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Wave Motion 44 (2006) Comment to qs-waves in a vicinit of the ais of smmetr of homogeneous transverse isotroic media, b M. Poov, G.F. Passos, and M.A. Boteho [Wave

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Wave Motion 44 (2006) Comment to qs-waves in a vicinit of the ais of smmetr of homogeneous transverse isotroic media, b M. Poov, G.F. Passos, and M.A. Boteho [Wave Motion 42 (2005) 9 20] Vácav Vavrču Geohsica Institute, Academ of Sciences of the Cech Reubic, Boční II, 4 3 Praha 4, Cech Reubic Received 4 Ju 2006; received in revised form 3 August 2006; acceted 3 August 2006 Avaiabe onine 9 October 2006 Abstract A new asmtotic formua for S waves roagating near the smmetr ais in transverse isotroic eastic media derived b Poov et a. [Wave Motion 42 (2005) 9 20] is discussed and commented. It is shown that the formua is a modification of the revious ubished formua b Vavrču [Geohs. J. Int. 38 (999) ]. The formua of Poov et a. is ess accurate and vaid under more restrictive conditions than that of Vavrču. Ó 2006 Esevier B.V. A rights reserved. A high-frequenc modeing of wavefieds in transverse isotroic (TI) media is more comicated in directions near the smmetr ais than in other directions, because we observe a so-caed S-wave iss singuarit aong the ais. The S-wave singuarit causes anomaies in the fied of oariation vectors (see Fig. ) and ossib aso in the shae of the sowness and wave surfaces. For these reasons, the standard eroth-order ra theor is inaicabe and more sohisticated methods must be used to mode the wavefied correct. For eame, Vavrču [] roosed to a higher-order ra theor, and Gridin [2] deveoed a high-frequenc asmtotics using a uniform stationar hase method. Other aroaches vaid for iss singuarities in an te of anisotro are resented in [3,4]. Recent, this toic was aso studied b Poov et a. [5], where the authors caim that the revious resuts resented in [,2] are either questionabe or incorrect, and the roose another high-frequenc asmtotics vaid for directions near the singuarit. In m comment, I wi show that the assertion of Poov et a. [5] regarding the revious ubished resuts is not justified and that the resuts in [5] are not in contradiction with the revious ubished resuts. Moreover, I wi show that the formuas in [5] are vaid under stronger restrictions than those derived in [,2]. E-mai address: /$ - see front matter Ó 2006 Esevier B.V. A rights reserved. doi:0.06/j.wavemoti V. Vavrču / Wave Motion 44 (2006) wave wave Fig.. Horionta rojections of oariation vectors of the and waves near the iss singuarit.. Eact soution The eact eastodnamic Green tensor G ij (,t) for unbounded, homogeneous, anisotroic, eastic media can be eressed as foows (Burridge [6], Eq. 4.6; Wang and Achenbach [7], Eq. 3): G ð; tþ ¼ HðtÞ X 3 Z g m gm d 8 2 q _ t N dsðnþ: ðþ m¼ SðNÞ ðc m Þ 3 c m Suerscrit m =,2,3 denotes the te of wave (P, S ands2), is the osition vector of an observation oint, t is time, g = g(n) denotes the unit oariation vector, c = c(n) is the hase veocit, q is the densit of the medium, H(t) is the Heaviside ste function, dðtþ _ is the time derivative of the Dirac deta function, and N is the direction of the sowness vector. The integration is over unit shere S(N). Formua () reresents the eact soution for homogeneous, wea as we as strong anisotroic media, containing far-fied as we as near-fied waves, and is vaid at a distances and directions incuding the shear-wave singuarities. For an isotroic medium, integra () can be evauated anatica to ied the we-nown Stoes soution (see Mura [8],. 6 63). For a transverse isotroic medium, integra () has aso been evauated anatica, but on for the smmetr ais direction [9,0]. For other directions, the integra has to be either evauated numerica, or eanded asmtotica. 2. Asmtotic soution of Vavrču [] Since the oariation fied is singuar aong the smmetr ais in TI, the standard asmtotics of (), which is equivaent to the eroth-order ra theor, fais aong the ais and in its vicinit. Therefore, Vavrču [,] roosed to a a more genera aroach and derived an asmtotic formua using higher-order ra theor. The formua consists of the eroth- and the first-order terms of the ra series and it is eressed as foows (Vavrču [], Eqs. 2 and 5): G S far ð; tþ ¼ m þ ffiffiffiffiffiffiffiffi K ffiffiffiffiffiffi g r g g dðt s Þþ? r 2 sin 2 #? m ffiffiffiffiffiffiffiffi K r ½Hðt s Þ Hðt s ÞŠ; dðt s Þ ð2þ where is the osition vector, t is time, q is the densit of the medium, r is the distance from the source to the receiver, g is the oariation vector, 30 V. Vavrču / Wave Motion 44 (2006) sin u cos u cos / cos u 6 7 ¼ 4 cos u 5;? 6 7 ¼ 4 sin u 5; g 6 7 ¼ 4 cos / sin u 5; ð3þ 0 0 sin / a ij are the densit-normaied eastic arameters in the Voigt notation, v is the grou veocit, s = r/v is the travetime, K = K() is the Gaussian curvature of the sowness surface, = N is the sowness vector, N is the sowness direction, =/c is sowness, and c is the hase veocit. The suerscrits in (2) denote the te of wave. The Gaussian curvature K can be cacuated in TI media as foows: K ¼ sin # d# 3 m sin h dh ; ð4þ where anges # and h define the incination of ra vector n and of sowness direction N from the vertica ais (see Fig. 2) sin # cos u 6 n ¼ 4 sin # sin u cos # 7 5; N ¼ 6 4 sin h cos u 7 sin h sin u 5: ð5þ cos h Ange / is the oar ange of the ra defined in the 2 ane. The -wave quantities in (2) can be eressed eicit in a cosed form as a function of ra vector n [2] c ¼ sin2 # þ cos2 # 2 sin 2 # þ cos2 # a 66 a 2 66 a 2 44 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s sin 2 # ¼r þ cos2 # ; K ¼ a 2 66 a 66 2 ; m ¼ sin2 # þ cos2 # 2 a 66 sin 2 # a 66 þ cos2 # 2 : ð6þ However, determining the -wave quantities is more invoved. First, we have to cacuate the sowness vector of the wave for a given ra vector n. This can be done anatica b soving a sstem of two agebraic equations of the fourth-order in two unnowns (see Musgrave [3], Eq ; Vavrču [4], Eq. 5.6), or numerica b iterations. Then, the needed quantities in (2) can be cacuated as a function of or N. In this wa, we obtain for the hase veocit c, h c ¼ ða 2 þ Þ sin 2 h þða 33 þ Þ cos 2 h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i A þ B cos 2 h þ C cos 4 h ; ð7þ where Sowness surface Z Wave surface Z θ ϑ n ϑ v θ N Fig. 2. Definitions of basic quantities. Vector is the sowness vector, N is the direction of the sowness vector, v is the grou veocit vector, and n is the ra vector. V. Vavrču / Wave Motion 44 (2006) A ¼ða Þ 2 ; B ¼ 4ða 3 þ Þ 2 2ða Þða þ a 33 2 Þ; C ¼ 4ða 3 þ Þ 2 þða þ a 33 Þ 2 4 ða þ a 33 Þ; and for the ange / defining the incination of oariation vector g from the horionta ane, qffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2/ ¼ 2D ða 3 þ Þ 3 2 þ 2 2; ð9þ D ¼ 2 ða þ Þð 2 þ 2 2 Þ ða 33 þ Þ 2 3 : The comonents of the grou veocit m read (see Musgrave [3], Eqs ; Červený et a. [5], Eqs ) m ¼ D fa þ 2a ð 2 þ 2 2 Þþ½ða 3 þ Þ 2 a a 33 a 2 44 Š2 3 g; m 2 ¼ D 2 fa þ 2a ð 2 þ 2 2 Þþ½ða 3 þ Þ 2 a a 33 a 2 44 Š2 3 g; m 3 ¼ D 3 fa 33 þ 2a þ½ða 3 þ Þ 2 a a 33 a 2 44 Šð2 þ 2 2 Þg: The most comicated quantit is the Gaussian curvature K, which is eressed using Eq. (4). Since the deendence between the ra direction n and the sowness direction N is comicated, the quantities g, m, K and s standing in (2) cannot be eressed eicit in terms of ra vector n. Instead, the shoud be evauated numerica or using aroimate formuas. 3. Phsica interretation of the soution Formua (2) consists of three terms. First two terms are the eroth-order terms of the ra series and describe the standard far-fied asmtotics of the and waves with conve sowness sheets at a given ra direction [6]. The third term is the first-order term of the ra series caed the couing term or the near-singuarit term. The couing term describes the interaction between the an waves and it is significant near the smmetr ais [6]. For the direction aong the ais, the amitude of the couing term diverges and its time duration goes to ero. Hence, the waveform of the couing term becomes the Dirac deta function simiar as the standard far-fied term (see [], Fig. 2). Formua (2) is vaid for a directions of ras, assuming that the wavefront has no triications and the receivers are far from the source. The essentia same formua has been obtained b Gridin [2], who derived the asmtotic Green function not on for directions near the smmetr ais but aso for other difficut situations such as the directions near cusida edges at the wavefront due to triications. The vaidit of (2) was furthermore confirmed direct b a numerica comarison of () and (2) assuming a smooth source-time function of a oint source situated in two different anisotroic homogeneous media (a craced medium and sandstone, see []). 4. Taor eansion near the smmetr ais Assuming directions near the smmetr ais, #, or eact aong the ais, # = 0, the wave quantities needed in Eq. (2) can further be simified. Aing the Taor eansion to (6 0) and negecting the thirdand higher-order terms in #, we obtain for the hase and grou veocities and for the travetime of the wave, c ¼ ffiffiffiffiffiffi þ a 44 sin 2 # 2 a 66 a 66 s ¼ r ffiffiffiffiffiffi 2 a 44 sin 2 # ; a 66 and subsequent of the wave, ; m ¼ ffiffiffiffiffiffi þ 2 a 66 sin 2 # ; ð8þ ð0þ ðþ 32 V. Vavrču / Wave Motion 44 (2006) c ¼ ffiffiffiffiffiffi E sin 2 # ; m ¼ ffiffiffiffiffiffi a 2E 2 44 E 2E sin2 # ; s ¼ r ffiffiffiffiffiffi þ E 2E sin2 # ; ð2þ where! E ¼ a ða 3 þ Þ 2 : ð3þ a 33 Quantit E must be ositive, otherwise transverse isotro disas an aia triication and the sowness surface is not conve aong and near the smmetr ais [3,7]. The directions of the sowness and oariation vectors of the wave are cacuated as sin h ¼ E sin #; sin / ¼ a 3 þ E sin #: ð4þ a 33 For the direction strict aong the smmetr ais, # = 0, the needed quantities in (2) can be evauated eact having the foowing form: / ¼ 0; ¼ ¼ ffiffiffiffiffiffi ; c ¼ c ¼ m ¼ m ¼ ffiffiffiffiffiffi ; s ¼ s ¼ r ffiffiffiffiffiffi ; K ¼ a2 66 ; K ¼ a ða 3 þ Þ 2 a 33! 2 ; ð5þ and the far-fied S-wave Green function reads [] 8 # 9 G S far ð; tþ ¼G S far 22 ð; tþ ¼ þ a ða 3 þ Þ 2 = 8q : a 66 a 33 ; jj d t jj ffiffiffiffiffiffi ð6þ the other comonents of the Green function being ero. This formua coincides with the far-fied aroimation of the eact Green function aong the smmetr ais ubished b Paton [9,0]. 5. Asmtotic soution of Poov et a. [5] First, et us comete the formua of Poov et a. [5] and transform it into the time domain to eress the Green function in a form comarabe with (2). Eressing Eqs. (2) and (3) of [5] vaid for the force aong the ais aso for forces aong the other aes, summing the contributions of the and waves, and transforming the equations into the time domain, we obtain the Green function in the foowing form: G S far ð; tþ ¼ þ ^g C r ffiffiffiffiffiffi ^g g dðt ^s Þþ? r 2 sin 2 #? C r dðt ^s Þ ½Hðt ^s Þ Hðt ^s ÞŠ; where 2 3 cos u ^g 6 7 ¼ 4 sin u 5; ð8þ 0 ! # ^s ¼ r ffiffiffiffiffiffi 2 þ ða 33 Þ ða 3 þ Þ 2 a ða 33 Þ sin 2 # ; ð9þ ^s ¼ r ffiffiffiffiffiffi 2 a 44 sin 2 # ; a 66 ð20þ ð7þ C ¼ a ða 3 þ Þ 2 ; a 33 ð2þ C ¼ a 66 ; ð22þ and the other quantities have the same meaning as in (2). Comaring formuas (2) and (7) we find that the are not ver different. The on difference is that some quantities such as the travetimes and the amitude factors of the and waves, and the oariation vector of the wave have sight different forms. A further comarison of Eqs. (8 22) with Eqs. ( 5) figures out that quantities ^s, ^s standing in (7) are identica with the Taor eansions of s, s eressed in () and (2). The quantities ^g, C and C standing in (7) are just vaues of g, m ffiffiffiffiffiffiffiffi K and m ffiffiffiffiffiffiffiffi K in (2) taen aong the smmetr ais, i.e., for # =0. This imies that Eq. (7) is not a soution inconsistent with Eq. (2), but it is rather its aroimation. Whie Eq. (2) is vaid for a ras, Eq. (7) is vaid just for ras near the smmetr ais. 6. Numerica eame V. Vavrču / Wave Motion 44 (2006) Here, I demonstrate a different range of vaidit of Eqs. (2) and (7) numerica. I use two transverse isotroic modes: first, the Mesaverde immature sandstone [8] with the densit-normaied eastic arameters (in m 2 /s 2 ): a = a 22 = 22.36, a 33 = 8.9, a 3 = 8.49, = a 55 = 6.6, a 66 = 8.00 and q = 2.46 g cm 3, and second, a theoretica mode of anisotro caused b thin aers (Batie et a. [9], mode PTL3) with the densit-normaied eastic arameters (in m 2 /s 2 ): a = a 22 = 0.99, a 33 = 6.68, a 3 = 2.56, = a 55 = 2.7, a 66 = 3.54 and q = 2.60 g cm 3. The Mesaverde immature sandstone disas anisotro of 8.3, 4.6 and 9.5% for the P, and waves; the mode of the aered medium disas anisotro of 24.8, 5.2 and 24.5% for the P, and waves. The wavefied is generated b a singe oint force f = (,,0) T. The source-time function is a one-sided use defined b Eq. (0) of []. The wavefied is cacuated b Eqs. (2) and (7) and is comared with the eact soution (). Fig. 3 shows waves roagating in the sandstone and recorded at two observation oints that ie in the 3 ane at distances from the source of 25.7 m (a) and 257. m (b). The distance corresonds to 0 and 00 waveengths of the S waves roagating in the direction of the smmetr ais, resective. The ras from the source to observation oints deviate from the smmetr ais b anges 0 and 30, resective. The waveforms at the observation oints are cacuated using a numerica integration of the eact soution () and using aroimate soution (2) roosed b Vavrču []. The comarison of both aroaches reveas that the coincidence of the S waveforms at arge distance is erfect and within the width of the ine (see Fig. 3b). However, visibe differences are detected at the shorter distance (see Fig. 3a). Since the P and S waves are not fu searated in this case, we comare the comete waveforms. The eact waveform is cacuated using Eq. (), and the far-fied aroimation is the sum of the far-fied S wave (2) and of the standard far-fied P wave [6,2]. The discreanc between the eact and aroimate soutions at the shorter distance is eected and aears main at times between the P and S waves. At these times, the near-fied waves are ronounced in the eact soution but negected b the far-fied aroimation. Obvious, as the source-receiver distance decreases, the discreanc between the eact and aroimate soutions becomes more rominent. Fig. 4 shows waveforms of the S waves roagating in the sandstone and in the aered medium. The sourcereceiver geometr is simiar to that in the revious eame. The distance of the observation oints from the source is 257. m for the sandstone and 47.2 m for the aered medium and corresonds to 00 waveengths of the S waves roagating aong the smmetr ais. The waveforms are cacuated using a three different aroaches: first, using a numerica integration of the eact soution (); second, using Eq. (2) roosed b Vavrču []; and third, using Eq. (7) derived from equations ubished b Poov et a. [5]. A comarison of the waveforms reveas that the aroimate waveforms cacuated using Eq. (2) coincide within the width of the ine with the eact soution (). The waveforms cacuated using Eq. (7) aroimate the eact soution we for ras deviating b 0. However, we detect remarabe differences in the waveforms for the ra deviating b 30. First, the aroimation (7) erroneous redicts no signa at the 3 -comonent; 34 V. Vavrču / Wave Motion 44 (2006) a o ϑ = 0 ϑ = 30 o P P b ϑ = 0 o ϑ = 30 o Fig. 3. The waves roagating in the Mesaverde immature sandstone. The source-receiver distance is 0 (a) and 00 (b) waveengths of the S wave roagating aong the smmetr ais. Soid ine waveforms cacuated using the eact formua (), dashed ine waveforms cacuated using the aroimate formua (2) roosed b Vavrču []. Ange # defines the deviation of a ra from the smmetr ais. For arameters of the source and of the medium, see the tet. second, the travetimes of the and waves are shifted; and third, the amitudes of the and waves are distorted at the horionta comonents. These differences increase with increasing strength of anisotro and with increasing deviation between the ra and the smmetr ais. Obvious, the ower accurac of Eq. (7) comared with Eq. (2) is roduced b using Taor eansions of oariation vectors, travetimes and amitudes instead of using the eact quantities. 7. Concusions The assertion of Poov et a. [5] that the asmtotic formuas resented b Vavrču [] and Gridin [2] are either questionabe or erroneous is unjustified and miseading. The soution ubished in [5] is not in contra- V. Vavrču / Wave Motion 44 (2006) a ϑ = 0 o ϑ = 30 o b o ϑ = ϑ = 30 o Fig. 4. Waveforms of S waves roagating in the Mesaverde immature sandstone (a) and in the aered medium (b). Soid ine waveforms cacuated using the eact formua () and using the aroimate formua (2) roosed b Vavrču []. Dashed ine waveforms cacuated using the aroimate formua (7) roosed b Poov et a. [5]. Ange # defines the deviation of a ra from the smmetr ais. For arameters of the source and for distances of observation oints from the source, see the tet. diction with the revious ubished resuts. The soution is just a modification of the formua resented in []. The modification consists in substituting some quantities standing in the formua of Vavrču [] b their Taor eansions vaid for directions near the smmetr ais. Some quantities are substituted direct b the vaues aong the smmetr ais. These substitutions are justified for sma deviations of a ra from the smmetr ais and are advantageous in cases, when we refer to avoid cacuating such quantities as the -wave travetime or the Gaussian curvature of the sowness surface. These quantities stand in the revious ubished formuas and evauating them is more invoved than evauating the other quantities needed. On the other hand, to aroimate formuas for the -wave travetime or the -wave Gaussian curvature as done b Poov et a. [5] brings no advantage, because these quantities can be cacuated using sime eact formuas. 36 V. Vavrču / Wave Motion 44 (2006) Acnowedgement The wor was suorted b the Grant Agenc of the Academ of Sciences of the Cech Reubic, Grant No. IAA References [] V. Vavrču, Proerties of S waves near a iss singuarit: a comarison of eact and ra soutions, Geohs. J. Int. 38 (999) [2] D. Gridin, Far-fied asmtotics of the Green s tensor for a transverse isotroic soid, Proc. R. Soc. Lond. A 456 (2000) [3] V.A. Boroviov, D. Gridin, Kiss singuarities of Green s functions of non-strict herboic equations, Proc. R. Soc. Lond. A 457 (200) [4] V. Vavrču, Asmtotic eastodnamic Green function in the iss singuarit in homogeneous anisotroic soids, Stud. Geohs. Geod. 46 (2002) [5] M. Poov, G.F. Passos, M.A. Boteho, qs-waves in a vicinit of the ais of smmetr of homogeneous transverse isotroic media, Wave Motion 42 (2005) 9 20, doi:0.06/j.wavemoti [6] R. Burridge, The singuarit on the ane ids of the wave surface of eastic media with cubic smmetr, Q.J. Mech. a. Math. 20 (967) [7] C.-Y. Wang, J.D. Achenbach, Eastodnamic fundamenta soutions for anisotroic soids, Geohs. J. Int. 8 (994) [8] T. Mura, Micromechanics of Defects in Soids, Kuwer Academic Pubishers, London, 99. [9] R.G. Paton, Smmetr-ais eastic waves for transverse isotroic media, Q. A. Math. 35 (977) [0] R.G. Paton, Eastic Wave Proagation in Transverse Isotroic Media, Martinus Nijhoff Pubishers, The Hague, 983. [] V. Vavrču, Aicabiit of higher-order ra theor for S wave roagation in inhomogeneous wea anisotroic eastic media, J. Geohs. Res. 04 (999) [2] V. Vavrču, K. Yomogida, -wave Green tensor for homogeneous transverse isotroic media b higher-order aroimations in asmtotic ra theor, Wave Motion 23 (996) [3] M.J.P. Musgrave, Crsta Acoustics, Hoden-Da, London, 970. [4] V. Vavrču, Cacuation of the sowness vector from the ra vector in anisotroic media, Proc. R. Soc. Lond. A 462 (2006) , doi:0.098/rsa [5] V. Červený, I.A. Mootov, I. Pšenčí, Ra Metod in Seismoog, Chares Universit Press, Praha, 977. [6] V. Vavrču, Eact eastodnamic Green functions for sime tes of anisotro derived from higher-order ra theor, Stud. Geohs. Geod. 45 (200) [7] V. Vavrču, Generation of triications in transverse isotroic media, Phs. Rev. B 68 (2003), doi:0.03/phsrevb , art.no [8] L. Thomsen, Wea eastic ani

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