Elastic Wave Signal Attenuation Characteristics in Layered Earth Channel

Jianjun HAO College of electric & Information engineering

Shandong University of Science & Technology Qingdao, China, 266510 E-mail: wfhjj@126.com

Yuan ZHAO College of electric & Information engineering

Shandong University of Science & Technology Qingdao, China, 266510

E-mail:zhaoyuan-0804@163.com

Abstract—In this paper, by analyzing the principles of elastic wave propagation in earth crust layer, the total attenuation model of longitudinal wave in simplified parallel layered earth medium channel is achieved, then a parallel layered channel and quasi-parallel layered channel model developed respectively. The propagation simulation is carried out using ray-trace method, the results show: (1) attenuation degraded with the incident angle increased; (2) whether in parallel layered earth medium or quasi-parallel earth medium, the variation of incident angle has slight effect on transmission attenuation, but can cause an enlarged arrived position offset; (3) medium absorption attenuation is lower than interface reflection loss of about 20dB at frequency of 1500Hz.

Keywords- elastic wave, through-the-earth communication, transmission attenuation

I. INTRODUCTION China is a country with high mine-accident frequency.

When mine disaster occurs, the existing communication channel is often destroyed, so in this case miners can’t know what had happened in time and contact with the people on ground after being trapped. Due to the stability of earth channel, which is not affected by accidents, through-the-earth communication becomes an effective communication method of keeping contact between the ground and underground, helps to realize urgent informing while the accident occurred and rescue communication post-disaster.

Currently, domestic and foreign researchers of through-the-earth communication often focus on low frequency electromagnetic wave. In China, Zhang Qing-yi[1] in China University of Mining and Technology, Tao Jin-yi[2] in Taiyuan University of technology and so on respectively have studied on underground electromagnetic wave propagation influence, modulation mode and weak signal receiving technology based on parameters such as the working frequency in the large loop antenna case. In abroad, in the mid 1990s Australian Mining Technology Company developed an PED (Personal Emergency Device)emergency command paging system working at the frequency of 3,000 ~ 8000Hz and can transmit through hundreds of meters’ rock; at 2006, a technical reports issued by American Los Alamos National Laboratory puts forward to use the very low frequency technology to solve the coal mine communication and the technical route of personnel positioning[3] and Barkand and other researchers[4], tested and studied the "Telemag" half-duplex through-the-earth

communication prototype system in a limestone ore and the operation of the coal mine, which show that the system can satisfy the requirement of emergency communication.

The defect of electromagnetic wave through-the-earth communication lies in demanding a large circle antenna with diameter of tens of kilometers. This can be avoided if the low-frequency elastic wave is used to through-the-earth communication. We need to know characteristics of the elastic wave as the carrier of information transporting under the ground before this communication system implemented. For elastic wave propagation in strata, experts in earthquakes and exploration have get many achievements, but most of those focus on source locating and metallic minerals distribution analysis, do not care about transmission characteristics of modulated elastic wave. In this paper we take advantage of research results in the seismic wave propagation and establish the multi-layer stratigraphic model according to the actual composition of the earth, then try to analyze the transmission characteristics of the earth channel, which can provide guidance for the study on the elastic wave through-the-earth communication technology.

II. ELASTIC WAVE’S ABSORPTION ATTENUATION IN EARTH MEDIUM

Several factors cause propagation attenuation and loss while elastic wave propagating in strata , such as the layered characteristics of the earth, the inhomogeneity of earth materials, and the non-absolute elasticity. Part to try to discuss transmission attenuation problem from several aspects citing below.

Absorption is one of the main losses while elastic wave propagating in strata. Due to the non-absolute elasticity of earth medium, the energy of the elastic wave would be dissipated as the absorption in the medium, and wave amplitude would decrease, which is mainly because of the internal friction of the elastic medium.

Suppose there is a plane harmonicwave transmitting along the x direction in the homogeneous medium, which displacement is )}(exp{)(),( tkxixAtxu ω−= . （1） Where A(x) is the wave's amplitude, k is the wave number. In isotropic medium the attenuation of the amplitude is

teAxA α−= 0)( , （2） where 00 )( == xxAA , is the amplitude of the signal source. Equation (2) shows that the amplitude attenuating exponentially with the increase of the distance, and the

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attenuation rate depends on α , which is called the attenuation coefficient of elastic wave.

Since for the non-absolute elasticity of earth materials, amplitude fades as the energy of the elastic wave is absorbed during transmitting in earth shell. In order to describe the degree of the medium absorption, a dimensionless factor Q called the quality factor of the medium is used. Q is used to measure the inelastic attenuation rate of vibration or wave energy, which is the medium’s inherent characteristic. Since the medium is elastic, the elastic kinetic energy is partly converted into wave propagation in the wave propagation process. When a periodic sine wave passes by rock type media, given the energy loss in a cycle is EΔ , the stored elastic energy is E, then define Q:

EE

QΔ=

π211 . （3）

Q is the quality factor of the rock (or medium). Obviously, the bigger the quality factor is, the smaller the loss of rock to elastic wave is, and the closer the characteristic is to elasticity. Generally believe that quality factor and frequency is irrelevant, the relationship between quality factor and attenuation coefficient is

Qπωα

2= . （4）

It is concluded that attenuation factor is the function of frequency.

According to Futterman model [5], assume elastic wave propagation speed in a certain layer is constant, the wave field of elastic wave after transmitting zΔ distance is

)2/exp()exp(),(),( QzvjzPzzP ωτωωω −Δ=Δ+ ,

（5） where τ is the time expense for travel zΔ in the layer, v is the phase velocity, Q is the quality factor of this layer, therefore, equation (5) can be written as

)},(/](2

1exp{[),(),( zvzjQjzPzzP ωωωω Δ+=Δ+

（6）

III. INTERFACE REFLECTION LOSS Wherever Times is specified, Times Roman or Times

New Roman may be used. If neither is available on your word processor, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.

Elastic wave will reflect at the stratum interface with different mechanical properties, causing the attenuation of the elastic wave amplitude. The reflectance ratio and transmissivity can be calculated by Zoeppritz equations, but solving the equations is very complex. Considering the elastic wave in the through-the-earth communication is vertical to the interface or nearly so, therefore, it can be thought to approximate for vertical incidence and calculate approximately the reflectance and transmission rate by (7)(8).

1122

1122

pp

pppp vv

vvR

ρρρρ

+−

= 0=psR , （7）

1 1

2 2 1 1

2 ppp

p p

vT

v vρ

ρ ρ=

+ 0=psT . （8）

where ppR and ppR are the reflect and transmit coefficient respectively. This is the energy distribution between reflection and transmission in the interface, and accordingly the elastic wave loss through dielectric layers interface can be calculated. If strata are obviously tilted, it needs to be calculated according to Zoeppritz equations. In view of Zoeppritz equation complex computation, R. Bortfeld gives the approximate simplified equation [6] of longitudinal wave transmission.

222

111

coscos

θρθρ

p

ppp v

vT ≈ . (9)

IV. WAVEFRONT DIFFUSION ATTENUATION With the elastic wave spreads from wave source to

nearby regions, the wavefront area become larger, and the amplitude forward become smaller, which is called wavefront diffusion (geometric diffusion).

Spherical wave is a common elastic wave in elastic method of exploration. In homogeneous isotropic elastic media, this spherical form transmitting in the medium is that in any sphere perpendicular to the propagation direction each point in same vibration phase, while the wave front is sphere. Far from signal source lead to a long r , and the displacement solution of spherical p-wave is

3

1 '4p

p p

r rU trV V rπ

⎛ ⎞≈ Φ −⎜ ⎟⎜ ⎟

⎝ ⎠. (10)

When far away from source, particle displacement (amplitude) is proportional to first-order derivative of elastic source function and inversely proportional to the distance.

Besides medium absorption, layer interface reflection and geometric diffusion, there are still medium scattering attenuation and others, which are not considered in this paper.

V. THE ESTABLISHMENT OF THE LAYERED EARTH CHANNEL MODEL

Due to the very complex channel model, different geological conditions from different places cause transmission characteristic very big different. Therefore, to facilitate discussion, we establish the parallel layered earth medium model and suppose channel is ideal strata, considering only medium absorption and interface reflection losses, the total transmission attenuation of the channel is

∏=

−+=

n

i

lii

iiieRa1

cos/)1,(

θα . (11)

The amplitude of the attenuation is not equal to loss situation, here we use another physical quantity, and the

239

energy flow density of elastic wave ispx VuI ρω 22

021= .

Because the frequency of elastic wave in different rock is constant, only can energy flow density be concerned with the amplitude 0u and the medium wave impedance pvρ , so total power loss is

∏=

−+

=n

i

iiliii eR

p

pnnp v

vd 1

cos/)1,(

1000

log2 θα

ρρ

(dB) , (12)

where the il is the length, iθ is the incident angle, iα is the

attenuation factor of layer i , nρ , pnv is the mass density

and wave velocity of bottom layer, and 0ρ , 0pv is the mass density and wave velocity of top layer.

Might as well simplify the earth materials: 1) Only considering interface reflection losses and

medium absorption attenuation, ignoring of scattering and other attenuation;

2) Each layer in the layered earth channel model is approximate for parallel layer (except the ground, the horizontal dip angle of other layer surfaces within 5 °);

3) Elastic wave is plane; 4) Under the circumstances of incident angle less than 5

°, don’t consider S wave division; Assume the earth medium is divided five layers and

parameters of each listed in Table I.

TABLE I. PARAMETERS OF THE LAYERED EARTH MEDIUM

Layer No. Lithologic

P-wave velocity (km/s)

Density of rock (g/cm3)

Thickness (m)

Quality factor

(Q) 1 siltstone 1.3 2.45 20 50 2 marlstone 2.0 2.5 20 90 3 argillite 2.4 2.5 20 100 4 dolomite 2.7. 2.55 20 130 5 limestone 3.0 2.6 20 140

VI. SIMULATION RESULTS AND ANALYSIS The simulation is presented on MATLAB 7.0, suppose

the layered channel is parallel layers with 20m thick, elastic wave propagating path is vertical to layer interfaces, the frequency varies from 0~2000Hz, the computed attenuation shows as Figure 1.

Figure 1. Propagation attenuation versus frequency

Figure 2 is the results when incident angle changes, the signal transmission gain curves are different with the frequency, the above is signal gain when frequency is 800Hz, the below is of 1000Hz.

Figure 2. Signal gain versus incident angle

Fig. 1 shows that the propagation attenuation variation is approximately linear to the frequency increasing, and the mean attenuation is nearly 0.4dB/m. Fig. 2 shows while the incident angle less than °5 , the attenuation curve is flat, while the incident angle is near °30 , total reflection occurs. Flat curve indicates the attenuation is not sensitive to a small incident angle variation, but a position offset will be occurred as shown in Fig. 3.

Figure 3. Arriving point offset versus incident angle

Figure 4. Magnitude attenuation versus incident angle

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Fig. 5 and Fig. 6 is the attenuation of absorption and reflection individually, the above is reflection loss. When the frequency is 800Hz, the attenuation caused by medium absorption is lower than interface reflection loss of 10dB, as shown in Fig. 5, and the attenuation difference rise to more than 20dB while f = 1500Hz.

Figure 5. Interface reflection attenuation (above) and absorption

attenuation (below), f=800Hz

Figure 6. Interface reflection attenuation (above) and absorption

attenuation (below), f = 1500Hz

VII. CONCLUSION Propagating in ideal layered earth medium channel

presented above, elastic wave propagation has some features: 1) The attenuation is degraded while frequency

increased, and interface reflection loss is lower than medium absorption attenuation of 2 orders of magnitude (f=1500Hz).

2) A small incident angle variation (less than °5 ) have nearly no influence on propagation attenuation, but can cause arriving position offset of energy flow；

3) Layered earth channel with tilt angle less than °5 slightly affect on transmission attenuation, but an enlarged arriving position offset should be considering.

ACKNOWLEDGMENT The paper is supported by the project of National Natural

Science Foundation of China (No. 61071087 )

REFERENCES [1] Zhang Qing-yi, Zhu Jian-ming, Study on Propagation Characteristics

of the VLF Through the Earth Communication Channel[J]. Chinese Journal of Radio Science. Vol. 14, 1999(1):36-40

[2] Tao Jin-yi. Discussion on Some Questions about through Earth Radio Communication System [J]. Journal of Taiyuan University of Technology. Vol,31, 2000 (6):47-50

[3] Chirdon D., Barkand T., Damiano N., Dolinar K., Dransite G., Hill J., Retzer P., and Shumaker W. “Emergency Communication and Tracking Committee Underground Communication and Tracking Systems Tests at CONSOL Energy Inc., McElroy Mine”. Mine Safety and Health Administration, Technical Support, Report of Findings, June 13, 2006.

[4] Hjelmstad K.E. and Pomroy W.H., “Ultra Low Frequency Electromagnetic Fire Alarm System for Underground Mines”, United States Department of the Interior, Bureau of Mines, Report of Investigations 9377,1991.

[5] Futterman W I. Dispersive body waves[J]. Geophys Res，1962，7(13)：5279-5291

[6] R. Bortfeld. Approximations to the reflection and transmission coefficients of plane longitudinal waves. Geophysical Prospecting, 1961, Vol. 9, No. 4:485-502

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