Dispersion of G type seismic waves in low velocity layer.

Chattopadhyay, A. ; Gupta, S. ; Kumari, Pato 等

Introduction

There have been numerous investigations of the radiation pattern of P and S waves based on the assumption that an earthquake source is represented by a double couple or a single couple. In these studies, the geometrical aspects of the couple have been emphasized, but the magnitude of its moment, the only physical quantity relevant to a couple, has not attracted due attention, if not neglected. There is a reason for this, that is, the amplitudes of the body waves with short wavelengths are influenced by fine structure of the crust-mantle, and it is difficult to eliminate their effect to isolate source factors. Further, the amplitude at short periods is too sensitive to minor fluctuation of the source time function. These difficulties may not exist in long period surface waves.

Surface waves in the earth are observed on seismograms of distant surface earthquakes as long trains of dispersed waves with large amplitudes. Dispersion is easily detected, first arrivals corresponding to waves of longer periods. As has been mentioned, the periods present on seismograms correspond to instantaneous frequencies. Love waves are registered only in the horizontal components whereas Rayleigh waves, which are polarized in the vertical plane, are registered both in horizontal and in vertical components. If we rotate the two horizontal components to make them coincide with the radial and transverse directions with respect to the orientation from the station to the epicentre, Love waves (LQ) are recorded only in the radial one. Love waves of long periods (60-300s) are also called G waves. For these periods, the dispersion curve is practically flat with a velocity of about 4.4 [kms.sup.-1] and waves have an almost impulsive form (e.g. Benioff 1958, Benioff and Press 1958, Press 1959). Because the group velocity of Love waves in the earth are nearly constant (4.4 km/s) over the period range from about 40 to 300s, their waveform is rather impulsive, and they have received this additional name. They are called G-waves after Gutenberg (1953, 1954). It takes about 2.5 hours for G waves to make a round trip of earth. After a large earthquake, a sequence of G-waves may be observed. Love and Rayleigh waves of short periods (8-12s) in continental trajectories are channelled in the upper crust and are known as [L.sub.g] and [R.sub.g] waves. For periods between 60 and 300s Love and Rayleigh waves travel mainly through the mantle and are called mantle waves. For large earthquakes, surface waves that travel around the Earth more than once are observed. These waves are designated with a subindex [G.sub.1], [G.sub.2], [G.sub.3], etc. for Love waves, and [R.sub.1], [R.sub.2], [R.sub.3], etc. for Rayleigh waves. Groups [G.sub.1] and [R.sub.1] are direct waves from the epicentre to the station and [G.sub.2] and [R.sub.2] are waves that arrive at the station travelling in the opposite direction. For higher subindexes, waves with an odd subindex circle the Earth, leaving the epicentre in the direction of the station; and those with an even subindex do it leaving in the opposite direction.

Literature on G type waves is not available in abundance. Although several researcher have done work in this field. Notable are Jeffreys (1959), Haskell (1964), Bhattachrya (1963), Brune et al (1963), Chattopadhyay (1978) etc. Aki (1966) discussed the generation and propagation of G waves from the Niigata earthquake of June 16, 1964. In part 2 he discussed the estimation of earthquake moment, released energy, and stress-strain drop from the G wave spectrum. Lehmann (1961) pointed out that the European as well as the north-eastern American observations of S waves at small epicentral distances indicates the presence of a low velocity layer. Bath (1957) studied the shadow zones, travel times and energies of longitudinal seismic waves in the presence of an asthenosphere low-velocity layer. Periods of G waves lie in the range from about 50 to 350s, corresponding to a flat portion of the group velocity curve for Love waves, which has been extended up to periods of around 700s (Burne, Benioff and Ewing 1961). As a consequence of their small dispersion, G waves have a transient pulse- like character in the records and are followed by a train of dispersed Love waves, especially for continental paths. An outstanding case of well developed G-type waves was provided by the earthquake of January 1960, in Peru. Bath and Arroyo (1962) presented the results obtained from this earthquake, especially with regard to absorption and velocity dispersion of G-waves. Mal (1962) studied the possibility of the generation of G-waves with the lower medium to be isotropic. Chattopadhyay and Keshri (1986) discussed the generation of G type waves under initial stress.

In this paper we have represented the low-velocity layer by assuming the rigidity and density in the inhomogeneous isotropic medium in the form

[[mu].sub.2] = [[mu].sub.0](1 - [epsilon] cos sZ),

[[rho].sub.2] = [[rho].sub.0](1 - [epsilon] cos sz)

where [epsilon] is small positive constant and s is real depth parameter. With this law of variation the equations of motion reduce to Hill's equation with periodic coefficients which has been solved by the method given by Valeev (1960). Valeev considered a certain class of system of linear differential equations with periodic coefficients which have the property that, by means of Laplace transformation, they may be converted to a system of linear difference equations, which in turn may be solved by the method of infinite determinants. This method of solving Hill's differential equation has been successfully employed by Bhattacharya (1963) and Mal (1962). Keeping terms up to first order in [epsilon], the Laplace transform F(p) of the displacement [V.sub.2](z) was obtained.

[FIGURE 1 OMITTED]

Formulation and solution of the problem

Let us consider isotropic homogeneous medium of thickness H with rigidity [[mu].sub.1] and density [[rho].sub.1] overlying a non-homogeneous medium with rigidity [[mu].sub.2] and density [[rho].sub.2] taken as

[[mu].sub.2] = [[mu].sub.0](1 - [epsilon] cos sz),

[[rho].sub.2] = [[rho].sub.0](1 - [epsilon] cos sz) (1)

where [epsilon] are small positive constant and s is real depth parameter. The x-axis is taken as horizontal axis, z-axis as vertically downwards and origin is taken at the interface (Fig. 1). We consider the propagation of horizontally polarized surface waves of shear type, propagating along x axis. So the displacement components are

u = [omega] = 0, v = v(x,z, t).

Therefore, the equation of motion for upper homogeneous isotropic layer is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

where [[beta].sub.1] = [square root of ([[mu].sub.1]/[[rho].sub.1] is the shear wave velocity in upper layer. If k is the wave number

and c is the wave velocity, [v.sub.1] can be taken as

[[upsilon].sub.1](x, z, t) = [V.sub.1](z)[e.sup.ik](x-et). (3)

Substituting (3) into (2) we have

[d.sup.2][V.sub.1]/d[z.sup.2] + [a.sup.2][V.sub.1] = 0 (4)

where

[a.sup.2] = [k.sup.2] ([c.sup.2]/[[beta].sup.2.sub.1] - 1). (5)

The solution of Eq.(4) is

[V.sub.1](z) = A cos az + B sin az

where A, B are constants. As the upper surface is stress free, so

d[V.sub.1]/dz = 0 at z = -H,

which gives

A sin aH + B cos aH = 0

Therefore,

[V.sub.1](z) = [a.sub.1] cos a (z+H). (6)

In the lower inhomogeneous medium the displacement [v.sub.2](x, z, t) satisfies differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

Writing

[v.sub.2](x, z, t) = [V.sub.2](z)[e.sup.ik(x-ct)]. (8)

Using Eqs.(7) and (8), the equation of motion for lower inhomogeneous medium may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

This is Hill's differential equation, which will be solved by the method given by Valeev (1960). We apply Laplace transform with respect to z, i.e. we multiply Eq.(9) by [e.sup.-pz] and integrate with respect to z from 0 to [infinity],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

Now the boundary conditions are

(i) [[upsilon].sub.1] = [[upsilon].sub.2] at z = 0,

(ii) [[mu].sub.1][delta][[upsilon].sub.1]/[delta]z = [[mu].sub.2] [delta][v.sub.2]/[delta]z at z=0,

(iii) [delta][[upsilon].sub.1]/[delta]z = 0 at z = -H. (11)

Using (3), (6) and (8) the boundary conditions (11) give

[V.sub.2](0) = [a.sub.1] cos aH, q(0) = [[mu].sub.1][a.sub.1]sin aH/[[mu].sub.2] (12)

where

q(0) = [(d[V.sub.2]/dz).sub.t=0], (13)

and

[[mu].sub.2] = [[mu].sub.0](1 - [epsilon]). (14)

We define the Laplace transform of [V.sub.2](z) as

F(p) = [[integral].sup.[infinity].sub.0][e.sup.-pz][V.sub.2](z)dz,

and using the boundary conditions, we obtain the following system of equations as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

where

[A.sub.1] = (1 - [epsilon])[V.sub.2](0), [A.sub.2] = (1 - [epsilon])q(0) (16)

and

[[omega].sup.2] = [k.sup.2](1 - [[rho].sub.0][c.sup.2]/[[mu].sub.0])

To find F(p) from (15) we replace p by p + ism and then divide throughout by [(ism).sup.n](m [not equal to] 0). We then obtain the following infinite system of linear algebraic equations in the quantities F(p+ism), (m=0, [+ or -] 1, [+ or -]2, ...)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

where p may be considered as a parameter in the coefficients. It should be noted that in order not to have to consider the special case m = 0 separately, we include (15) in (17) by agreeing to regard [(ism).sup.-n] =1 when m = 0. Solving the system of difference equations (17) we obtain F(p) as the ratio of two infinite determinants, viz.,

F(p) = [[DELTA].sub.1]/[[DELTA].sub.2] (18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The first approximation of the Eq. (18) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The second approximation of Eq. (18) is

F(p) = [[DELTA].sub.3]/[[DELTA].sub.4] (19)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Neglecting the terms containing [[epsilon].sup.2] and higher powers, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[[s.sup.2n][[DELTA].sub.4] = ([p.sup.2] - [[omega].sup.2]{[(p - is).sup.2] - [[omega].sup.2]}{[(p + is).sup.2] + [[omega].sup.2]}.

Hence Eq.(19) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

Then [V.sub.2](z) will be given by the inversion formula as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

The residues [R.sub.1], [R.sub.2], [R.sub.3] at the poles p = [omega], p = [omega]+is, p = [omega]-is, are given, respectively, by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (24)

where

[P.sub.1] = [[rho].sub.0][k.sup.2][c.sup.2]/[[mu].sub.0] - [k.sup.2].

Eqs.(22), (23) and (24) show that the conditions for a large amount of energy to be confined near the surface are

[omega][V.sub.2](0) + q(0) = 0 (25)

2[[omega].sup.2] + [s.sup.2] = 0 (26)

and

q(0) - [omega][V.sub.2](0) = 0 (27)

Eqs.(25) and (27) both give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (28)

where

[[beta].sup.2] = [[square root of ([[mu].sub.0]/[[rho].sub.0)]

Taking positive sign, Eq.(28) gives the dispersion equation for Love type waves for isotropic homogeneous layer overlying inhomogeneous isotropic half space. As a special case when [epsilon] = 0 Eq. (28) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which is the usual dispersion equation for Love waves with [[beta].sub.1] < c < [[beta].sub.2] [9]. Eq.(26) gives

kc = [square root of ([[mu].sub.0]/2[[rho].sub.0](2[k.sup.2] + [s.sup.2]))].

Then the group velocity U is given by

U = d/dk(kc) = [[beta].sub.2 [square root of (2k))]/[square roof of (2[k.sup.2] + [s.sup.2])] (29)

It follows from Eq.(29) that U < [[beta].sub.2] i.e. the group velocity is less than the shear wave velocity in the upper mantle.

Numerical calculations and discussions

Data of upper isotropic homogeneous medium [11]

[[rho].sub.1] = 3364 Kg/[m.sup.3], [[mu].sub.1] = 6.34x[10.sup.10] N/[m.sup.2]

Data of lower isotropic homogeneous medium [11]

[[rho].sub.0] = 3535 Kg/[m.sup.3], [[mu].sub.0] = 7.84x[10.sup.10] N/[[mu].sup.2].

Results and discussion

In Fig. 2 we have shown the variation in dimensionless phase velocity c/[[beta].sub.1] against dimensionless wave number kH for different value of inhomogeneity parameter [epsilon] and in Fig. 3 we have plotted the variation in dimensionless group velocity U/[[beta].sub.2] with respect to scaled wave number k/s. In Fig. 4 we have drawn the surface plot of group velocity U for variation in parameter s and wave number k. It is clear from Figs. 2 and 3 that phase velocity decreases slightly with increase in inhomogeneity parameter ?, and group velocity U approaches [[beta].sub.2] asymptotically with increase in scaled wave number k/s. From Fig. 4 it can be concluded that group velocity is depending significantly on wave number k and depth parameter s. Also Eq.(25) shows that if horizontally polarized S waves propagate, a large fraction of energy may flow along the surface only when the wave length and the period satisfy the ordinary dispersion equation for Love waves in uniform media, with the group velocity U satisfying relation (29). Also it is clear from (29) that the group velocity is lower than the Shear wave velocity in the upper mantle.

Hence finally it can be concluded that the energy carried by long period surface waves relative to the total seismic energy indicates that earthquakes show a greater amount of long period surface waves than explosions. The present results may become useful in deep earthquake and predicting the nature of long period Love waves.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Acknowledgement

The authors are thankful to DST, New Delhi (India) for providing financial support through Project no. SR/S4/ES-246/2006, Project title: "Investigation of torsional surface waves in non-homogeneous layered earth".

References

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[2] Bath M., 1957, Shadow zone, travel times and energies of longitudinal seismic waves in the presence of an asthenosphere low-velocity layer, Trans. Amer. Geophys. Union, 38, 529-538.

[3] Bath M., Arroyo A. L., 1962, Attenuation and dispersion of G-waves, J. Geophys. Res., 67, 1933-1942.

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[5] Benioff H., Press F., 1958, Progress report on long period seismographs, Geophys. J. R. Astron. Soc., 1, 3, 208-215.

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[7] Burne J. N., Benioff H., Ewing M., 1961, Long period surface waves from the Chilean earthquake of May 22, 1960, recorded on linear strain seismographs, J. Geophys. Res., 66, 2895-2910.

[8] Brune, J., A. Espinosa and J. Oliver (1963): Relative excitation of surface waves by earthquakes and underground explosions in the California-Nevada region. J. Geophys. Res., 68, 3501-3513.

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[10] Chattopadhyay A. and Keshri A., 1986, Generation of G-type seismic waves under initial stress, Indian J. Pure Appl. Math., 17(8), 1042-1055.

[11] Gubbins D., 1990, Seismology and plate tectonics, Cambridge University press, Cambridge, New York, p-170.

[12] Gutenberg B., 1953, Wave velocities at depths between 50 and 600 kilometers, Bull. Seismol. Soc. Am., 43, 223-232.

[13] Gutenberg B., 1954, Effects of low-velocity layers, Geof. Pura e Appl., 29, 1-10.

[14] Haskell N. A., 1964, Radiation pattern of surface waves from point sources in multi-layered medium, 54, 377-393.

[15] Jeffreys H.,1959, The Earth 4th edition, Cambridge University press, pp.195-210.

[16] Lehmann I., 1961, S-waves and the structure of the upper mantle, Geophys. J. R. Astron. Soc., 3, 529-538.

[17] Mal A. K., 1962, on the generation of G-waves, Gerlands Beitr. Geophys., 72, 82-88.

[18] Press F., 1959, Some implications on mantle and crustal structure from G-waves and Love-waves, J. Geophys. Res., 64, 565-568.

[19] Valeev K.G., 1960, On Hill's method in the theory of linear differential equations with periodic coefficients, J. Appl. Math. A. Mech., 24, 1493-1505.

A. Chattopadhyay, S. Gupta, Pato Kumari and V. K. Sharma

Department of Applied Mathematics, Indian School of Mines University, Dhanbad-826004, Jharkhand, India E-mail: amares.c@gmail