrm{x}}_{\mathrm{d}}^2+{\mathrm{y}}_{\mathrm{d}}^2\right)\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}}{2\left({\mathrm{x}}_{\mathrm{b}}-{\mathrm{x}}_{\mathrm{d}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}\right)}+\frac{{\mathrm{v}}^2{\mathrm{t}}_1{\mathrm{t}}_2-\left({\mathrm{x}}_{\mathrm{c}}^2+{\mathrm{y}}_{\mathrm{c}}^2\right)\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}}{2\left({\mathrm{x}}_{\mathrm{b}}-{\mathrm{x}}_{\mathrm{c}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}\right)}}{\frac{{\mathrm{y}}_{\mathrm{b}}-{\mathrm{y}}_{\mathrm{d}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}}{{\mathrm{x}}_{\mathrm{b}}-{\mathrm{x}}_{\mathrm{d}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_3}}-\frac{{\mathrm{y}}_{\mathrm{b}}-{\mathrm{y}}_{\mathrm{c}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}}{{\mathrm{x}}_{\mathrm{b}}-{\mathrm{x}}_{\mathrm{c}}\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}}}$$

(8)

3. Results

3.1. Structure of Software Implementation of the System for Automatic Identification of Sound Source Position Coordinates

The system involves three parts: server program, client program and sensor program. These programs are implemented in the Node.js environment.

The server program is constantly running and waiting for a message from the sensor program. This program is responsible for the calculation of sound source geolocation based on data coming from the sensor program. It also sends audio source location data to all customers.

The client program is connected to the server program, and as soon as the calculation is completed on the server, this program receives results. There may be many client programs and they will all receive data from the server instantly.

The sensor program simulates sound wave movement. There are programmed sensor objects inside, and the algorithm sends data to the server. Thus, running the simulation, the server will receive data as if from real sensors.

The results of the determined values of the coordinates of the location of the sound source in the simulated environment on the Node.js platform are shown in Figure 6.

Based on the simulation created on the Node.js platform, you can start a sound wave, add sound sensors (which can be placed in different geometries) and move sound sensors. After activating the sound source, you can see how the wave comes from the sound source, and as soon as it hits the sensor (Figure 6), the server immediately receives a data packet from this sensor.

3.2. Calculation of Relative Error Identification of x and y Coordinates of Sound Sources Depending on Temperature, Humidity and Wind Direction and Its Speed

To calculate relative errors in the determination of the x and y coordinates depending on temperature, humidity and time measurement errors, it is first required for their absolute errors to be found:

$$\begin{gathered} \mathrm{d}\mathrm{x}=\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_1}\mathrm{d}{\mathrm{t}}_1+\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_2}\mathrm{d}{\mathrm{t}}_2+\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_3}\mathrm{d}{\mathrm{t}}_3+\frac{\partial \mathrm{x}}{\partial \mathrm{v}}\mathrm{d}\mathrm{v} \\ \mathrm{d}\mathrm{y}=\frac{\partial \mathrm{y}}{\partial {\mathrm{t}}_1}\mathrm{d}{\mathrm{t}}_1+\frac{\partial \mathrm{y}}{\partial {\mathrm{t}}_2}\mathrm{d}{\mathrm{t}}_2+\frac{\partial \mathrm{y}}{\partial {\mathrm{t}}_3}\mathrm{d}{\mathrm{t}}_3+\frac{\partial \mathrm{y}}{\partial \mathrm{v}}\mathrm{d}\mathrm{v} \end{gathered}$$

Absolute errors of measurement sensors (time t₁, t₂, t₃) are equal to $\mathrm{d}{\mathrm{t}}_1=\mathrm{d}{\mathrm{t}}_2=\mathrm{d}{\mathrm{t}}_3=\mathrm{d}\mathrm{t}$.

Derivatives $\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_1},\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_2},\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_3},\frac{\partial \mathrm{x}}{\partial \mathrm{v}},\frac{\partial \mathrm{y}}{\partial {\mathrm{t}}_1},\frac{\partial \mathrm{y}}{\partial {\mathrm{t}}_2},\frac{\partial \mathrm{y}}{\partial {\mathrm{t}}_3},\frac{\partial \mathrm{y}}{\partial \mathrm{v}}$ are found by the formulas (7) and (8).

Absolute error dv in determining the speed of sound as a function of temperature and humidity is found on the basis of sound wave speed dependence on temperature and relative humidity.

$${\mathrm{v}}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}=20.067\cdot \sqrt{\mathrm{t}+273}\cdot \left(1+0.14\frac{\mathsf{\phi }{\mathrm{p}}_{\mathrm{s}\mathrm{v}}}{{\mathrm{p}}_{\mathrm{a}\mathrm{t}\mathrm{m}}}\right)$$

(9)

$${\mathrm{p}}_{\mathrm{s}\mathrm{v}}=0.0155{\mathrm{t}}^3+2.6041{\mathrm{t}}^2+21.3098\mathrm{t}+746.5601$$

(10)

t—air temperature, C; $\mathsf{\phi }$—relative air humidity, %; ${\mathrm{p}}_{\mathrm{s}\mathrm{v}}$—saturated water vapor pressure (Pa), depending on t is described by a polynomial (4); ${\mathrm{p}}_{\mathrm{a}\mathrm{t}\mathrm{m}}$—atmospheric pressure (Pa).

In conditions of wind, the phase speed of sound for the stationary observer is determined in the following way: $\mathrm{v}={\mathrm{v}}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}+{\mathrm{V}}_{\mathrm{b}}\cos \mathsf{\psi }$.

${\mathrm{V}}_{\mathrm{b}}$—wind velocity; $\mathsf{\psi }=\overline{\mathrm{n}}^{\overline{\mathrm{V}}}_{\mathrm{b}}$—the angle between the normal of point to the front of the acoustic wave and wind direction.

The expression for the relative error in determining the speed of sound with changes in temperature and air humidity, taking into account (10), is as follows:

• We logarithmize the expression (9).
• We take the differential from the natural logarithm and group the terms that contain the same differential
• We will replace all minuses with pluses

As a result, the relative error for determining the speed of sound with changes in temperature and air humidity will be calculated by the following formula:

$$\frac{\mathrm{d}\mathrm{v}}{\mathrm{v}}=\frac{1}{2}\frac{\mathrm{d}\mathrm{t}}{\left(\mathrm{t}+273\right)}+\frac{0.14\left({\mathsf{\phi }}^{\prime }\left(\mathrm{t}\right){\mathrm{p}}_{\mathrm{s}\mathrm{v}}+\mathsf{\phi }\left(\mathrm{t}\right){{\mathrm{p}}^{\prime }}_{\mathrm{s}\mathrm{v}}\left(\mathrm{t}\right)\right)}{{\mathrm{p}}_{\mathrm{a}\mathrm{t}\mathrm{m}}\left(1+0.14\frac{\mathsf{\phi }\cdot {\mathrm{p}}_{\mathrm{s}\mathrm{v}}}{{\mathrm{p}}_{\mathrm{a}\mathrm{t}\mathrm{m}}}\right)}$$

(11)

$${{\mathrm{p}}^{\prime }}_{\mathrm{s}\mathrm{v}}=\left(0.0465{\mathrm{t}}^2-5.2082\mathrm{t}+21.3098\right)\mathrm{d}\mathrm{t}$$

${\mathsf{\phi }}^{\prime }\left(\mathrm{t}\right)$, ${{\mathrm{p}}^{\prime }}_{\mathrm{s}\mathrm{v}}\left(\mathrm{t}\right)$–derivatives of relative humidity and saturated vapor pressure by temperature t.

To find the derivative of relative humidity ${\mathsf{\phi }}^{\prime }\left(\mathrm{t}\right)$,

Table 1. Experimental data on the relative humidity of air at different values of dry air temperature and the difference between values of temperatures of “dry” and “wet” thermometers.

“Dry” Thermometer Temperature, C	Difference between Values of Temperatures of “Dry” and “Wet” Thermometers
	0	1	2	3	4	5	6	7	8	9	10
	Relative Air Humidity, %
12	100	89	78	68	57	48	38	29	20	11	-
13	100	89	79	69	59	49	40	31	23	14	6
14	100	89	79	70	60	51	42	34	25	17	9
15	100	90	80	71	61	52	44	36	27	20	12
16	100	90	81	71	62	54	46	37	30	22	15
17	100	90	81	72	64	55	47	39	32	24	17
18	100	91	82	73	65	56	49	41	34	27	20
19	100	91	82	74	65	58	50	43	35	29	22
20	100	91	83	74	66	59	51	44	37	30	24
21	100	91	83	75	67	60	52	46	39	32	26
22	100	92	83	76	68	61	54	47	40	34	28
23	100	92	84	76	69	61	55	48	42	36	30
24	100	92	84	77	69	62	56	49	43	37	31
25	100	92	84	77	70	63	57	50	44	38	33

is used, which shows experimental data on the relative humidity of air at different values of dry air temperature and the difference between values of temperatures $\left(\Delta \mathrm{t}\right)$ of “dry” and “wet” thermometers.

According to Table 1 interpolate function $\mathsf{\phi }\left({\mathrm{t}}_{\mathrm{d}\mathrm{r}\mathrm{y}}\right)$ and its derivative ${\mathsf{\phi }}^{\prime }\left({\mathrm{t}}_{\mathrm{d}\mathrm{r}\mathrm{y}}\right)$ under different values $\Delta \mathrm{t}$ have been found (

Table 2. Function value $\mathsf{\phi }\left({\mathrm{t}}_{\mathrm{d}\mathrm{r}\mathrm{y}}\right)$ and its derivative value $\mathsf{\phi }\left({\mathrm{t}}_{\mathrm{d}\mathrm{r}\mathrm{y}}\right)$ under different values $\Delta \mathrm{t}$.

$\Delta \mathbf{t}$	$\mathsf{\phi }\left({\mathbf{t}}_{\mathbf{d}\mathbf{r}\mathbf{y}}\right)$	${\mathsf{\phi }}^{\prime }\left({\mathbf{t}}_{\mathbf{d}\mathbf{r}\mathbf{y}}\right)$
1	$-\frac{1}{1944}{\mathrm{t}}^4+\frac{35}{975}{\mathrm{t}}^3-\frac{605}{648}{\mathrm{t}}^2+\frac{10649}{972}\mathrm{t}+\frac{9706}{243}$	$-\frac{1}{486}{\mathrm{t}}^3+\frac{35}{325}{\mathrm{t}}^2-\frac{605}{324}\mathrm{t}+\frac{10649}{972}$
2	$-\frac{1}{1944}{\mathrm{t}}^4+\frac{41}{972}{\mathrm{t}}^3-\frac{833}{648}{\mathrm{t}}^2+\frac{17093}{972}\mathrm{t}-\frac{2513}{243}$	$-\frac{1}{486}{\mathrm{t}}^3+\frac{41}{324}{\mathrm{t}}^2-\frac{833}{324}\mathrm{t}+\frac{17093}{972}$
3	$\frac{1}{972}{\mathrm{t}}^4-\frac{41}{486}{\mathrm{t}}^3+\frac{815}{324}{\mathrm{t}}^2-\frac{7669}{243}\mathrm{t}+\frac{51061}{243}$	$\frac{1}{243}{\mathrm{t}}^3-\frac{41}{162}{\mathrm{t}}^2+\frac{815}{162}\mathrm{t}-\frac{7669}{243}$
4	$-\frac{1}{1944}{\mathrm{t}}^4+\frac{35}{972}{\mathrm{t}}^3-\frac{605}{648}{\mathrm{t}}^2+\frac{11279}{972}\mathrm{t}+\frac{310}{243}$	$-\frac{1}{486}{\mathrm{t}}^3+\frac{35}{324}{\mathrm{t}}^2-\frac{605}{324}\mathrm{t}+\frac{11279}{972}$
5	$-\frac{1}{18}{\mathrm{t}}^2+\frac{59}{18}\mathrm{t}+\frac{142}{9}$	$-\frac{1}{9}\mathrm{t}-\frac{59}{18}$
6	$-\frac{1}{648}{\mathrm{t}}^4+\frac{13}{108}{\mathrm{t}}^3-\frac{757}{216}{\mathrm{t}}^2+\frac{15053}{324}\mathrm{t}-\frac{5186}{27}$	$-\frac{1}{162}{\mathrm{t}}^3+\frac{13}{36}{\mathrm{t}}^2-\frac{757}{108}\mathrm{t}+\frac{15053}{324}$
7	$\frac{1}{648}{\mathrm{t}}^4-\frac{13}{108}{\mathrm{t}}^3+\frac{733}{216}{\mathrm{t}}^2-\frac{12713}{324}\mathrm{t}+\frac{5075}{27}$	$\frac{1}{162}{\mathrm{t}}^3-\frac{13}{36}{\mathrm{t}}^2+\frac{733}{108}\mathrm{t}-\frac{12713}{324}$
8	$-\frac{1}{648}{\mathrm{t}}^4+\frac{13}{108}{\mathrm{t}}^3-\frac{757}{216}{\mathrm{t}}^2+\frac{15053}{324}\mathrm{t}-\frac{5186}{27}$	$-\frac{1}{162}{\mathrm{t}}^3+\frac{13}{36}{\mathrm{t}}^2-\frac{757}{108}\mathrm{t}+\frac{15053}{324}$
9	$\frac{1}{972}{\mathrm{t}}^4-\frac{19}{243}{\mathrm{t}}^3+\frac{683}{324}{\mathrm{t}}^2-\frac{10523}{486}\mathrm{t}+\frac{19834}{243}$	$\frac{1}{243}{\mathrm{t}}^3-\frac{19}{81}{\mathrm{t}}^2+\frac{683}{162}\mathrm{t}-\frac{10523}{486}$
10	$-\frac{1}{1944}{\mathrm{t}}^4+\frac{41}{972}{\mathrm{t}}^3-\frac{869}{648}{\mathrm{t}}^2+\frac{20927}{972}\mathrm{t}-\frac{30431}{243}$	$-\frac{1}{486}{\mathrm{t}}^3+\frac{41}{324}{\mathrm{t}}^2-\frac{869}{324}\mathrm{t}+\frac{20927}{972}$

):

The following real example has been investigated: Sensor receivers A, B, C, D have such values: ${\mathrm{x}}_{\mathrm{a}}=0\mathrm{m}$; ${\mathrm{y}}_{\mathrm{a}}=0\mathrm{m}$; ${\mathrm{x}}_{\mathrm{b}}=-200\mathrm{m}$; ${\mathrm{y}}_{\mathrm{b}}=0\mathrm{m}$; ${\mathrm{x}}_{\mathrm{c}}=0\mathrm{m}$; ${\mathrm{y}}_{\mathrm{c}}=-200\mathrm{m}$; ${\mathrm{x}}_{\mathrm{d}}=-200\mathrm{m}$; ${\mathrm{y}}_{\mathrm{d}}=-200\mathrm{m}$.

Air temperature $\mathrm{t}={20}^{°}\mathrm{C}$; atmospheric pressure ${\mathrm{p}}_{\mathrm{a}\mathrm{t}\mathrm{m}}=101325$ Pa; the difference between values of “dry” and “wet” thermometers is $\Delta \mathrm{t}=5$ degrees and, accordingly, the relative humidity is $\mathsf{\phi }=60\%$; saturated vapor pressure when $\mathrm{t}={20}^{°}\mathrm{C}$ is equal to ${\mathrm{p}}_{\mathrm{s}\mathrm{v}}=2338$ Pa.

Stopwatch measurement devices error is $\mathrm{d}\mathrm{t}=0.0005\mathrm{s}$ and thermometer error is $\mathrm{d}\mathrm{T}={0.05}^{°}\mathrm{C}$. x–the coordinate of the position of the sound source (point O) relative to the sensor of receiver A, which is located at the origin of the coordinates, which we chose to be 5000 m.

The general calculation of the x coordinate absolute error is:

$$\begin{array}{c}\mathrm{d}\mathrm{x}=\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_1}\mathrm{d}{\mathrm{t}}_1+\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_2}\mathrm{d}{\mathrm{t}}_2+\frac{\partial \mathrm{x}}{\partial {\mathrm{t}}_3}\mathrm{d}{\mathrm{t}}_3+\frac{\partial \mathrm{x}}{\partial \mathrm{v}}\mathrm{d}\mathrm{v}=35748.4292\times 0.0005+13583.7553\\ \times 0.0005+11644.3133\times 0.0005+0.000015\times 0.000888=30.4882\mathrm{m}\end{array}$$

Then relative error of the x coordinate is the following: $\frac{\mathrm{d}\mathrm{x}}{\mathrm{x}}=\frac{30.488249}{5000}=0.61\%$.

4. Discussion

Table 3. Results of numerical calculations of the relative error of determining the coordinates x and y of the position of the sound source depending on the distance between the four sensor-receivers.

№	The Distance between the Sensor-Receivers (Side of the Square), m	The Relative Error of Determining the Coordinate of the Position of the Sound Source, %
1	20	7.37
2	50	2.88
3	70	2.02
4	100	1.37
5	200	0.61
6	300	0.35
7	500	0.18
8	1000	0.15
9	2000	0.18
10	5000	0.25

shows the results of numerical calculations of the relative error of determining the coordinates x and y of the position of the sound source depending on the distance between the four sensor-receivers, which are located at the vertices of the square, at a temperature $\mathrm{t}={20}^{°}\mathrm{C}$ and relative humidity $\mathsf{\phi }=60\%$.

As can be seen from Table 3, with an increase in the distance between the sensor-receivers from 20 m to 1000 m, the relative error in determining the coordinates of the position of the sound source decreases from 7.37% to 0.15%. When sensor receivers are located at distances greater than 1000 m, the relative error in determining the coordinates of the position of the sound source increases. The main part of the error lies in the time measurement error [28].

The analysis of Table 3 shows that at a distance between the sensor-receivers of 200 m, there is a significant decrease in the relative uncertainty of determining the coordinates of the sound source. Further analysis of the influence of air temperature and relative humidity is carried out when four sensor-receivers are located at a distance of 200 m from each other. With an increase in temperature from $-10°\mathrm{C}$ to $+35°\mathrm{C}$ and relative humidity $\mathsf{\phi }=60\%$, the relative error in determining the coordinates of the sound source increases by 0.06% (

Table 4. Analysis of the effect of air temperature on the relative error of determining the coordinates of the sound source.

№	Air Temperature, ℃	The Relative Error of Determining the Coordinate of the Position of the Sound Source, %
1	−10	0.57
2	−5	0.58
3	0	0.58
4	5	0.59
5	10	0.60
6	15	0.60
7	20	0.61
8	25	0.62
9	30	0.62
10	35	0.63

), while when the humidity changes from 10% to 90% at an air temperature $\mathrm{t}=20°\mathrm{C}$, the relative error in determining the coordinates of the sound source does not change (

Table 5. Analysis of the influence of relative air humidity on the relative error of determining the coordinates of the sound source.

№	Relative Humidity, %	The Relative Error of Determining the Coordinate of the Position of the Sound Source, %
1	10	0.60
2	20	0.61
3	30	0.61
4	40	0.61
5	50	0.61
6	60	0.61
7	70	0.61
8	80	0.61
9	90	0.61

).

As shown in the previous section, sound source position coordinates determination relative error for the case of sensors’ positions in vertices of a square with a side of 200 m is 0.61%.

Table 6. Relative error research results of the x and y coordinates depending on the distance between sensors, which are located at the square vertices for the above parameters of temperature and humidity.

№	Distance between the Sensors (the Side of the Square), m	Relative Error of the Coordinate Finding, %
1	20	7.37
2	50	2.88
3	70	2.02
4	100	1.37
5	200	0.61
6	300	0.35
7	500	0.18
8	1000	0.15
9	2000	0.18
10	5000	0.25

shows relative error research results of the x and y coordinates depending on the distance between sensors, which are located at the square vertices for the above parameters of temperature and humidity.

It should be noted that for the minimum relative error of sound source position coordinates determination, sound receiver sensors should be located at a distance of 1000 m (side of the square) (Table 6), because the main error part lies in time measurement error.

Therefore, the study of the relative error of determining the coordinates of the position of the sound source depending on the symmetry of the placement of the sensors showed that in order to obtain the minimum relative error, the receiver sensors should be placed at the vertices of a flat symmetrical figure (square) with 4 axes of symmetry of the second order. As the symmetry of placement of the sensor-receivers decreases, the relative error of determining the coordinates of the position of the sound source increases.

This study can be found at the following link [31].

Importantly, in the existing algorithms [25,26,27,28,29], the method of determining the coordinates of the sound source is based on the hyperbola equation, which describes the non-linear relationship between the difference in the time of arrival of sound signals to sensors located in a given geometry. This approach is based on a piecewise linear approximation of the second-order curve, which ultimately leads to an error in measuring the direction angle to the target. Contrary to existing algorithms, the method proposed in this article does not approximate the second-order curve. The basis of our algorithm is a system of three equations of the second order describing the dynamics of the propagation of the spherical acoustic wavefront. Moreover, the proposed algorithm considers the change of the phase velocity of the sound wave depending on the parameters that were not included in the works [25,28,29], i.e., temperature, humidity, wind direction and speed.

5. Conclusions

In this article, a new algorithm for identifying the coordinates of the position of sound sources, based on a system of equations describing the propagation dynamics of the acoustic wavefront, has been developed. The algorithm considers parameters such as temperature, humidity, wind direction and speed. Taking these parameters into account is very important because the speed of sound depends on weather conditions [30]. Moreover, to avoid an error in measuring the direction angle to the target, the proposed algorithm does not approximate the second-order curve and is based on a system of three second-order equations describing the dynamics of the propagation of the spherical acoustic wavefront.

The system for automatically determining the coordinates of the sound source location consists of three parts: the server program, the client program and the sensor program, and the experiment with real input data was carried out in a simulated environment, which was created on the Node.js platform.

In summary, the scientific novelty of the obtained results is as follows:

• A new algorithm for determining the coordinates of the location of sound sources has been developed. The algorithm takes into account the change in the phase speed of the sound wave when the temperature, air humidity, wind direction and wind speed change and does not lead to the appearance of a systematic error in the measurement of the direction angle to the target.
• A system for automatically determining the coordinates of the position of the sound source, containing the server, client and sensor-receiver subroutines, has been implemented in software.
• It was calculated that at a temperature of $\mathrm{t}={20}^{°}\mathrm{C}$ and relative humidity $\mathsf{\phi }=60\%$, with an increase in the distance between the sensor-receivers from 20 m to 1000 m, located at the vertices of the square, the relative error in determining the coordinates of the sound source location decreases from 7.37% to 0.15%.
• The analysis of the influence of temperature and relative air humidity showed that with an increase in temperature from $-{10}^{°}\mathrm{C}$ to $+{35}^{°}\mathrm{C}$ and an increase in relative air humidity from 10% to 90% with a fixed location of the sensor-receivers, the relative error in determining the position of the coordinates of the sound source increases from 0.57% to 0.63%.
• It was recognized that the relative error of determining the coordinates of the position of the sound sources is minimal if the sensor-receivers are placed at the vertices of a flat figure with 4 axes of symmetry of the second order.

In our model, only one type of sound signal is taken into account. In future research, we intend to deal with the problem of determining the coordinates of the position of sound sources, taking into account the recognition of different types of sound signals. Recognition of different types of sound signals can be implemented by the neural network method of blind distribution of signals.