APPLIED GEOPHYSICS, Vol.11, No.1 (March 2014), P. 107-114, 5 Figures.DOI: 10.1007/s11770-014-0409-5Analysis of quality factors for Rayleigh channel waves*Yang Xiao-Hui*, Cao Si-Yuan', Li De-Chun, Yu Peng-Fei, and Zhang Hao-Ran'Abstract: To facilitate investigation of the effect of imperfect elastic dissipation on thepropagation of Rayleigh-type channel waves and use of their quality factors in investigationsof the properties of coal seams, a simple method for calculating the quality factor Qr isproposed in this paper. Introduction of complex velocities into the dispersion function allowscalculation of the dispersion function of Rayleigh-type channel waves in coal seams. By thecontrol variable method, we analyzed changes in Qr with changes in coal seam thickness andP- and S-wave Q factors within the coal seam and adjacent rock layers. The numerical resultsshow that the trend of the Qr curve is consistent with the group velocity curve. The minimumQk value occurs at the Airy phase frequency; the Airy phase frequency decreases as coal seamthickness increases. The value of Qr increases with increasing Qs2 (quality factor for S wavein coal seam). We can compensate for the absorption of Rayleigh-type channel waves usingthe computed Qr curve. Inversion of the LR curve can also be used to predict the thicknessesand lithologies of coal seams.Keywords: Coal seam, Rayleigh channel wave, dispersion curve, attenuation, quality factorIntroductionface, whereas for reflection surveys, shots and receiversare located along one side (the tunnel wall) of the coalseam working face.There are several geological factors affecting the safetyWhen seismic waves are excited within a coal seam,of production during mining, such as minor geologicalthe waves are reflected from both its top surface and itsfaults, collapse columns, and mutations of coal structurebottom surface. Seismic waves are trapped in the coalfeatures. Abandoned workings may also affect safety.seam and spread within it; these waves are called channelHowever, it is difficult to detect such anomalies throughwaves. ISS uses artificially generated channel waves toground exploration. For this reason, several minelocate minor geological disturbances within coal seams.geophysical exploration techniques have been developed,The waveforms of channel waves are easily identifiable.of which in-seam seismic (ISS) surveys, a type of channelIn addition, channel waves carry information about thewave seismic survey, is a promising method.construction and properties of the coal seam once theyISS surveys can be divided into two types based on theare reflected or transmitted.observation method: transmission surveys and reflectionChannel waves are divided into Rayleigh-type channelsurveys. For transmission surveys, shots and receiverswaves and Love-type channel waves according to theirare located along both sides of the coal seam workingpolarization characteristics and physical composition.Manuscript received by the Editor March 31, 2013; revised manuscript received October 29, 2013.*This work is supported by the National Natural Science Foundation of China (No. 41 140033).1. Sate Key Laboratory of Petroleum Resource and Prospecting, China University of Petroleum, Beijing 102249, China2. School of Resource and Geosciences, China University of Mining And Techn中國煤化工a.3. State Key Laboratory of Marine Geology, Tongji University, Shanghai 200092C 2014 The Editorial Department of APPLIED GEOPHYSICS. All rights reserved.f YHCNMH G107Quality factors for Rayleigh channel wavesRayleigh-type channel waves are formed by P- and SV-medium using complex velocities, and investigate thewaves, while Love-type channel waves consist purelyeffect of imperfect elastic dissipation on the propagationof SH waves (Krey, 1963). In 1955, Evison generatedof Rayleigh-type channel waves. Numerical calculationsand recorded seam waves for the first time. Krey (1963)show that the attenuation of the Rayleigh-type channelderived the dispersion functions of Rayleigh-type .waves is dependent on frequency, which is mainlychannel waves and Love-type channel waves based onaffected by the thickness of the coal seam and its S-wave,a three-layer symmetrical model. He also used the ISS-which is affected by parameters of formation such astransmission record to verify the dispersion characteristicsporosity, fluid content and pressure. The estimated valueand study the amplitude-depth distributions of Love-typeof the quality factor (Qk) can be inverted to obtain theand Rayleigh-type channel waves. Rader et al. (1985)dirt bands, fracture, and thickness of the coal seam.calculated dispersion curves for Love-type channel wavesin multiply layered media using a fast and stable phaserecursion algorithm. Kerner and Dresen (1985) studiedPrinciples and methodsthe influence of dirt bands and faults on the dispersioncharacteristics of Love-type channel waves.These previous studies used the assumption of a perfect1. Attenuation of seismic waveselastic medium, without considering the attenuation ofThe displacement field of a plane harmonic wave thatchannel waves. Three factors have been found to betravels along the positive x-direction in a homogeneousresponsible for the amplitude decay of channel waves withperfect elastic medium is usually expressed as (Aki anddistance: the cylindrical spreading pattern, geometricalRichards, 1980; Minister, 1980).dispersion and imperfect elastic dissipation. To use theISS method for detecting a wide range of geologicalu(x,t)= A(x)exp[i(kx - 0ot)],(1)anomalies within coal seam working faces, it is necessaryto compensate for amplitude attenuation by recordingwhere u(x,t) is the displacement field, A(x) is the planedata. As using a geometrical spreading factor can easilywave amplitude that attenuates over x distance due to theachieve diffusion wavefront amplitude correction,absorption of the medium, k is the angle wave number,imperfect elastic dissipation is the focus of absorptionk= o/c.0 is the angular frequency and∞=2πf.research (Buchanan, 1978; Krey et al., 1982; Buchanan etAs the plane wave travels from x to x+△x, its .al, 1983; Liu et al, 1993; Li et al, 1995). Buchanan (1978)amplitude changes from A(x) to A(x+Ar). We can writederived the dispersion function of Love-type channelan expression thus:waves based on the study of Krey (1963) and computedthe dispersion curves and attenuation coefficient curvesA(x+Ar)- 4(x) =-aAr,(2)A(x)based on a three-layer symmetrical model. Krey et al.(1982) derived a simpler dispersion function for Love-where a is the absorption cofficient. It is defined by:type channel waves, computed a quality factor, andanalyzed the influence of absorption on the recording ofa = o(2Qc),(3)Love-type channel waves. Li et al. (1995) studied theinfluence of Qsi and Qs2 on the absorption of Love-typewhere c is the phase velocity, Q is the quality factor,channel waves and estimated their quality factors fromand Q' is proportional to the energy loss per cyclesynthetic seismograms using a spectral ratio method. Inwith respect to the energy stored in the oscillation in anthat study, the estimation of quality factors for Love-typeexperiment with harmonic loading (Aki and Richardswaves is consistent with the theoretical quality factor.1980; Minister 1980).After estimating the quality factor from synthetic data,As Or-→0, equation (2) becomes:they removed the effect of the attenuation for real data.However, there has been lttle research on the attenuationAA(x)__ a4(). o___ 0A(x)__ 0ln[4(x)]4)features of Rayleigh-type channel waves. To compensateaxA(x)0x2xfor the amplitude of Rayleigh-type channel waves, it isBy integrating from zero to x, equation (4) can beimportant to study their attenuation.In this paper, we build upon previous studies to proposeexpressed as:中國煤化工a simple method for computing the dispersion functionfor Rayleigh-type channel waves in an imperfect elasticMYHCNMH G(5)108Yang et al.The solution of equation (5) has the following expression:are assumed to be plane waves. The velocities of theprimary waves are denoted by Vp1 Vp2, and those of theax= ln A(x)|x=o-In A(x).(6)shear waves are expressed as Vs1, V's2 for rock and coal,Thenrespectively. The densities of rock and coal are denotedA(x)=Ane -x,(7by P1 and P2, respectively. The thickness of the coal seamis 2d, and μ1 and μ2 are the shear moduli for the rock andwhere Ao= A(x)|x=o is the energy of the source.for the coal seam, respectively.Equation (7) indicates that the amplitude of theseismic wave decays exponentially with increasingRockHP1 Vsu Vpldistance. Substituting equation (7) into equation (1), the :displacement of the plane wave can be expressed thuslyCoal從2Pz Vs Vp2(Liu et al, 1994):Rock從P1 V'snVplu(x,)= 40 exp{[(k + ia)x - on]}= Ao exp[(Kx- ot)], (8)Fig.1 The perfect elastic geological model for calculationwhere K =k + ia is the complex wave number.of the dispersion function.A coal seam with a low velocity is sandwiched between two infinite2. Computation of QRrock half spaces with higher velocities. The thickness of the coal seamis 2d, μ is the Lame constant, Vs and Vp are the velocities for S- and2.1 Dispersion equation for Rayleigh channel waves in aP-waves, respectively, and p is the density.perfect elastic mediumWe now consider the sequence of layers shown inIf the displacement field of the seismic waves and theFigurel, with the z- direction downward. In this simplestress field on the different medium interfaces(z= -d, zmodel, a coal seam is embedded between two elastic= d) satisfy the boundary conditions (Ewing et al, 1957),half spaces i.e., surrounding rock, thus neglecting thethe Rayleigh wave dispersion equation for the mediuminfluence of the earth's surface on wave propagationmodel is obtained:along the deep-lying coal seam. The surroundingdet(M)=0,(9rock and coal are assumed to be perfectly elastic, .homogeneous and isotropic. All interfering body waves M can be expanded as- iksh(azd) Brsh(B2d)- ikB0a2ch(a2d)| ikch(β2d) | -a1B2ch(a2d) i- A2sh(B2d)- BPvsv! APvv號(hào)M=|- D:sh(a2d)- C2sh(B2d) -PvD2 - Cρv唱v(10)! 2ikch(azd)- 2pzsh(B2d) - ik-β2a2sh(a2d)j 2iksh(B2d) | a-ik02Bzsh(a2d)- 2A2sh(Bzd)Bρv V3ApwVEv0 :2D,ch(a2d) 2C2ch(β2d)!- PvD2 Cpv3v」where a, =[k2 - (/yvy)1]'/, β,=[k2 -(0/*.)]”2D.=2k2 - (o/v32), D2 = 2(kvsv)2 - (@/v8s2)-.(11)with(l=1, 2, 3)The subscripts (I = 1,2,3) represent the upperk=o/c, @=2對", Pv=p|/P2, Vsv =vs/vs2，surrounding rock, the coal seam, and the lower surroundingrock, respectively. The phase velocity is denoted by c,A=k2+β}, A2=k2+陘,the symbols sh and ch are the hyperbolic sine and thehyperbolic cos中國煤化工of -1.B= 2ika, B2 = 2ika2,In equation心.nown variables,C = 2ikB, C2 = 2ikBz,f and C, as thelMYH.CNMHGwn, as shown in109Quality factors for Rayleigh channel wavesFigure 1. We can obtain the relationship between themiddle coal seam. From the above discussion, it can beunknown variables (f and c) and the dispersion curvesseen that, in comparison with a perfect elastic medium,from equation (9) by the enumeration method.the energy attenuation of seismic waves propagating inThe group velocity is usually computed by numericalan imperfect medium has a wave propagation velocitydifferentiation of the phase velocity dispersion curveequivalent to the complex velocity. Then, for an imperfectusing (Dersen and Riter, 1994)elastic medium, the complex P-wave velocity (VvpQ), thecomplex S-wave velocity (Vso), and the complex phaseU=c+k-(12)velocity for Rayleigh channel waves (CRO) are defined by:dl, =Vpnx(1- i&p),Using equation (12), we can obtain the correspondinggroup velocity from the phase velocity.'sne ='snx(1-i&gx)，2.2 Dispersion equation for Rayleigh channel waves in anVp20 =Vp2 x(1- i&p2), .(14)imperfect elastic mediumVvs2e =Vvs2x(1-ies2),As mentioned above, absorption of wave energy in anCRo=cx(1-i&x),imperfect elastic medium erax requires the introduction ofthe complex wave number to the harmonic propagationwithfactor exp[i(kx-ot)]. Combining equations (3) and (8),εp1 =1/(2Qp), Ep2 = 1/(2Q2p2),the complex wave number K can be changed into:εs1 = 1/(2Qs), Es2 = 1/(2Q3s2), .)_K=k+ia=k+i-2Qc(1-i/2Q)<(1+2εp =1/(2Qr).(15)In this, Vpl and Vs(l= 1, 2) are the P-wave velocity and(13)the S-wave velocity in a perfect elastic medium. Qp andQs(l= 1, 2) are the quality factors for P- and S-waves,where k is the wave number in the elastic medium, a isrespectively, and considered to be constant, and QRthe absorption coefficient, Q is the quality factor, c isdenotes the quality factor for the Rayleigh channel wave,the phase velocity in the elastic medium, and c' is thewhich is in general frequency-dependent. For thesecomplex phase velocity, defined by c' = c*(1-i/2Q). Then,variables, the subscriptsl= 1, 2 mean the surroundingthe absorption- dispersion equation for a homogeneousrock and the coal seam, respectively.isotropic structure with weakly imperfect elastic layersSubstituting the complex wave number equationcan be obtained from the dispersion equation (equation 9)(13) and the complex velocity equations (14 and 15)using the complex velocity and complex wave numberinto equation (9), we obtain the absorption- dispersion(Dersen and Riter, 1994).function of Rayleigh waves for a seam with a structureFigure 2 shows a symmetrical three-layered model ofof homogeneous isotropic, weakly imperfect elastican imperfect elastic medium for studying the absorptionlayers, as shown in Figure 2:-dispersion equation. In it, a coal seam is sandwicheddet(M')=0.(16)between two homogeneous half spaces, which are thesurrounding rocks with the same parameters as thewhere- iKsh(a2d)| β2sh(β2d)| - iKB0a2ch(azd) ! iK ch(B2d) - 01- iK)B2ch(azd) - A2sh(β2d)- BpvvEw! APvv5v .M'=- D:sh(a2d)j- C2sh(B2d) -PwD2 - CPvEN(17):2iKch(a2d)- 2β2sh(β2d) - iK2a 2sh(a2d) 2iKsh(B2d) a-iK2B2sh(a2d)-2/中國煤化工SN0 :2D,ch(a2d) 2CMYHCNMHGsV|110Yang et al.K =o/cRQ,∞=2πf,done in two steps. The parameters required are the P-and S-wave velocities, the densities and Q-factors for P-and S-waves in the coal seam and the surrounding rock,β=|K2-(°)and the thickness of the coal seam. The first step is toVPIQVsQsubstitute the parameters for P- and S-wave velocitiesand coal seam density and thickness into equation (9)PN = Pi/P2, Vsv = V'siq/vs2Q,to obtain the phase velocity of the Rayleigh channelA=K2+β, A2=K2+陘,waves and the dispersion curves. The second step isto substitute the phase velocities and Q-factors intoB:= 2iKa，B2 = 2iKa,equation (16), thus obtaining QR.q= 2iKB，C2 = 2iKB2,Qualitative analysis of QkD\ =2K2-(@/vs2o)", D2 = 2(Kvsv)2 -(@/vs20)2.(18)1. Generating Qr curves using model dataTo analyze the effect of coal seam thickness on QR,Rock從P V'g1 VpI Qp Qs1we generate QR curves using the three-layer modelsshown in Figures 1 and 2, using model parameters of:Coal hP2 Vx Vp QnQxVpl = 28008 m/s, VsI =1800 m/s,p1= 2.6 g/cm', Qp1 =375, Qs1= 150; V'pz= 1710 m/s, Vs2 = 900 m/s,P2= 1.3Rock HP Vsl Vp Qn Qsg/cm', Qs2 = 50, and Qr2 = 120 (Anderson et al, 1965;Fig.2 Geological model for calculation of the absorption-Liu et al, 1994). The thickness of the coal seam is 3 m.Solving equation (9), we obtain the dispersion curvesdispersion equation.A coal seam with a lower velocity is embedded between two infinite rockshown in Figure 3a. Solving equation (16) gives thehalf spaces with higher velocities. The thickness of the coal seam is 2d,Qk curves shown in Figure 3b. The dispersion functionμ is the Lame constant, Vs and Vp are the velocities for S- and P-waves,(equation 16) is a complex function. The real part ofrespectively, p is the density, and Qs and Qr are the quality factors for S-the dispersion function gives the relationship betweenand P-waves, respectively.the phase velocity and the frequency, which is equal tothe dispersion function for Rayleigh channel waves in a3. Calculation of QRperfect elastic medium (equation 9). The imaginary partIf the values of the parameters of the geological modelis a key function, which gives the relationship betweenin Figure 2 are known, then the calculation of Qr can beQr and frequency.間1800、12 mode. Phase velocity16Group velocity |1600120-冒140010012008(10006C806002000200400Frequency (Hz)Fig.3 Comparative analysis of the theoretical dispersion curves (a) and the Qr curves (b) of Rayleigh-typechannel waves.(a) Dispersion curves for the geological model in Figure 1. (b) Lr curves for the ge中國煤化ifrentcolored lines represent the dferent modes, i.e. the fundamental, first, and second mTYHCNMHGQuality factors for Rayleigh channel wavesFigure 3a shows the dispersion curves of the Rayleighphase appears at the end of the Rayleigh-type channelchannel waves based on the three-layer model forwave array. This is the dominant seismic phase formodes zero, one and two. The different mode dispersionchannel waves with dispersion. Therefore, the Qr for thecurves correspond to different cut-off frequencies. TheAiry phase of the Rayleigh channel waves is significantcut-off-frequency defining the low-frequency limit offor analysis of the Qk values for different layer structures.the dispersion curves corresponds to the highest phaseTo understand the relationship between the geologicalvelocity of the dispersion curve. The cut-off frequencyparameters of coal seams and QR, we studied the Qkof the fundamental mode is close to zero (see the zero-curves of a symmetrical three-layer model. We used amode curve).varying coal seam thickness 2d, withd= 1.5 m, 2.0 m,The cut-off frequency for a higher mode is higherand 2.5 m, a P-wave quality factor Qp = 375, 365, 355than for a lower mode. The phase velocity is always inin the adjacent rocks and Qr2= 120, 110, 100 in the coalthe range of the S-wave velocities of the surroundingseam, and an S-wave quality factor Qs1= 150, 140, 130 inrock and the coal seam. For each phase curve mode, thethe adjacent rocks and Qs2 = 50, 40, 30 in the coal seam.phase velocity CR. _.Vs1 as the frequency f approaches theIn this study we focus only on the Qk curve for the firstcut-off frequency. The phase velocity decreases as themode Rayleigh wave. Figure 4 shows the Qk curves forfrequency increases, and finally CR- _Vs2. In general, thethe one mode with different coal thicknesses, from whichgroup velocity is always lower than the phase velocitywe can see that the Qk curves change as the coal seamfor each mode. When the frequency is close to the cut-thickness changes. At low frequency, the thinner the coaloff frequency or∞, the group velocity approaches theseam thickness is, the higher Qk is; at high frequency, Qkphase velocity. In each mode of group velocity curves,is smaller for smaller coal seam thickness. For the variousthere is a velocity minimum (indicated by red points incoal seams, the frequency at which the minimum pointFigure 3a), which corresponds to an abnormal seismicof the Qr curve appears is also the Airy phase frequency.phase in the channel wave array. It occurs at the end ofThe Airy phase frequency is smaller for larger coal seamthe channel wave array in the form of a high amplitude.thicknesses, and its value drops suddenly as the coal seamThis special seismic phase is called an Airy phase. Thethickness increases. Although the Qr curves are differentfrequency at which the minimum velocity occurs isfor different coal seam thicknesses, the minimum value ofcalled the Airy phase frequency. The group velocitythe Qk curves is almost equal for all the curves.changes with frequency. At first, the group velocitydecreases and achieves a velocity minimum (shown by160 [- 2d=3m]arrows in Figure 3a) when the frequency increases to2d=4 m|the Airy-phase frequency from the cut-off frequency.12_2d= 5m |Subsequently, the group velocity increases as the。100frequency increases, and finally approaches Vs2. Figure803b displays the Qr curve for modes zero, one and two.From Figures 3a and 3b, we know that the shape of the60Qr curve is consistent with the group velocity curve40(the green dashed line in Figure 3a), both of which are20dependent on frequency. Each mode of the Qr curve40000corresponds to its own cut-off frequency. Qk is alwaysFrequency (Hz)lower than QS1 (the Q-factor for S-waves in the adjacentFig.4 Effect of coal seam thickness on QR.rock) and the Qr minimum occurs at the Airy phaseGeological parameters: Qpi = 375, Qsi= 150; Q3z=50, Qr= 12(咖p= 28008 m/s, Vsi= 1800 m/s, P1 =2.6 glcm?, Vp2= 1710 m/s,frequency. For frequency values lower than the AiryVs = 900 m/s, P2= 1.3 g/cm?.phase frequency, Qr decreases to the Q-factor minimumas the frequency increase. For frequency values largerFigure 5 shows how Qk varies with QpI, Qr2, QsI, andthan the Airy phase frequency, QR is increases to beQs2. first- mode QR curves for different Qp1 are plotted inequal to Qs2 as the frequency increases.Figure 5a. The three curves overlap, there is almost nodifference between Qr for different Qri, meaning that2. Relationships between the geologicalQr is not affected by Qpr. The first-mode Qr curves withdiferent values中國煤化工in Figures 5bparameters of coal and Qrand 5c. From b一MChat changes inAnalysis of Figures 3a and 3b shows that the AiryQr2 and Qs1 onlMYHCNMH(! H_-ncies, and the112Yang et al.Qr value reduces with decreasing Qr2 or Qs1. Figure 5dcontrolled by Qs2.shows how the Qk curve changes with different values ofAnalyzing Figures 4 and 5, we can see that the Airycoal seam S-wave Qs2. From comparative analysis of thephase frequency (the Qr minimum) is closely related tofour panels in Figure 5 we found that Qr is influencedthe coal seam thickness (2d). Therefore, this relationshipgreatly by Qs2, but the effects of the other three qualitycan be used to predict the coal seam thickness. Moreover,factors (Qsi, Qr2, and Qr2) can be disregarded, because ofthe QR of the Airy phase frequency also is closely relatedtheir small effects. Thus, the Qk minimum is determinedto the S-wave quality factor in the coal seam (Qs2). So,mainly by Qs2. In other words, for an imperfect elasticfor a given model, the coal lithology can be predicted bymedium, the quality factor for the Airy phase isinversion of Qs2 using the Airy phase QR.(a)(b60「160Qp2= 120價(jià)=3652n2=110 |40「Q。=35514002=100120并100sf 1008030 t6C4Co一2000600200400.1000Frequency (Hz)d)[c)s0 r2，-1502s2= 50 |2s=14040 t2二4l23=20 t100 I占80-8C20 Fo200 400800 1000400 600Fig.5 The effects of rock and coal quality factors on QR.(a) Quality factors Qs1 = 150, Qs2= 50, Qp = 120 held constant, Qr1 varies from 355 to 375 in increments of 10. (b)Quality factors Qr = 375, Qst = 150, Q32 = 50 held constant, Qp varies from 100 to 120 in increments of 10. (C) Qualityfactors Qpi = 375, Qs2 = 50, Qr2 = 120 held constant, Qsi varies from 130 to 150 in increments of 10. (d) Quality factorsQr = 375, QsI = 150, Qrz = 120 held constant, Qs2 varies from 30 to 50 in increments of 10.Conclusionsvariable in turn. We found that the trend of the Qrcurves is consistent with those of group velocity againstfrequency. Qs2 and coal seam thickness both stronglyBased on the theory of attenuation of seismic wavesinfluence QR, and the quality factors Qp1, QsI, and Qpin an imperfect elastic medium we have proposed ahave a lesser effect on Qk. Qk increases as Qs2 increases.method to calculate the dispersion function for Rayleigh-The Airy phase frequency, which is the minimum Qrtype channel waves using complex velocities. Wevalue, decrea中國煤化工Seam hicknesalso studied the effect of coal seam thickness on theIn this paper,channel wavequality factors QrI, Qs1, Qr2, and Qs2 by varying eachquality factorsMHCNMHGthreelayersof113Quality factors for Rayleigh channel wavessymmetrical horizontally isotropic media. However, realKerner, C, and Dresen, L, 1985, The influence of dirtrock formations are more complex than the idealizedbands and faults on the propagation of Love seam waves:ones considered herein; this will be addressed in futureJournal of Geophysics, 57, 77- 89.research.Li, X. P., Schott, W, and Riter, H, 1995, Frequency-dependent Q-estimate of Love-type channel wavesand the application of Q-correction to seismograms:ReferencesGeophysics, 60(6), 1773 - 1789.Liu, T. F, Cheng, J. L, Pan, D. M., and Li, D. C., 1993,Attenuation of channel wave: Journal of China CoalAki, K., and Richards, P. G.，1980, QuantitiativeSociety, 18(5), 83 - 86.seismology: Theory and methods: W. H. Freeman andLiu, T. F, Pan, D. 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