Download Distinctive Particle Motion of Surface Waves as a Diagnostic of

Geophys. J.
R. astr. SOC.(1975) 40, 177-186.
Distinctive Particle Motion of Surface Waves
as a Diagnostic of Anisotropic Layering
Stuart Crampin
(Received 1974 October 21)*
Summary
Surface waves propagating along particular directions in anisotropic
media may have one of three distinctive particle motions indicating
orientations of planes of anisotropic symmetry with respect to the
direction of propagation and the free surface. The particle motion of
surface waves in a medium with a horizontal plane of symmetry has been
observed on seismograms and used to indicate the presence of such
anisotropy within the Upper Mantle.
Introduction
Aligned crystalline anisotropy patently exists on a small scale within the Earth.
The presence of anisotropy on a larger scale is hard to establish because of the
difficulty in finding a seismological parameter, which is sufficiently sensitive to
anisotropy to be used to determine its scale and perhaps its nature. Seismic waves
passing through an anisotropic region would be affected in many ways. Anomalies
would be present in travel times, seismic velocities, the relationship between P, SV,
and S H , body wave amplitudes, and surface wave dispersion, among others, but only
in exceptional circumstances would the anomaly in these parameters be so marked
that it could unequivocally be assigned to the presence of anisotropy, and not to
some unknown inhomogeneity. In this paper we shall show that surface waves have
distinctive particle motion when propagating in a structure, which contains a layer of
anisotropic material with certain symmetry relations, and that t h s anomaly is
sufficiently large to be used as a diagnostic for the presence of anisotropy.
We consider the propagation of surface waves in media which contains a layer
having crystalline anistropy. Except along directions of crystal symmetry, the three
body waves in an anisotropic media have independent velocities, and particle motions
which are a combination of linear oscillations in three dimensions. This phenomenon
is mirrored by the surface waves, which do not have the degenerate form of two
independent families of normal modes, but form one family of normal geneialized
modes each with elliptical particle-motion in three dimensions (Crampin 1970,
hereafter called Paper I). Along directions of symmetry, this generalized family
degenerates into two independent families, one having the elliptical particle motion
in the sagittal plane of the familiar Rayleigh wave, and the other the horizontally
polarized transverse motion of the Love wave.
Crystals have various types of symmetry, which considerably reduce the number
of elastic constants below the maximum of 21. Nevertheless, analytical examination
of the effects of anisotropy is confined to problems in unlayered homogeneous
* Received in original form
1974 July 26.
177
C
178
S . Crampin
Table 1
Relative phases of particle motion of surface waves (xl propagating, x3free surface) in
anisotropic half-spaces for various symmetry conditions.
Each component has oscillatory motion of the same frequency but dgerent phase.
Crystal symmetry
Phase differences between
particle motion relative
to phase u1
Planes of symmetry
u1
1. No symmetry
2. x1 = 0
0
a
B
0
+a12
+ff/2
=0
0
3.
4.
XJ
XI
=0
u2
0
-
a
Type of particle motion
u3
+,I2
B
-
5. x1 = 0, x2 = 0,
and xj = 0
Generalized
Ti1ted-Rayleigh
Inclined-Rayleigh
Sloping-Rayleigh
Love
Rayleigh
Love
6. Transversely
isotropic
7. Isotropic
Where a and /3 are constants not equal to 0 or f n / 2 , and ' - ' denotes no motion
for that component. The word Ruyleigh refers to elliptical particle motion in a plane,
where the plane is not necessarily the sagittal plane (see Fig. 1).
(a)
A
b
.... .............9. Propagation
direction
FIG. 1. Three types of particle motion characteristic of particular symmetry
orientations: (a) Inclined-Rayleigh motion-horizontal plane of symmetry,
(b) Tilted-Rayleigh motion-propagation at right angles to vertical plane of
symmetry, and (c) Sloping-Rayleigh motion-propagation with the sagittal plane
a plane of symmetry.
Distinctive particle motion of sudace waves
179
Rotations about x-axis
4.0
'0
30'
60°
90'
4.0
0"
(001)
Angle from (001)
60'
90°
Plane of cut
(010)
30'
(001)
Plane of cut
FIG. 2. Surface waves on an olivine half-space for three ranges of orientations.
The upper figures show the variation with orientation of the phase velocities
of the generalized surface wave, and the two slower body waves (the quasi-shear
waves). The lower figures show the deviations of the particle motion from
elliptic motion in the sagittal plane (0 in Fig. 1). The crosses and circles in the
first two diagrams mark positions along directions of symmetry where the
generalized wave degenerates into Rayleigh and Love type motion, respectively.
QSH, QSV, QSl, and QS2 refer to quasi-SH, quasi-SY, and two undifferentiated quasi-shear waves, respectively.
Table 2
Structural models for Figs 2-6.
Olivine half-space
Fig. 2(a) (010)cut, propagating over a range of directions between (001) and (100).
Fig, 2(b) (100)-propagating for a range of cuts between (001) and (010).
Fig. 2(c) x-propagating for a range of cuts between (001) and (100).
Layered structure
Figs 3-5
h
10
20
30
P
2.7
2.9
3 * 324
3.6
a
B
3.4
6.6
3.8
olivine layer
9.0
5.4
5.8
The orientation of the olivine is given
in the Figure captions.
(100)
180
S. Crampin
half-spaces, and frequently only to directions of symmetry within such a half space.
The program used for the numerical analysis in this paper is based on the work in
Paper I, and in Crampin & Taylor (1971, hereafter called Paper II), which has been
further developed by Armstrong & Crampin (1972, 1973) to calculate surface wave
motion in piezoelectric structures for microwave acoustic device applications.
Surface-wave particle-motion in a homogeneous anisotropic half-space
The particle motion of surface waves of a given frequency in a randomly orientated
anisotropic crystalline half-space is oscillatory motion in three dimensions, in which
each component has a different angle of phase. The orientation of the anisotropy with
respect to the direction of propagation and the free surface has characteristic effects
on the particle motion of the generalized normal-mode surface wave. It is found
by numerical experiment (and can be shown theoretically, Crampin & Taylor, in
preparation) that surface waves, propagating parallel or perpendicular to planes of
symmetry, have components of displacement, which are in phase, out of phase by
& n/2, or completely decoupled as the horizontally polarized transverse motion of
Love waves is decoupled from Rayleigh wave motion in isotropic media. These phase
differences persist in a layered structure if each layer contains similar symmetries.
In particular, isotropic layers, which being wholly symmetric, are transparent to
particle motion and do not affect the anisotropic phase differences.
The effects on the particle motion of propagation parallel or perpendicular
to planes of symmetry are summarized in Table 1. The terms inclined-Rayleigh,
tilted-Rayleigh, and sloping-Rayleigh refer to motion of the types shown in Fig. 1.
Fig. 2 shows some numerical evaluations of the deviation from isotropic Rayieigh
wave motion of surface waves propagating on olivine half-spaces with three ranges of
orientation. Olivine is orthorhombic with three mutually orthogonal planes of
symmetry: (loo), (OlO), and (001). The elastic constants are taken from Verma
(1960), and the orientations are given in Table 2.
Fig. 2(a) shows the variation in motion over a range of directions when the free
surface is a plane of symmetry.
Fig. 2(b) shows the variation for propagation at right angles to a plane of symmetry
as the plane is rotated. As the direction of propagation does not change, the velocities
of the body waves are constant, and velocity of the generalized surface wave changes
very little.
Fig. 2(c) shows the variation of motion when the sagittal plane is a plane of
symmetry as the plane is rotated. The generalized surface wave is independent of
one of the two quasi-shear body waves (in this case the lowest), and this body wave
identically satisfies the equations for Love waves over the whole range.
In all three cases, the deviations from the isotropic Rayleigh-wave particlemotion are comparatively small, although no attempt was made to search for
orientations where the particle motion would have larger deviations.
Surface-wave particle-motion in layered earth models
The layered models in Table 2 consist of a two-layered isotropic crust overlying an
isotropic mantle with a layer of anisotropic olivine in the upper mantle beneath the
crust. The earth models are not realistic but are intended to indicate that aligned
anisotropy can have marked effects on particle motion at the surface, which are
diEcult if not impossible to reproduce with any other type of non-isotropic
phenomenon.
Figs 3, 4 and 5 show the variation of the particle motion with period of the four
gravest modes of generalized surface waves propagating in the layered structure
Distinctive particle motion of surface waves
I
\
I
I
\
\/
3,
1 2c
0
I
I
181
I
/
/
(D
w
9(
b
H
.-s
-
6C
(D
n
e
n
e
Etn
.L
0
w
.-5
z
.--
3c
0
-
a
I
I
I
I
I
I
-30
1
10
1
I
30
Period in seconds
20
I
40
FIG.3. The variation with period of the deviations from isotropic Rayleigh particlemotion of generalized waves, having inclined-Rayleigh particle motion, propagating in the layered structure of Table 2, when the olivine has a horizontal plane of
symmetry. The olivine layer is (010) cut and propagation is at 30" (solid line)
and 60" (dashed line) measured from (001) towards (100).
when the olivine layer is 30 km thick. These four modes are equivalent to the fundamental and second modes of Rayleigh and Love waves in an isotropic structure.
The correspondence between the modes in an anisotropic structure and those in an
isotropic structure is not always simple: there may be occasions when two modes
pinch together (Paper 11) and the modes exchange characteristics, and the orthogonal
nature of the particle motion of Rayleigh and Love modes in an isotropic structure
does not exist between the diffeient normal modes in an anisotropic structure.
182
S. Crampin
I
I
I
1
I
I
I
12c
I
mode4
9c
-m
5
c)
6(
r
.E" . " I
c
I
I
I
r
0
._
L
0
c
._
I
I
I
c
c
E
g
ln
I
I
I
3c
I
P
c
I
I
c
30
10
1
1
I
20
30
40
Period in seconds
FIG.4. The variation with period of the deviations of generalized waves, having
tilted-Rayleigh particle-motion, propagating in the layered structure at rightangles to a plane of symmetry. The Olivine layer is (100)-propagating at cuts of
30" (solid line) and 60" (dashed line) measured from (001) towards (010).
The orientations of the Olivine layers in the models used for Figs 3, 4 and 5 are
two directions or cuts within the ranges of orientations used for Fig. 2(a), 2(b) and 2(c),
respectively, and the waves have the correspondingly inclined-, tilted-, and slopingRayleigh particle-motion.
The size of the deviations in Figs 3 and 4 are generally similar. The deviations of the
first and second generalized modes peak near a period of 30 s, with a maximum
deviation of 20" or less. The higher modes, in particular the third mode (usually
Distinctive particle motion of surface waves
-10
30
40
183
50
60
50
60
70
I
-10-
-20Y)
7
-30-
2
-40-
:
al
a
0 -50m
-60-
L
-L
I
10
20
30
2
40
I
.
I
70
PerirJd in seconds
FIG.5. The variation with period of the deviations of generalized waves, having
sloping-Rayleigh particle-motion, propagating in the layered structure, when the
sagittal plane is a plane of symmetry. The olivine layer is x-propagating at cuts
of 30" (solid line) and 60" (dashed line) measured from (001) towards (100).
equivalent to the second Rayleigh mode in isotropic media), show a much larger
range of deviations. There are two contributing factors to the large deviations:
(1) The variation with depth of the particle motion of third modes has a large
lobe of amplitude below the crust near the depth of the olivine layer (the exact depth
of the lobe depending on the period) so that most of the energy of the third mode
travels in the anisotropic layer. Consequently, the anisotropy usually has a large
effect on this mode.
(2) In these anisotropic earth models, the third and four generalized modes
have similar dispersion (as have the corresponding modes, the second Rayleigh
and second Love modes, in an isotropic earth model). Pinching occurs when two
modes approach each other in period and velocity. The closeness of the third- and
fourth-mode dispersion curves makes them liable to pinching, and each pinch will
result in a rapid transition in properties, in particular the particle motion may show a
rapid change.
These factors result in the plane elliptical particle motion of the third and fourth
modes changing in just a few seconds period, from being nearly in the sagittal plane
to being nearly in the plane of the wavefront (inclined-Rayleigh motion of Fig. 3), or
nearly in the horizontal plane (tilted-Rayleigh motion of Fig. 4).
The sloping-Rayleigh wave motion in Fig. 5 behaves rather differently from the
motion in the previous two figures. Rapid deviations due to pinching are not possible,
as, for this orientation of olivine, the generalized family of modes has degenerated
into the two independent families of the sloping-Rayleigh waves and the Love waves
(see Table 1, Symmetry type 4), and pinching usually occurs only between adjacent
pairs of generalized modes. Thus, in Fig. 5, the second sloping-Rayleigh mode shows
little deviation, and the elliptical motion of the fundamental mode shows, particularly
for the 60" cut, a change at 40-s period from being vertically elongated to being
horizontally elongated, which is a possible fluctuation of isotropic Rayleigh motion.
Fig. 6 shows the deviations in the inclined-Rayleigh motion as the thickness of the
olivine layer is varied from 30 to 5 km with the orientation kept constant. The
S. Crampin
propagating with the same structure and orientation as for Fig. 3, but fo; a range
of thicknesses of the olivine layer: 30 km, solid line; 20 km, dashed line; 10 km,
dash/dot line; and 5 km, dotted line. The Olivine layer is (010) cut and 45"
propagating (001 towards 100). The deviations for mode 3 have not been plotted
for periods less than 9.5 s, as they show rapid fluctuations which could be followed
only with difficulty.
deviations of Mode 1 are reduced as the thickness of the olivine layer decreases, but
even a thickness of only 5 km produces a peak deviation of some 13" at 23-s period.
The particle motion of Mode 2 is little effected by the layer of anisotropy and the
peak deviation does not exceed 7". The fourth mode also shows little effect. The
third mode shows rapid fluctuations for the whole range of thicknesses of the olivine
layer; in particular there is still rapid change at the lowest thickness of 5 km. The
Distinctive particle motion of surface waves
185
deviation at this thickness ranges over 7112 in 6-s period, which includes a section
between 10- and 14-s period where the deviation only varies between 35" and 50".
Conclusions
The figures show that for some orientations even quite small thicknesses of
anisotropy in the upper mantle produce marked effects on the particle motion of
surface waves, which are characteristic not only of anisotropy but also of the
orientations of some of the planes of symmetry. If the upper mantle, or more particularly, a continental plate, contains any crystalline material, the long-persisting
horizontal stresses required to move plates will be sufficient to align crystals over a
large area of the plate. The forces likely to align crystals will be the stress causing the
plate movement acting in a horizontal plane, and vertical forces such as gravity, and
forces associated with differentiating processes, resulting in two orthogonal planes of
symmetry: horizontal and vertical. Thus, of the orientations resulting in the three
distinctive particle motions discussed here, only orientations producing inclinedRayleigh motion appears to be possible in the Earth, except perhaps on a limited local
scale.
The effects on particle motion of anisotropy in the upper mantle are most marked
for the motion of the third generalized mode (second Rayleigh mode), because these
modes have most of their energy travelling in the upper mantle, and because the third
and fourth modes have similar dispersion making mode pinching liable to occur in
earth models containing anistiopy.
Paper I1 presented calculations of several surface-wave parameters for similar
structures. From Paper I1 and other numerical examinations of the effects of
anisotropy, it appears that particle motion is probably the most sensitive observable
parameter for the unambiguous recognition of anisotropy in the upper mantle at a
single station. This distinction belongs to particle motion because one of the major
effects of anisotropy on elastic wave propagation in a homogeneous medium is the
loss of independence of P and S waves and the consequent breakdown in polarity
between motion in the sagittal plane and SH motion characteristic of horizontally
layered isotropic media.
Inclined-Rayleigh particle motion has been observed by Crampin (1966, 1967)
for Second Rayleigh/Second Love modes (Third Generalized mode) over many
paths across the Russian shield. The observations show many different angles of
inclination as we would expect from Fig 3 and 6. The persistence of the inclined
motion over long paths, and the known sensitivity of higher modes to changes in
structure, with consequent attenuation, suggest that the only explanation for the
inclined particle motion is the presence of anisotropy in the upper mantle.
Acknowledgments
This work was undertaken as part of the research programme of the Institute
of Geological Sciences and is published with approval of the Director, IGS.
Global Seismology Unit,
Imtitute of Geological Sciences,
Edinburgh.
References
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Armstrong, G. A. & Crampin, S . , 1973. Preferential excitation of 2nd-mode piezoelectric surface waves in Zinc-oxide-layered substrates, Electron. Lett., 9, 322-323,
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S. Crampin
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Bull. seism. SOC.Am., 56, 1227-1239.
Crampin, S., 1967. Coupled Rayleigh-Love second modes, Geophys. J. R. astr. SOC.,
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