Emura Kinya

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Emura, Kinya

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Effects of Transition Zone and Low VelocityLayer on Frequency Spectra of Plane SH Waves著者 Emura  Kinya雑誌名 Science reports of the Tohoku University. Ser.5, Geophysics巻 16号 3ページ 164-172発行年 1965-03URL http://hdl.handle.net/10097/44664Effects of Transition Zone and Low  Velocity Layer on       Frequency Spectra of Plane SH  W  ayes            By KINYA EMURA Geophysical Institute,  TOboku University, Sendai, Japan              (Received March 8, 1965) Abstract     Transition zone, which is sometimes assumed to be at the Mohorovicic discon-tinuity, and  low velocity layer within the upper mantle are simulated by two velocity functions, of which the derivatives are continuous and finite. Frequency spectra of reflected and transmitted plane SH waves from each transition zone are theoretically studied.    The Moho transition zone of more than 0.2 km in its effective thickness has a stronger effect on the frequency content in near-vertical reflections than the absorption of energy in the crustal layer with Q>250. The low velocity layer amplifies reflected waves, although the effect may be too small to be recognized at the frequencies above 0.1 cps. Frequency  dependence of transmitted waves for both types of the transi-tion zones considered is almost negligible. 1 Introduction    The velocity of seismic waves just below the Mohorovicic discontinuity has been studied extensively in explosion seismology, and known to be considerably uniform in the world. Results of explosion studies, however, are not so accurate to solve the problem whether or not there exists a transition at the Moho.    Some geochemists suggest hat the Moho is a zone of gradual transition based on the theory of phase transition (e.g., LOVERING, 1958; MACDONALD and NESS, 1960). If this is the actual case, the Moho should affect the frequency contents in reflected, refracted and transmitted waves.  SATO (1957) studied reflection and transmission of plane SH waves at the transition layer, in which the velocity increases exponentially with depth, for various angles of incidence, and his conclusion is that the layer plays a role of low pass filter for the reflected waves. NAKAMURA and  HOWELL, Jr. (1964) made the Fourier analysis of refraction arrivals obtained in the Maine explosion experiment for the frequency range of 2.6 to 25 cps. They showed that the assumption of linear velocity transition at the Moho gave the maximum possible thickness of  1/2 km for the transition layer.    In thepresent study, a transition zone at the Moho is assumed to be simulated by a downward increase in velocity, of which the derivative is continuous and finite throughout he medium. The low velocity  layer, which may possibly be in the upper mantle is represented by a transition zone including a low velocity region. The effects of each transition zone on the frequency spectra of reflected and transmitted plane SHEFFECTS OF TRANSITION ZONE AND LOW VELOCITY LAYERinfinite medium where the density is constant and the shear wave velocity varies continuously along one axis, say z-axis (Fig. 1). Since the displacement v of plane SH waves is assumed to be independent of  y-axis, the equation of motion for v is given  by  a2 a2 aa v   P  a t2 - = (z)  a,c2  + azr-(z) az (1) where p is the density, t the time and  it the rigidity.    If we put the displacement of simple harmonic plane  Z  (z)                                  v —   -  e'(r,'t-kx)                          1/4u (z) Z(z) is the solution of the equation     d2kId)2                     "2 (z) k2  d21                           4,1G2^ dz2 It where  ks,---olvs,  w being the angular frequency and v,  th( the case where the velocity gradient in the medium is  s• frequency  f=o)/27r is taken into account in the present stuc bracket are negligible compared with the first two terms.165waves are theoretically studied, comparing with the effect of the absorption energy due to an internal friction within the crustal layer. 2 Velocity Distribution and Reflection and Transmission  Coefficients    The rectangular coordinate system is taken in an i inite edium ere e nsity  nstant d e   xr_tA•t• (z)Q - CONST.waves as z Fig. 1.(2)of         2Z-               +L.kc2 (z) 24,4  ^   zdit)221d2                                  pdz2 Z  = 0  (3)   z21/4 =olvs,  co e shear wave velocity. As ufficiently smaller than the co/ n dy, the last two terms in the     We now assume that the refraction index n(z) in the medium is represented by  n2 (z) =  1N emz —  4  M (4)                            1+e(1 + e'n-)2 where m is a constant, and                       M = 1  —  n2 (0) —  Nf2  N  =  1  —  n2  (+  oc) Any higher derivative of n(z) is finite. The incident wave is assumed  for be originated in the homogeneous region far from the transition zone. Considering the reflected and transmitted waves in the far regions, the moduli of the reflection and transmission coefficients, i.e.  I R  I and  I  T  I respectively, are given as follows, by use of the same procedure of analytic continuation of hypergeometric function as was adopted by BREKHOVSKIKH (1960).              I R1=cos27rd•ch2nsA + sin2nd • sh2 nsA                                             (5)           cos2 7rd • ch2n-sBsin2 nd • sB 166  K.  EMURA         (11 N2) cos doIsh (7-rs cosdo)  •  sh (Its-1/cos'do N) T/cos2/V (6)                790 — cos2  Itd  •  eh'  7tsB +  sin2  7rd  •sh2zsB where  do is the angle of incidence, and  2  A  =  cos  do  i/cos2  73,  —  N                       2 B  --cos  6'0  +  1/cos2  61, — N 2d-1/  I  —  4  s2M,  M  -‹  0  s  =  4  Ir  f  Int  v  v, being the velocity of the incident waves. The negative M is applied to the low velocity layer. For the incident waves with unit amplitude,  IR  I and  IT  I are taken as the amplitudes of frequency spectra of reflected and transmitted waves. 3 Transition Zone of Gradually Increasing Velocity             Vs(km/sec 1The Moho may be represented by the                                  transition zone in which the velocity increases  3.8 4.2 4.6                                   withz. The refraction index in such a zone                               is given  by  -1.0   en,  n2  (z)  I —  N(7)  1enz.,-. e  . NII-I I I 1- 0 0 0Fig. 2. Shear velocity distribution in a        transition zone assumed to be        at the Moho. ; seen in Fig. 2.The gradient of  n2(z) is symmetric in z and has the maximum value at  z=0.    "Effective thickness h" is defined as a measure of the thickness of transition zone so that the gradient of n2(z) decreases down to a half of its maximum at  z  h  2  .  It is inde-pendent of the values of the velocities in the homogeneous regions far from the transition zone and is obtained to be  3.526/m from (7). The shear velocity distribution for  h  =0.5 km,  7',  (--  00  )—3.8 km/sec and  71,  ( -I-  00)  —  4.6 km/secIs ri . z. 3.1  Reflection from the Transition Zone    The amplitudes of the reflected waves for the SH wave incidence from the lowest velocity region to the transition zone, are shown in Fig. 3 as the function of frequency, the angle of incidence being taken as parameter. The critical angle for the total reflec-tion is 55°42'. General trend below few cycles per second is relatively flat for interme-diate angles. The amplitudes decrease rapidly with frequency at smaller angles. The transition zone plays a role of low pass filter for the reflected waves, as in the case ofEFFECTS OF TRANSITION ZONE AND LOW VELOCITY LAYER 167cc  w  -J  (-) 55°42' KO40-a0 X  x.  X XXX  \ \\\ \ \\\ \ \\\ so*- 2d1II o-°  X \\  \  \ \ \ \ 11  X  -  N 1 N  \  \-  '  ^\ \ 0  5  10  15 FREOUENCY  IN CPS Fig. 3. Spectral amplitudes of reflected waves from the Moho  "transition         zone" of 0.5 km in effective  thickness.transition layer studied by  SATO (1957).    The wave length of incidentwave at 10 cps is 0.38 km, which is comparable to the effective thickness of 0.50km. For reference, the ratios of the amplitude at 10 cps to that of reflected waves for the Moho "discontinuity  (h=0)," are shown in Fig. 4 as the function of the effective thickness. Comparing Figs. 3 and 4, it is seen that the transition zone of 0.15 km or less in its effective thickness does not reduce the amplitude by more than a factor of tenth for frequencies below 10 cps.    The effect of the thickness on the frequency content is compared with that of the absorption of energy due to the internal friction within the crustal ayer. The effect of absorption is written in the expression of displacement by the factor of exp  (-7rfrl  Qvs), where  r is the distance along the ray path and  11Q the dissipation function. It is known that Q is almost independent of frequency over the range of frequencies in seismic waves. The value of Q in the crust is estimated to be of the order of 102 (PRESS, 1964). The two cases of  Q=250 and 500 will be considered, although Q=250 might be somewhat overestimation of the absorption. We assume the homogeneous crustal layer whose shear velocity and thickness are  3.6 km/sec and 40 km respectively, and an earthquake168  1. EMURAFig. 4. ...- 0  1  _  cc  ,.. 0 —  cc  ua a  I-  I-:  13.  Z  a  w >  1.- < —1  41  CY X55°42' \ )4\W.\ 40°  ..---  50°\ \ \  \ \ \  \ \\   1  \ \\  \ \ \  \ \\0 02  0.4                    EFFECTIVE THICKNESS IN KM Amplitude ratios of reflected waves at 10 cps from the zone" and those from "discontinuity  (h  =  0)"  . Open show the values of dissipation factor for  Q=250 and in a situation seen  in Fig. 5.Moho "transition and solid circles 500 respectively Vs- 3.6 km/ 0 .): 250                 Fig. 5. Reflected ray paths in the crustal  layer assumed. with the focal depth of 20 km is taken into consideration (Fig.  5). The values of dissipation factors at 10 cps for Q=250 and 500 are shown by open and solid circles in Fig. 4 respectively, the Moho being here taken as the first order discontinuity  (h=0).    For the reflection at normal incidence, the effect of the absorption for Q=250 iscomparable to that of the transition zone of 0.15 km in the effective thickness. The transition zone of more than 0.2 km affects the frequency content in near-vertical re-flections much more than the absorption for  Q=250, which may be too low to be in actual crust. If this is the actual case, it seems to be difficult to identify the near-vertical reflection from the Moho on the seismograms. Little evidence of the reflections fromEFFECTS  OF TRANSITION ZONE AND LOW VELOCITY LAYER 169the Moho have been reported  (BuLLEN, 1963). One of the causes may be the small amplitude in high  frequency range in the reflection due to the "transition zone" at the Moho. 3.2 Transmission through the Transition Zone    The frequency dependence of the amplitudes in transmission is seen in Table 1. The spectra of transmitted waves are almost flat in both cases where the incident waves come from the lowest  and the highest velocity regions. It may be expected, therefore, that the Moho "transition zone" of the order of 1 km in its effective thickness does not so much affect he transmitted waves.        Table 1. Frequency dependence of transmitted waves for the Moho transition                    zone  / 4/cos2  0-0—N  1 T  1, for SH wave incidence  (1—N2) cos  170            (a) from  u  s=3  .8 km/sec region (b) from  vs  —4.6 km/sec region     f  19,-0°  40° 60°  40=-0° 40° 60°  0 0.996 0.981  0 0.996 0.990  0.968   1.0 97 85  0 97 94 72   2.0 99 93 0 99 98 84  4.0 1.00 98  0 1.00 99 96  cc 1.00 1.00  0 1.00 1.00 1.003.3 The Case of  f>  1 cps    For the frequencies far about 1 cps and the angles of incidence below the critical angle, we have  I  R  I  exp  (—  4  17'2  f  cos2  bto  N  lmv,) (8)                      (1 —N2)cosz5(°  Ll —0(IRI)j (9)                         cos2 —N The first term in (9) indicates that the amount of transmitted energy into the highest velocity region passing through the cross-section  CD=a1/(cos2z510—N)1(1—N2)  is the same as the energy of the incident wave, which gets into the transition zone through the cross section  AB=  f   a cosdo, a being a length in the direction parallel to theA x-axis (Fig. 6). As almost all energy in the incident waves at higher frequencies are transmitted throughTRANSITION  ZONE the transition zone, it may be expected that the Moho "transition zone" is transparent for short period SH waves. 4 Transition Zone Including Low Velocity LayerFig. 6. Ray paths in                                                           regions far from the    The velocity distribution, which corresponds to thetransition zone.                                                                      AB=  a cos0-0.low velocity layer in the upper mantle, is assumed as  CD=  as/(cos20—  N)I(1—N2). in  Fig. 7. The velocity tends to 4.60 km/sec above the170 -100E  '` 0  NJFig. 1007 Vs  (  KM  /  Sec  1  457---7— I  1 5.0  1  1 1 Shear velocity distributions in transition zones with (full line) and without (broken line) a low velocity layer.Fig. w  8 1—.  ci.  x  ..2  -.1  .2  CC  )- 0  W  CI-COK. EMURA  low velocity region and 5.00  km/sec below    it. The minimum velocity in the low     velocity layer is 4.29 km/sec. Full line    shows the case where the low velocity    layer exists and broken line the case of     absence of the layer.        Fig. 8 shows the amplitudes of reflected    waves for the SH wave incidence from the    4.60 km/sec region. Full and broken lines    in Fig. 8 correspond to those in Fig. 7. The    critical angle for the total reflection is     66°56'. General trend of the broken lines    is, of course, similar to that for the Moho     "transition zone" considered in the previous     section. The spectra demonstrate maxima    and minima when a low velocity layer     exists, while this feature is not seen in the     case of absence of it. The discrepancy 0 0 6o° /ii  N N -N \  \ -N. • \ \I\ - .00 \  \\V--  30' sc  A  A  \ \ \ \ 3  4x10-2  8x10-2                      FREQUENCY  iN CPS  8. Spectral amplitudes of reflected waves from transition zones    with (full line) and without (broken line) the low velocity layer.EFFECTS OF TRANSITION ZONE AND LOW VELOCITY LAYER 171between the full and broken 1 The lower the frequency, the velocity layer.    At the frequencyines decreases with the larger the amplification  fn     m Tr  1/ LAI  [(2  —1)2  — IR  I in the case of presence of the low velocity layer has the  n th minimum value, which coincides with that in the case of absence of it.  f„ is independent of angle of incidence and is given in Table 2. The frequency intervals between the successive minima decrease with  n and tend to  m  vii4n-1M  I as  n  00    For the case     2  7r2f(cos  Vo  1/cos2  z —  N)Im  v1  > 1 (11) we have  IR exp  (  —  4  7r2  f 1/cos2  —1increasing angle of the amplitude11Tableof incidence. by the low2.(10) Nim  v1) It 1 2 3 4 5 00In0 5.50 9.53 13.5 17.4 x10-2 3.89  XIIwhich is the same factor as that for the absence of the low velocity layer. As the condition (11) corresponds to the frequencies  f>0.1 cps, the effect of the low velocity layer may hardly be recognized for the higher frequencies than 0.1 cps.    The frequency dependence of the transmitted waves is shown in Table 3. The difference between the spectra for two types of zone, with and without the low velocityTable 3. Frequency dependence of transmitted waves I/     T  ,  (1  —N2)  cos  t5-0   for the transition zones with (A) and without (B) the low velocitylayer                         for SH wave  incidence,                    (a) from  vs=4.6 km/sec region .0-a=0°0  1.0x10-2 2.0 3.0 4.0 5.0  Co 30° A 0.9992   63    89    98 1.000  1.000 1.000 0  1  1  1  1  1  1B 9992 000 000 000 000 000 0001 A 0.9984    41    69    95    97    99 1.000(b) from  vs-5.0  km/sec 60c B 0.9984    95    98 1.000 1.000 1.000 1.000region A_ 0.9721   452   335  637   907  988 1.000 B 0.9721   62   846   919   968   988 1.0000 1. 2. 3. 4. 5.  co 0><10-2  0  0 0 00.9992   65    93 1.000 1.000 1.000 1.0000.9992    96 1.000 1.000 1.000 1.000 1.0000.9986    45    68    91    94    99 1.000 0.9986   90    92 1.000 1.000 1.000  1.0000.9906 0.9906  759 24  753 60  914 92  980 94 1.000 1.000 1.000 1.000 172  X.  tWURA layer, is so little over full ranges of frequency and angle of incidence that it might be impossible to discriminate one type from another. 5 Conclusions    The spectral amplitudes of reflected and transmitted plane SH waves for two types of transition zone, are theoretically studied. A downward increase in seismic wave velocity simulates a transition zone assumed to be at the Mohorovicic discon-tinuity. A transition zone including a  low velocity region represents the low velocity layer, which may possibly be in the upper mantle. The derivatives of two velocity functions are continuous and finite throughout he medium.    The main results of the present study are summarized as  follows;    The Moho "transition zone" of 0.15 km or less in its effective thickness does not reduce the amplitude of the reflected waves by more than a factor of tenth for frequencies below 10 cps. The Moho with more than 0.2 km thick affects the frequency content in near-vertical reflections much more than the absorption of energy due to the internal friction in the crustal layer with Q  >250. The low velocity layer in the upper mantle amplifies the reflected waves, although the effect is too small to be recognized at the frequencies above 0.1 cps. Frequency spectra of the transmitted waves are almost flat for both types of the transition zone considered. The zone is transparent for short period SH waves in both cases.    Acknowledgement: The author wishes to express his hearty thanks to Prof. Z.  SUZUKI, Drs. K. NAKAMURA and A. TAKAGI for their valuable discussions.                               References    BULLEN,  K.E.  (1963): "An Introduction tothe Theory of Seismology".  BREKBOVSKIKH,  L.M. (1960): "Waves in Layered Media".  LOVERING, J.F.  (1958): The Nature of the Mohorovicic Discontinuity. Trans. Amer. Geophys.         Un., 39, 947-955.    MACDONALD, G.J.F., and N.F. NESS  (1960): Stability of Phase Transitions within the Earth.  fount. Geophys. Res., 65, 2173-2190.    NAKAMURA, Y. and B.F. HOWELL, Jr.  (1964): Maine Seismic Experiment: Frequency        Spectra of Refraction Arrivals and the Nature of the  Mohorovicic Discontinuity. Bull.         Seism. Soc. Amer., 54,  9-18.    PRESS, F.  (1964): SeismicWave Attenuation i the Crust.  Jour  n. Geophys. Res., 69,  4417—       4418.  SATO, R.  (1957):  Reflection and Transmission f Elastic Waves at a Discontinuity with an        Intervenient Transient Layer. Zisin, Ser. 2, 10, 141-153.  (in. Japanese with English abstract).

Effects of Transition Zone and Low Velocity Layer on Frequency Spectra of Plane SH Waves

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Effects of Transition Zone and Low Velocity Layer on Frequency Spectra of Plane SH Waves

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