A Theoretical Investigation of Channel Wave Multipath Propagation in a Coal Seam

+is study investigates the mechanism of channel wave multipath propagation to determine features of a channel wave in a coal seam. For this purpose, deduction formulas were used for different time delays and propagation paths and, subsequently, a multipath fading channel wave pattern was built. MATLAB software was used to simulate the model. To this end, the characteristics of the coal seam and the surrounding rock were considered.+e simulation results were compared with results obtained from the model contained in COMSOL software to verify the accuracy of the model. According to the results obtained from the simulation, the proposed model shows a good match with the COMSOL model, which is a common simulation comparison standard. In addition, the channel wave energy approximates to exponential decay at a constant distance from the source, and the multipath propagation considerably affects its energy attenuation. Furthermore, the multipath reflections’ travel time derivation is accomplished properly in the coal seam.


Introduction
When using an elastic wave to probe a coal seam, a specific elastic wave is connected to the coal seam using a transmitter [1,2]. According to the weak elastic wave signal detected by the receiver, the detection system records the geological structure of the coal seam through signal processing and analysis [3,4]. When elastic waves propagate in the formation, the elastic wave reflection [5], refraction [6], and transmission [7] characteristics make the composition of the elastic wave very complicated, resulting in a multipath effect [8]. Under the influence of this effect, the signal is transmitted via various paths to reach the receiving end. Multipath propagation creates two problems: fading and dispersion [9,10]. e various propagation velocities that form the frequency of the elastic waves cause dispersion. Meanwhile, channel wave losses in the formation cause fading [11]. In particular, the multipath fading caused by signal reflection, which may vary greatly [12], may change the transmitted signal.
ere are many studies on the multipath propagation phenomenon of electromagnetic waves at home and abroad [13][14][15], but there are few studies on the multipath effect of elastic waves in a geological formation. Zhang et al. established the attenuation and time delay model of a multipath channel by analyzing the attenuation and delay characteristics of multipath propagation in a tunnel [16]. Hao et al. obtained a multipath channel model for elastic wave signal propagation in a homogeneous medium by analyzing the attenuation characteristics of elastic waves propagating in parallel layers [17]. Gao and Huang observed that the multipath effect on elastic waves is determined by the characteristics of the earth channel during transmission and is the main factor affecting the performance of a through-the-earth communication system [18]. e main contribution of this paper is to try to explore the signal fading problem in the fine detection of coal channel transmission trough wave. e characteristics of elastic wave multipath fading are analyzed based on the multipath effect which has great influence on fading. A new multipath propagation model is established, and the multipath propagation of elastic waves is simulated by MATLAB and COMSOL. e correctness of the model was verified.
rough the work of this paper, it can provide theoretical basis and fundamental guarantee for the study of the elastic wave receiving method of formation propagation. Meanwhile, it provides a new method for detecting stratum structure during coal mining, which is of great significance for coal seam exploration. is paper is divided into four parts. is first part discusses the role and characteristics of elastic waves and the research status of multipath effects on elastic waves propagating in a formation. e second part discusses the mechanism of elastic wave multipath propagation, the path difference of multipath propagation and time delay, and the energy attenuation of multipath signals. In the third part, based on the established multipath propagation model of elastic waves in a coal seam, the propagation of elastic waves in a coal seam is simulated. Finally, our conclusions are summarized in the fourth part.

Mechanism of Elastic Wave Multipath Propagation.
In the current study, the coal seam is assumed to be a homogeneous medium that is in parallel to the rock around it. A coal seam channel wave multipath propagation model is developed in which the diffraction propagation of the surrounding rock is ignored. It is assumed that the receiver detection sensor R and the transmitter transducer T are positioned in the way indicated in Figure 1. e height of the coal seam is denoted by H, and the heights of T and R from the bottom of the seam are both H/2. d denotes the transmitter-receiver interval. In the first situation, waves of the channel are reflected onto the lower interface. is situation is considered first as when waves reflect at both the higher and lower interfaces, the same interaction occurs.

e Collineation
Path. T launches the signal, and they are received by R though the path L; this is shown as the direct path signal. e L value is expressed as follows: (1)

e Initial Reflection
Path. T launches the signal, and they arrive at P 11 by L 11 . Some signals travel into the surrounding rock (this type of signal is not considered here), while other signals are reflected by P 11 , and they are received by R through L 12 . e total path L 1 for these reflection signals is expressed as follows: (2) θ 1 is the incidence angle of this signal. Hence,

e Secondary Reflection
Path. T sends the signal, and it arrives at P 21 by L 21 . Some signals are transferred into the enclosing rock, while other signals reach P 22 by L 22 . A part of the latter signals is transferred into the surrounding rock, and R receives the remaining part. is signal is referred to as the secondary reflection signal, and the total path L 2 is expressed as follows: (4) θ 2 represents this signal's incidence angle. Hence, (5)

e Triple Reflection
Path. T sends the signal, and it arrives at P 31 by L 31 . Some signals are transferred into the enclosing rock, while other signals reach P 32 by L 32 . A part of the latter signals is transferred into the enclosing rock, and the remaining part reaches P 33 by L 33 . In addition, one part of the latter signals is partially transferred into the rock, whereas R receives the other part through L 34 . is signal is referred to as the triple reflection signal total path, of which L 2 is expressed as follows: θ 3 represents the signal's incidence angle. Hence, is process continues in the same way. e assumed coal seam contains a gentle and flat interface parallel to the enclosing rock. erefore, when scattering and reflection are not considered in its microstructure, the following process is used [16].

Surrounding rock
Coal seam d Surrounding rock Figure 1: e diagram of a multipath signal that reflects the higher and lower interfaces. 2 Mathematical Problems in Engineering e received signal includes a signal reflected from the bottom and top of the coal seam, and a direct path signal. It is assumed that the coal seam includes Z paths (each reflected signal has one path), and N denotes the number of received signals. Hence, N � 1 + 2Z.
For a path with multiple reflections (the first originating from the interface) in a coal seam, it is assumed that n (n � 1, 2, 3, . . .) denotes the number of reflections. us, the path includes n + 1 broken lines, and the following relation expresses the total path length: where H denotes the height of the coal seam. d denotes the transmitter-receiver interval. θ n represents the assumed incident angle of the nth reflected signal. Hence, (9)

Path Difference and Time Delay of Multipath
Propagation. Path difference occurs among multipath signals at the time of propagation of channel waves in a coal seam, causing signal delay. Multipath fading develops as a result of the multipath propagation of channel waves at the time of reception of the signal by the receiver. is can have a significant impact on the received signal's quality [19][20][21]. It is possible to deduce the path difference between the multipath signal and the direct signal, along with the time delay formula in the multipath signal, by considering the direct wave based on the path formula and multipath propagation model given above. Δd 1 denotes the difference between the primary reflection path and the direct path difference at the bottom of the coal seam, expressed as follows: Δt 1 denotes the one-time reflection signal propagation that is expressed as follows: ] denotes the channel waves' propagation velocity in the coal seam. e height is equal to H/2 for both the transceivers. It means that there is a relationship between propagation delay and transmitter-receiver distance, and it is not dependent on the initial radiation direction at the height of the transceivers.
Δd 2 represents the path difference between the secondary reflection path and the direct path difference at the bottom of the coal seam, expressed as follows: Δt 2 denotes the second reflection signal propagation delay that is expressed as follows: e equation given above indicates that there is a close relationship between the time delay, the transmitter-receiver distance, and the thickness of the coal seam. erefore, it is not dependent on the initial departure angle.
Δd 3 denotes the path difference between the third reflection path and the direct path difference at the bottom of the coal seam: Δt 3 denotes the third reflection signal propagation delay that is expressed as follows: According to the mechanism of the multipath propagation, the path difference between the nth reflection L n path and the direct path difference L at the bottom of the coal seam is Δd n (n � 1, 2, 3, . . .), expressed as follows: e following equation gives the nth propagation delay between the reflected path and the direct path: where n � 1, 2, 3, . . . and ] denotes the channel wave propagation velocity in the coal seam. Considering equation (17), if the height from the bottom of the coal seam is H/2 for both the transmitter antenna and the receiver antenna, the propagation delay of the channel wave depends on H, d, and v.

Multipath Fading Problem.
Several factors cause the attenuation and loss of channel wave propagation within a coal seam. e main factors include dielectric absorption loss due to the incomplete elasticity of the coal seam, and interface reflection loss due to channel wave reflection through the mechanical properties of the coal seam interface [22]. If Δl represents the channel wave propagation distance in the coal seam, then the 0-1 variable k is written as follows: Mathematical Problems in Engineering k � 0, The elastic wave does not reflect in the propagation Δl, 1, The elastic wave has reflection in the propagation Δl. (18) According to the change in energy of channel wave, it is possible to obtain the following equation: where E(Δl) denotes the energy of the channel wave propagation distance after Δl; E 0 denotes the primary energy of channel waves; and η α denotes the coefficient of the absorption loss. It can be calculated using properties of the coal seam medium. η β denotes the interface reflection coefficient, calculated from the features of the coal seam and the rock enclosing it. e units are in dB, and the negative symbol is used. e absorption loss coefficient: for a clear description of the absorption level of the medium, a dimensionless quality factor Q is presented [17]. is factor is defined as the ratio of storage of energy to dissipation of energy in the system. It is expressed as follows: where ΔE denotes the energy lost through the unit crosssectional area within one cycle; E denotes the energy that is stored as elastic energy; and Q denotes the medium (coal) quality factor. is factor is an essential property of the medium. e following equation indicates a relationship between the attenuation coefficient and the quality factor: Here, ] indicates the wave velocity and f indicates the channel wave frequency. η α denotes the coefficient of the absorption loss. Q denotes the medium (coal) quality factor. e reflection loss coefficient: channel waves have the ability to create reflection at the interface of the coal seam.
e Zoeppritz equation can be used for calculation of reflectivity. However, it is a highly complicated equation. It is assumed that channel waves are plane harmonic waves.
us, the Zoeppritz equation can be solved in a simplified way, and the reflection coefficient can be expressed as follows: where ρ is the density of the coal seam; ] denotes the propagation velocity of the channel wave in the coal seam; θ is the incidence angle; ρ ′ denotes the enclosing rock density; v ′ denotes the channel wave propagation velocity in the enclosing rock; and θ ′ denotes the angle of refraction. η β denotes the interface reflection coefficient. Equations (19), (21), and (22) can be solved simultaneously so that E is obtained, which is expressed as follows: where k 1 represents a constant; E 0 denotes the primary energy of the channel waves; and l represents the channel wave propagation distance in the coal seam.

Simulating Channel Wave Multipath Transmission in a Coal Seam
A channel wave transmission was simulated according to the multipath transmission model. Table 1 gives the main parameters of the coal seam and the rock enclosing it [23]. ere is a common belief that the Ricker wavelet represents the seismic wave model. erefore, the Ricker wavelet is used as a transmitting wave with 200 Hz frequency. e distance between the receiver detection sensor and the transmitter transducer is 50 m. e shape of the coal seam is rectangle with thicknesses of 6, 8, and 10 m. e distances between the transmitter transducer and the receiver detection sensor are 3, 4, and 5 m. COMSOL was used for running the simulation. COMSOL is a leading numerical simulation software package capable of simulating a variety of physical processes. Its simulation data provided by this software are usually utilized as a standard [24][25][26]. Reflected wave time delays are associated with direct waves in the model, and the simulation. Figure 2 indicates these time delays.
As observed in Figure 2, the reflected wave time delay associated with the direct wave increases with the increase in thickness of coal seam, both in the simulation and in the model. ere is some discrepancy between results of the model and the simulation, as the peak of the receipt signal is used as the time of the receipt multipath signal in the simulation.
ere are different factors influencing the peak signal arrival time, the most significant of which is the compound nature of the channel wave signal spreading from the enclosing rock and all reflection signals. Nevertheless, the reflection signal's time delay trend shows consistency with the increase in the thickness of the coal seam. Figure 3 indicates the signal time disparity between the multipath transmission model and COMSOL simulation.
From the comparison of the time delays obtained from the multipath transmission model and COMSOL simulation shown in Figure 3, it is found that the difference in path times is always between − 0.002 s and 0.002 s. Hence, there is consistency between the model given in COMSOL and the model proposed in the current study. e thickness of the coal seam is considered as 8 m, and the height of the enclosing rock is 10 m for calculating the pressure attenuation decline of all path signals, i. It is calculated by the use of MATLAB and COMSOL according to the multipath transmission model and the energy loss model. It can be observed in Figure 4.
It is observed that there is consistency between the decline tendencies of the proposed model and the COMSOL simulation results. Hence, the validity of the channel wave attenuation model in the current research is confirmed. Fluctuations are larger in the COMSOL simulation illustrated in Figure 4 due to the impact of channel waves from the enclosing rock. In addition, there is a similarity between the channel wave energy response and an attenuation following the index law over time in 50 m.  The height of the coal seam is 6 m The height of the coal seam is 8m The height of the coal seam is 10 m

Conclusion
In this paper, the multipath propagation mechanism is analyzed based on simplifying the coal seam and surrounding rock, and the path difference and delay difference formulas of multipath propagation are derived. e elastic wave multipath in the formation is based on the simple experiment of the transmitter and receiver. e effect problem was mathematically modeled with MATLAB and COMSOL simulation, and the multipath fading model of a channel wave was given. It is concluded that through model simulation and comparison with COMSOL simulation results, although there are certain differences, the development trend is consistent, which verifies the correctness of the model built in this paper; the time delay of elastic wave propagation in a coal seam increases with the increase of coal seam thickness; as time increases, the elastic wave energy at a fixed distance from the source is approximately exponentially attenuated. e conclusions reached are of great significance for coal seam exploration.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.