Geophys. J . Int. (1996)126, 173-183
Surface waves in layered anisotropic structures
Jeffrey Park
Department of Geology and Geophysics, Yale University, PO Box 208109, New Haven, CT 06520-8109, USA
Accepted 1996 February 29. Received 1996 February 20; in original form 1995 October 20
SUMMARY
We report a method to calculate surface waves in layered anisotropic media using a
reflectivity algorithm. The method posits a finite stack of constant anisotropic layers
over an isotropic half-space. The parametrization within each layer allows for anisotropy
with magnitude scaled by three
with an arbitrarily oriented axis of symmetry
constants related to wave-speed variations of P and S waves with respect to the
Within each layer, eigensolutions to the equations of motion
azimuth relative to
can be expressed as upgoing and downgoing oscillatory and/or evanescent plane waves,
except possibly within narrow intervals of horizontal slowness p where the governing
matrix becomes near-defective and an alternative set of eigensolutions is used to avoid
numerical instability. Synthetic seismograms for simple anisotropic crustal models
demonstrate that significant Love-Rayleigh coupling occurs between adjacent overtone
branches, leading to significant ‘scattering’ in the absence of 3-D velocity structure.
can cause the LoveThe dependence of this scattering on azimuth relative to
Rayleigh scattering to appear to be related to a spurious component of the source
mechanism.
+,
+.
+
Key words: anisotropy, surface waves, synthetic seismograms, wave propagation.
INTRODUCTION
Surface waves confined to the Earth’s crustal waveguide
typically exhibit strong scattering effects, most notably where
explosive sources generate large amplitudes on the transverse
component. Since such scattering does not occur for surface
waves in a laterally homogeneous isotropic earth model, efforts
at modelling have examined the effects of 3-D structure and
anisotropy. Large-amplitude small-scale structure seems to be
required if attention is restricted to 3-D isotropic models, and
methods based on phase-screens (Wu 1994; Fisk & McCartor
1993) and finite differences (Levander & Hollinger 1992) have
been applied, typically using a stochastically generated velocity
model. The justification for this approach is the large amount
of small-scale variation in rock composition, and properties
evident in outcrops (Goff, Hollinger & Levander 1994).
Significant amounts of elastic anisotropy (>20 per cent) are
evident in many crustal minerals, for example hornblendes
and micas (Babuska & Cara 1991), to which must be added
the velocity anisotropy associated with aligned cracks within
rocks (Crampin 1984), and the directional effects of small-scale
compositional layering on wave speed (Backus 1962). Using a
perturbation formalism, Maupin ( 1989, 1990) showed that
modest amounts of crustal anisotropy could generate significant Love-Rayleigh coupling, leading to large transverse
motion in an explosion-generated LB phase. The anisotropy
used by Maupin was laterally homogeneous, and much smaller
0 1996 RAS
than the 3-D isotropic perturbations needed to produce similar
wave-propagation effects.
Dispersion curves and synthetic seismograms for a 1-D
anisotropic medium need not be computed with a coupledmode perturbation algorithm, but can be generated with a
direct layered-structure reflectivity solution (Crampin 1970
Crampin & Taylor 1971; Booth & Crampin 1983a, b). The
models considered by Maupin ( 1990) involved either vertical
cracks or horizontal foliation planes, so the effects of tilted
structures were not considered. Evidence for upper-mantle
anisotropy with a tilted axis of symmetry has been cited in
P-traveltime residuals in the Adirondack region by Levin,
Lerner-Lam & Menke (1995) and Levin, Menke & LernerLam (1996), and might be expected in any crustal region that
has undergone extensive horizontal compression or extension.
Seismologists have lacked simple formulas to express the effects
of anisotropic structures on the stress-strain relation, and so
have often limited investigations to a subset of possible model
geometries. On a theoretical level, wave propagation in a
layered (i.e. laterally homogeneous) anisotropic structure has
been shown to enjoy many of the simplifying properties and
symmetries of wave propagation in isotropic layered structures
(Garmany 1983; Fryer & Frazer 1984, 1987; Frazer & Fryer
1989; Thomson, Clarke & Garmany 1986; Chapman 1994).
Reflectivity codes for anisotropic structures have been applied
at high frequencies to investigate the interaction of body waves
with crustal anisotropy (e.g. Mallick & Frazer 1991), and for
173
J. Park
174
guided waves within thin anisotropic layers (Lou & Crampin
1991). Unusual behaviour in the numerical solutions can occur
when the quasi-S waves in an anisotropic layer are evanescent,
which makes a surface-wave synthesis for layered anisotropic
media more difficult than a body-wave synthesis.
In the next section we express the equations of motion for
waves within an elastic anisotropic layer with an axis of
symmetry, arbitrarily oriented. We next outline modifications
to the eigenmode solver of Chen ( 1993) needed to incorporate
anisotropic effects, including those where the governing matrix
for plane-wave polarization within a layer becomes defective.
In the penultimate section we present formulas, valid in the far
field, to calculate synthetic seismograms in weakly anisotropic
structures, and give examples for simple layered anisotropic
crustal structures.
P L A N E WAVES IN ANISOTROPIC LAYERS
We express the interaction of upgoing and downgoing plane
waves in an anisotropic layered medium with an axis of
symmetry &. Elastic properties are constant within each of a
stack of layers over a uniform half-space, but the axis of
symmetry can vary among the layers. We express the elastic
properties as a function of depth as A(z), where Aijklis the
fourth-order stress-strain tensor. If the axis of symmetry is
horizontal, we can express the azimuthal dependence of the
squared P and SV velocities for horizontal propagation in
terms of the angle 5 from &, according to formulas similar to
the head-wave formulas of Backus (1965) and Park (1993):
+ B cos 25 + c cos 45,
p D 2 ( ( ) = D + E cos 25.
pa2(5) = A
(1)
If density perturbations are neglected, knowledge of A, B, C, D
and E is sufficient to determine the stress-strain tensor (Shearer
& Orcutt 1986). In an isotropic medium, B = C = E = 0 and
A = 1 + Zp and D = p, where A, p are the Lam6 parameters.
Park ( 1993) showed how these azimuthal relations generalize
to other orientations of &. We assume a flat earth, z = 0 at the
free surface, and z increasing downwards. We assume a planewave solution of the form U(x, t ) = u(x)ei(k'x-ot).The strain
tensor in a homogeneous layer is
1
e = - [Vu
2
i
+ ( V U ) ~ =] ;(k
@u
+ u @ k) ei(k'x-Wt).
(2)
I
In each layer, the elastic tensor can be expressed
h = AA,
+ BAB + CAC + DAD + EA,,
(3)
where
combination of anisotropic deviations from an isotropic reference model, each with its own axis of symmetry
This would
be useful for media with both oriented cracks and oriented
minerals, if the orientations differ. (Such a linear combination
can be carried through the derivations below.)
The defining properties of the anisotropic parameters B,
C and E can be confirmed by direct calculation using
the Christoffel equations. Assuming Cartesian coordinates
in a constant layer, the momentum equation becomes
pd:ui = Aijk[Uk,i[. Consider a plane-wave solution in the form
+.
u ( x , t ) = u,, ei*x-wf)
with k real-valued. The gradients are straightforward, and the
solution is an eigenvector of the Christoffel matrix C = 1; * A * t
according to the matrix equation
(k2C-p21).uo=0,
where k = Ikl and
t-~,.1;= (1;0 + + \;v 0 1;)cos 5 - t 0 ic,
t . A c - k = 867 @ & COS' 5
- 4 ( t @ & + & @ t)cos 5 + t @ t ,
t * A , , - k = I -1; @ k,
k - A , - k = 2& @ & + I cos 25
-2 ( t@
Ac = 8W @ W - I @ I ,
(4)
A D = (1 3 ) I @ I + (14)IOI - 2 1 0 1 ,
AE = 2[( 13)&
+ ( 14)&
-~
A B+
] AD,
where '0' is the tensor product operation, and W = + @ %-21,
where I is the identity tensor. The permutation ( i j ) indicates
the interchange of the ith and jth tensor indices, for example
{ ( 1 3 ) I O I}ijk[= & j a i l . An isotropic elastic tensor A(') contains
only terms proportional to the isotropic tensors AA and AD,
as neither depends on &. It is also possible to form a linear
+ + % @ 1;)cos 5 + 1; @ 1;.
Define Cartesian coordinates %, 9, 1. Specify a horizontally
propagating plane wave with wavenumber orientation & = 2.
If & = 2 cos 5 + 9 sin 5, the Christoffel matrix can be expressed
as
A
+ B cos 25 + C cos 41'
25
1
+ Csin45
1 .
- B sin 25 + C sin 45
2
D + C( 1 -COS 4 5 ) + ~
0
0
O
1
0
D + E cos 251
(7)
The particle motions of plane waves are the eigenvectors of C.
To first order, the wave speeds are the square roots of the
terms along the diagonal of C. Note that the P and SH
polarizations (u parallel to k and 1 x k, respectively) will couple
as a result of the off-diagonal terms. Assuming B, C << A, the
wave speed for the quasi-P wave is approximately
(A/p)"'
W @ I+ I @ W ,
6 = k/k. Assume that 1;* % = cos 5. Then
1 ; . A A 4= t @ t ,
AA= I @ I ,
AB=
(5)
(+
1
1B
cos 25 + - - cos 45
2A
2A
--
In the classical situation of head waves in marine refraction,
with 1; and & in the horizontal plane, this corresponds to the
well-known Backus formulas, with B/A and C/A scaling the
peak-to-peak relative azimuthal velocity variations. The SV
wave has particle motion parallel to 1. Assuming that E << D,
the SV wave speed is approximately
(D/p)"'
( + ;:
1
- -COS
4
-
.
The SH wave speed has a cos 45 dependence in this geometry.
0 1996 RAS, GJI 126, 173-183
175
Surface waves in layered anisotropic media
For shear-wave splitting at vertical incidence with horizontal
a = 2,
a - ic = cos 5
fTb)= Lux,u,,
=0
and
rD+E
I
c=i 0
Lo
0
D-E
0
K,, K,, K,I
eiwvz.
: 1.
4 v A :=
~ [ ( k . u)(A- B + C - 2 0 + 2 E )
(8)
A-B+C
+ VPS + V ~ R ) - U= 0 ,
(9)
where
T=(A - B
u,,
The traction components can be expressed as a linear
combination of the vectors 2, a, k and u:
The particle motions are uncoupled, obtaining the maximum
difference in S wave speed. For weak anisotropy, the total
relative range in S wave speed is approximately E/D. For
oblique incidence in shear-wave splitting, again assuming a
horizontal axis of symmetry, the fast S polarization will mix
with the P polarization, and suffer a cos 49 velocity perturbation from the C term. Similar behaviour occurs for vertical
incidence if the axis of symmetry is tilted from the horizontal
plane.
P, SL' and S H motions can couple at all interfaces in a
layered anisotropic structure with general axis of symmetry G.
All interacting upgoing and downgoing plane waves share the
same horizontal wavenumber. Let k/w = pf + v2, where p and
v are the horizontal and vertical slownesses, respectively. In a
general anisotropic medium, there are three upgoing and three
downgoing plane waves. [If the axis of symmetry is tilted, it
is possible for a downgoing wave to possess an 'upgoing' phase
velocity, and vice versa-see Germany (1983).] For fixed p ,
the Christoffel relations lead to six possible solutions for v, as
determined by the quadratic eigenvalue problem
( P ~ -PI
T
the propagation factor exp[i(k . x - O X ) ] implicit,
+ C - D + E)f 0f + (D + ( 2 ~ 2 -1)E)I
+ (8w2C+ 2E)G 0G + ( B - 4C 2E)wx(G02 + f 0+),
S = ( A - B + C - D + E)(f 04+ 2 0%)
+ ~w,w,EI
+ l6w,w2Cfi @ vir + ( B - 4C - 2E)
(10)
x [w,(* 0f + a 0G)+ w,(G 04 + 2 03 1
R = ( A -B + C - D + E)4@ 1 + ( D + ( 2 ~ 2 1)E)I
+(8w;C + 2E)G 0G + ( B -4C 2E)w,(G 04+ 2 0G).
-
+ ( k . G)(u*G)(B- 4C - 4E)]2
+[~,(D-E)+2(u*+)w,E]k
+ [wv(D- E ) + 2(k.+)w,E]u
+ { ( k u)w,(B - 4C - 4E) + 8(k %)(u
+2[(k - G)uz+ wv(u fi)]E}fi,
(12)
+)w, C
*
where the phase factor i exp[i(k * x - wt)] is implicit.
In anisotropic media, the group velocity of a plane wave is
not necessarily parallel to the wavenumber vector k. This also
holds true for trapped resonances in the surface waveguide.
For isotropic layered media in the far field, an accurate
synthetic seismogram can be constructed from a single integral
over horizontal wavenumber. Because the phase- and groupvelocity directions differ, an integral over a 2-D azimuthal
range of horizontal wavenumber is necessary to represent
surface-wave motion in anisotropic media accurately. Nolte,
Frazer & Mallick (1992) showed that a single integration is
adequate for body-wave synthesis in 'weak' anisotropy-their
test example possessed roughly 6 per cent azimuthal variation
in S wave speed. Although we focus on weak anisotropy in
this report, this is not appropriate in many situations of
interest. Therefore, the more general case will be considered
briefly.
Consider a trapped resonance in k homogeneous layers over
a half-space, composed of plane waves with wavenumber
vectors of the form k = k,f k,Y + wv2. The horizontal wavenumbers k,, k, are common to all plane waves and the vertical
slownesses v differ. The Lagrangian of motion can be expressed
as
+
B = ( w 2 1 1 - k ~ I , , - k ~ I v y - k , k y I x y - k , l x z - k y I y z - I,,),
(13)
2
-
-
Define
= R-'
- (p2T -PI)
(T + v s + ?I)- u = 0. This
s
-
and
= pR-' S to obtain
3 x 3 system can be solved with
the 6 x 6 system
Solving (11) will obtain three upgoing and three downgoing waves, aside from exceptional cases, for example if p
equals the P or S slowness in a layer and the associated wave
travels horizontally. (In this case the matrix system is formally
defective.) To solve the equation of motion in a stack of
constant-property layers, one matches boundary conditions
for traction and displacement continuity for these six waves at
each interface. We define the stress-displacement vector f (z)
for each plane-wave polarization associated with an eigenvector u and eigenvalue v of the matrix system (9). Leaving
0 1996 RAS, G J I 126, 173-183
where the I,, are integrals from z = 0 to z = cc involving the
particle motion u(x, t ) , density p and the stress-strain tensor
A. The horizontal wavenumber k , = 0 for the solutions in the
last section, but the first variation with respect to k, is not
generally zero for layered structures in which P-SV and S H
motions couple. Because the Lagrangian is stationary to small
perturbations in the model solutions, the group velocity can
be found by taking the first variation of 9 with respect to w,
k, and k,, and considering the two cases Sk, = 0 and Sky = 0:
where c = w/k, is the phase velocity. We express u(x) as a sum
of upgoing and downgoing plane waves, because the integral
between any two of the six allowed waves in the kth layer
can be evaluated analytically. Consider two plane waves
uexp[i(k-x-wt)], u'exp[i(k.x-ot)], where u,u'areconstant vectors, k = k,f kyY wv2 and k = k,P + k,jr wv'2.
+
+
+
J. Park
176
The kinetic energy integral I, for these waves within the kth
layer is
-
p(u* u’)exp[iw(v‘ - v*)z] dz
=p(u*.u‘)f(w,
(15)
v, V’,=k-lrZk),
where p is the density within the kth layer and
v, v’, zk-l,
-
it
zk)
exp[iw(v’ - v*)zk] - exp[io(v’ - v*)zk-,]
2iw(v‘ - v*)
(=k
if v* # v’,
if v* = v’.
- zk- 1)/2
(16)
The asterisks denote complex conjugation. The integral in the
half-space extends from zK to co, where zk is the depth of the
top of the half-space. All plane-wave interaction terms possess
a factor of f with suitable parameters.
The potential-energy integrals I,, in (14) are evaluated
by calculating the plane-wave interaction inner products
Vu* : A :Vu‘ term by term. Each term is of the following form,
with the factor f ( w , v, v‘, z k - , , z k ) implicit:
I,, + ( A - B + C - D
+ E)u:u: + [D + ( 2 ~ : - l)E](u* * u ’ )
+(8wpC+ 2 E ) ( 3 7 - u * ) ( \ i * u r )
+(B-~C-~E)W,[U:(\;~.U’)+(*.U*)U:]
I,, - - + ( A- B + C - 2 0 + ~E)(v*u:u:
+ 2Ew,w,(v* + v’)(u*
+ ( D -E)(v*u:u:
*
+ w’u:u~)
u’)+ 8Cw,w,(v*
+ w’)(*-
-
u*)(+ u’)
+ v’u;u:)+ ~E[(v’u,*w,+v*u:w,)(**u’)
+ v’u:w,)] + ( B - 4 c -4E)
+ (**u*)(v*u:w,
x [(v‘u: w, + v* u; w,)(% - u’)+ (* u*)(v*u:w, + v’u:w,)]
*
I,,-+(A-B+ C-D+E)(U:U:+U:U;)+~EW,W,,(U*.U’)
+ 16Cw,w,(Vi*u*)(B~u‘)
+(B-~C-~E)[(U:W,+U:W,)(*.U’)
+(*~u*)(u:w,+u;w,)]
I,,
( A - B + C - 2 0 + ~E)(v*u:u;
-i
+ V’U:U:)
ues occur in conjugate pairs, so the cohorts of upgoing and
downgoing evanescent waves are equal. Near the turning
points for propagating waves, however, it is possible for pairs
of real eigenvalues to be both positive or both negative, and
hence represent pairs of either both upgoing or both downgoing
waves. (Since the group velocity need not be aligned with the
wavenumber vector, the directions of energy transport need
not be the same.) To calculate the generalized reflection and
transmission coefficients in the next section, the upgoing and
downgoing cohorts of plane waves must each have three waves.
When the real eigenvalues of 9 do not divide into equal
cohorts, we remedy this by switching the identification of the
plane wave(s) with the smallest real eigenvalue(s), ‘upgoing’ to
‘downgoing’ or vice versa.
Another peculiarity of anisotropic wave propagation is that
the matrix .T can be formally defective at isolated, and somewhat unpredictable, values of horizontal slowness p . A matrix
is defective if it has a repeated root in its determinant, but
only has a single associated eigenvector. If a matrix is nearly
defective, it possesses two closely spaced eigenvalues whose
eigenvectors are nearly parallel. For isotropic media, 9 is
defective at the turning points of P and S, where upgoing and
downgoing plane waves both travel horizontally. Because
reflectivity techniques typically manipulate upgoing and downgoing stressdisplacement vectors in a layer separately (see the
next section), this instance of a defective matrix does not lead
to numerical problems. In constant anisotropic layers, however,
pairs of upgoing (or downgoing) evanescent plane waves,
represented by eigenvectors of 9,
can have parallel or nearly
parallel particle motions. In practice, with double-precision
arithmetic, this leads to numerical problems if the condition
number of the two eigenvectors exceeds 10’. Although such
behaviour is typically restricted to narrow intervals of horizontal slowness, it can sabotage root-finding algorithms that
determine dispersion curves.
In the slowness interval where one of the layer matrices
9 is near-defective, we generalize our ‘plane-wave’ solutions
to avoid numerical instability. At the value po where F is
defective, there is a repeated eigenvalue vo but only a single
polarization vector uo that satisfies (1 1). Analogous to methods
used with systems of linear ordinary differential equations
(Finney & Ostberg 1976), we seek solutions to (9) in the form
-
+ vo(z
t)] ,
+ 2Ew,w,(v* + v’)(u* * u ‘ ) + 8Cw,w,(v* + v’)(*-u*)($i..u’)
+ ( D - E)(v*u:u: + V ’ U ~ U ; ) + ~E[(v’u,*w,+v*u;w~)(& u’)
U(x, t ) = uo exp[io(p,x
+(**u*)(v*u:w,+
where Z is a reference depth, either at the top or the bottom
of the layer. The polarization vector u, satisfies
6 ( x , t ) = [u,
- 5) -
+ ( z - Z”)U~]exp[iw(p,x + vo(z - z“)
-
t)] ,
(18)
*
x [(v’u;w,
v’u;w,)] +(B-4C-4E)
+ V*U,*W,)(*’U’) + (a.u*)(v*u;w, + v’u:w,)]
(17)
Note that I,, and I,, vanish for eigenmodes of an isotropic layered structure, for which particle displacement divides
into either S H ( u , = u , = O ) or P-SV (u,=O) motion, and
B = C = E = 0. For the surface-wave modes discussed in the
next section, the maximum deflection of group velocity, relative
to the horizontal wavenumber vector, is roughly 1”.
If the axis of symmetry
is tilted between the horizontal
and the vertical within a layer, there is an interval of horizontal
slowness p in which the six eigenvectors of 9 do not neatly
divide into three upgoing and three downgoing vertical slownesses. Since 9 is real-valued, any complex-valued eigenval-
*
(T + v , s
+ v;I).
u, =
i
-(s
+ 2v;I).u0.
(19)
0
The vector u1is indeterminate by a linear multiple of uo. Once
this multiple is chosen, the linear combination of the solutions
( 18) is uniquely determined by the boundary conditions of
the problem.
In practice, numerical round-off transforms a defective matrix
.T into a nearly defective matrix with closely spaced eigenvalues
v + , v - , so the solutions (18) are typically approximate. In the
interval of slowness p where the condition number of the two
near-parallel polarization vectors u+, u- is high ( 2lo’), we
use the defective matrix solutions by setting v o = ( v + v-)/2
+
0 1996 RAS, GJI 126, 173-183
Surface waves in layered anisotropic media
and solving
(T+ ~ 0 +9 v ~ I u0
) = B * u0
*
0
(20)
by identifying uo with the smallest eigenvalue 1 of the matrix
B. Because vo is fixed and the eigenproblem (B - 21) uo linear
rather than quadratic, this eigenvalue is isolated and uo can
be determined stably. We then solve for u1 using (19).
To calculate the stress-displacement vector associated with
6,we note the extra terms in its vector gradient:
+
V 6 = [ik @
( ~ 1I
+(Z - i)uO)
2 0 uO]
x exp[iw(p,x
+ v&
- 2) - t)] ,
(21)
where ik = iw(po%+ vo2). At the reference depth z" the traction
associated with 6 involves two terms that can be calculated
from (12), ope using ik, u1 in place of ik, u, respectively, the
other using 2, uo in place of ik, u. At z # 5, this expression
incorporates a linear multiple of the traction associated with
the true plane-wave solution U. We show in the next section
that the propagation of this not-quite plane-wave solution
away from its reference depth z" can be accommodated with
minor adjustments to the reflectivity algorithm. Similarly,
the terms of V 6 that involve uo lead to extra terms in
formula (17) for group velocity. These correction terms are
straightforward variants of (17), but lengthy, so we omit them.
H Y B R I D S U R F A C E WAVES
Using displacements and tractions derived from (11) and (12),
we can set up stress-displacement vectors for upgoing and
downgoing plane waves within each anisotropic layer. We
obtain surface-wave modes by forcing an evanescent condition
in the half-space, propagating the stress-displacement vector
to the free surface, and enforcing a zero-traction condition.
There are several schemes for doing this. We have adapted the
computational algorithm of Chen (1993), which forces all
evanescent waves in the recursive algorithm of Kennett (1983)
to 'decay' away from interfaces, thereby decreasing round-off
error. At a fixed frequency, the algorithm searches the allowable
interval of horizontal slowness for roots to the secular function
that describes free-surface traction. Aside from cases where a
layer matrix Y is nearly defective, the computational formulas
are quite similar to the Rayleigh-wave formulas of Chen (1993),
except that the reflection and transmission matrices are 3 x 3,
rather than 2 x 2. However, because the near-defective case
requires modification of the algorithm, we outline it briefly.
Consider a stack of K elastic anisotropic layers over an
isotropic half-space. Beneath the free surface z = zo = 0, the
bottom of the kth layer is at depth zk. The elastic properties
of the kth layer are determined by an axis of symmetry i%k and
Assume oscillatory
elastic parameters A(k),B(k),C(k),lYk),
dependence exp [iw(px - t ) ] at a fixed slowness p and frequency
w . Within the kth layer, the stress-displacement vector f'k)(z)
of wave motion, where
uy(z), U A 4 , T z m TrY(Z)>T,,(z)l >
can be expressed as a linear combination of stress-displacement
vectors associated with the solutions of (11). To promote
numerical stability, the exponential factors of 'upgoing' solutions in the kth layer are referenced to zk,that is, they have
a z dependence proportional to exp[iwv(z - zk)] and, in the
near-defective case, to (z - zk) exp[iov(z - zk)]. The exponential
factors of 'downgoing' solutions are referenced to zk-1. Only
177
downgoing evanescent waves are considered in the half-space,
referenced to depth zK. If E(k)is a matrix whose columns are
the stress-displacement vectors of the solutions of ( l l ) , and
c is a vector that defines the linear combination of these solutions within the layer, referenced to either its upper or lower
boundaries, then
(22)
f'k)(z)=E(k).r(k)(z).C(k)
for zk- < z < zk. w e group the three 'downgoing' waves in the
first three columns of E(k),and the three 'upgoing' waves in
the last three columns. Away from slowness intervals where
.T(k)
is near-defective, the matrix Ytk)is diagonal:
y(k)(z)
= diag {eivl(z- z k - d
eiv2(z -zk-
eiv5(z-zk),eiv6(z-zk) } .
eiv3(z-zk-l)
eiv4(z
-Ik)
(23)
In cases where the near-defective solution is appropriate,
Y(k)has off-diagonal terms. If the lth and mth downgoing
solutions correspond to U, 6 in (18), respectively, the relevant
components of Y(') are
yjf)= y ( rnm
k ) = eivo(~-zk--l)
y!:
= (z- z k - l )
eivo(z-zk-l)
(24)
r$j=o.
The off-diagonal component governs the propagation of the
second term of the solution 6 in (18). Similar relations govern
the case where two upgoing solutions are of the form (18).
The partition of solutions of (11) into upgoing and downgoing cohorts motivates a partitioning of E(k)and YS into
3 x 3 submatrices and dk)into 3 x 1 subvectors to distinguish
reflection and transmission effects and to separate stress
and displacement behaviour. Because the stress4isplacement
vector must be continuous at layer interfaces, at Zk we have
the relation
E(k) E(k)
[Eii E;i][
Yf'(zk) 0
0
I][::]
where the subscripts '8 and 'u' denote downgoing and upgoing,
respectively. The sum of stress-displacement vectors that comprises this boundary condition can be rearranged into a cohort
of waves that approach the interface at zk, and a cohort of
waves that depart from it. This rearrangement can be used to
obtain a simple expression for the reflection and transmission
of incoming waves at the interface:
f(k)T(Z)
= CUX(Z),
0 1996 RAS, GJI 126, 173-183
From here, the recursive calculation of generalized reflection
and transmission coefficient submatrices fik),
Rfd follows
Chen (1993) and Kennett (1983) closely. By definition, these
generalized coefficients incorporate the reverberation within
178
J. Park
the layered structure, so the amplitudes of waves that leave an
interface can be expressed in terms of downgoing waves in the
layer above:
c&k + 1) = .f.W c&k) ,
,.;k) = & ( k ) .C ( k )
du
d ’
(27)
.
Table 1. Isotropic reference crustal model, adapted from Kanamori
& Hadley’s (1975) P-wave model for southern California.
Layer No.
Top (km)
1
2
3
4
0.0
4.0
27.4
32.4
Only downgoing evanescent solutions are allowed in the halfspace, so c:’)=O at the deepest interface z = z K , and (26)
reduces so that TiK)= TiK)and Rst)= RS:). From this starting
point we step upwards through the interfaces, combining (26)
and (27) to calculate
--
(a)
CI
(m s-l)
p (m s-’)
p (kg m - 3 )
5550
6300
6800
7800
3175
3637
3925
4500
2600
2800
2900
3200
Dispersion: Isotropic Crust
m
To enforce the traction-free boundary condition at the free
surface, we note that an application of (25) at the free surface
z,, = 0 leads to the relation
0 = ESl] .cil)+ E$Y.yII)(O) .~ ( 1 )
(29)
or, equivalently,
cis)= -(E$:))-I .E$Y. ~ ~ ~ ) ~ O ) . c (ud~ ~ =u R. ( o ) . c ( ~ ) (30)
By combining this with (27), one obtains cg) = Rfd Ri! ci’).
-
2
4.4
x
v
2
4> 3.6
3.2
L2
-
This has a solution only if the secular determinant vanishes,
1.e.
det(I-Rf~.&&~)=detS=O
(31)
and c6’) is a right-eigenvector of S with zero eigenvalue. The
algorithm ‘finds’ a zero of (31) when the determinant is less
than some tolerance (typically
We compute ci’) as the
eigenvector of the smallest eigenvalue of S. Once this is determined, the remaining coefficient vectors are found recursively
with (27).
The secular function that arises from the free-surface condition has closely spaced zeroes where hybrid Love and
Rayleigh waves have nearly identical phase velocities. In
addition, the secular function has a zero at the p that corresponds to a horizontally propagating wave in the top layer,
and occasional near-zeroes at other phase velocities. We found
it easiest to cull these spurious solutions after running a rootfinder over a frequency interval, saving the culled discretized
dispersion curves to generate resonant wave solutions for
synthetic seismograms. We used an extrapolation formula for
the dispersion curves to predict successive values of horizontal
phase velocity c,(w) of the nth overtone branch for equally
spaced frequencies:
where U , is the group velocity of the nth overtone surface
wave. In this manner the search for secular-function roots can
be pre-restricted to narrow intervals of slowness.
We adapted the P-velocity model of Kanamori 8z Hadley
(1975) for the southern California crust as our isotropic
reference. We scaled the reference shear velocity by assuming
a Poisson solid, and chose densities appropriate for crustal
rocks (Table 1). Dispersion curves for the isotropic crustal
model (Fig. 1) separate into paired Love-Rayleigh branches.
The fundamental Love and Rayleigh curves are well-separated
in phase and group velocity because the Rayleigh fundamental
can be evanescent in the uppermost layer. Love and Rayleigh
overtone branches are principally shear waves trapped in the
crustal waveguide with S H and SV polarizations, respectively.
4.0
0.0
(b)
h
0.1
0.2
0.3
0.4
Frequency (Hz)
0.5
Dispersion: Isotropic Crust
r
0.0
0.1
0.2
0.3
0.4
Frequency (Hz)
0.5
Figure 1. Dispersion characteristics of surface waves for the isotropic
crustal structure given in Table 1. The fundamental and five overtone
branches are present at f = 0.5 Hz for both Rayleigh- and Lovepolarized motion. (a) Phase velocity c = w/k, versus frequency 1.
(b) Group velocity U = dw/dk, versus frequency f .
Each overtone pair has similar dispersion, especially as the
frequency increases. This similarity tends to favour the coupling
of Love and Rayleigh overtone branches by anisotropy in the
crust. We calculated eigenmodes with 0 < f 0.5 Hz, with a
frequency spacing of 0.0005 Hz. With Fortran optimization,
this required 15-20 min on a Sparc 10 workstation.
Consider anisotropy in the lowermost 5 km of the crust with
a horizontal symmetry axis angled 45” to the direction of
surface-wave phase velocity f. This ‘Model A’ is consistent
with a lateral shear zone at the base of the crust, and we
specify B/A = 0.06, C / A = 0.00, E / D = 0.03, consistent with
sheared ultramafic rocks. The dispersion for this model is
similar to the dispersion of the reference model, but note the
intertwined group-velocity curves of the first and second LoveRayleigh overtone pairs (Fig. 2). The additional points of
intersection occur where the phase velocities of the Love and
Rayleigh overtones in the isotropic model nearly coincide,
becoming quasi-degenerate. If anisotropy is specified in the
upper and middle crust of our reference model (hereafter
‘Model B’), the coupling becomes stronger, and the sinuous
0 1996 RAS, GJI 126, 173-183
Suvface waves in layered anisotropic media
(a) Dispersion: Lowermost Crustal Anisotropy
h
8 4.4
v
2
(a) Dispersion: Mid and Upper Crustal Anisotropy
2s 4.4
h
v
4.0
R
179
p,
%
5 3.6
4.0
2 3.6
$
f 3.2
#
;3.2
cl
if!
-
a
5.0
0.1
0.3
0.4
F r e q u e n c y (Hz)
0.2
0.5
0.0
0.1
0.2
0.3
0.4
0.5
Frequency (Hz)
-
(b) Single Slowness Integral Approximation
(b) Single Slowness Integral Approximation
h
0.0
0.1
0.3
0.4
Frequency (Hz)
0.2
0.5
Figure 2. Dispersion characteristics of surface waves for anisotropic
crustal model A, obtained by specifying B/A = 0.06, C/A = 0.00,
E/D=O.O3 in layer 3 of the isotropic model in Table 1, with axis of
symmetry \ir horizontal and oriented 45" counterclockwise from %.
The fundamental and five overtone branches are present at f = 0.5 Hz
for both Rayleigh- and Love-polarized motion. (a) Phase velocity
c = w/k, versus frequency f . (b) Group velocity U, = dw/dk, versus
frequency f . The misalignment of group and phase velocities for these
modes, represented by non-zero U,=dw/dk,, is no more than 1.5",
and typically much smaller.
group-velocity curves of the first and second Love-Rayleigh
overtone pairs become more tangled (Fig. 3). Model B has
B/A = 0.04, C / A = 0.00, E / D = 0.02 from the surface to 27.4 km
depth, with i% at 45" strike to 8, the direction of phase
propagation. In the mid-crust, the axis of symmetry i% is
horizontal. In the uppermost 4 km we specify i%to tilt 30"
above the horizontal. Such anisotropy might be appropriate
for a crustal block undergoing extension, with tilted shear
from low-angle normal faulting near the surface.
The extensive hybridization of the overtone surface waves is
evident in Figs 4 and 5, where the size of the SH-polarized
component at the free surface z = 0 is plotted, relative to the
total amplitude. In an isotropic crust, this quantity would be
unity or zero, depending on whether the mode was Love or
Rayleigh, respectively. In an anisotropic crust, most modes
can be identified as quasi-Love or quasi-Rayleigh, depending
on the.dominant amplitude fraction. [Park & Yu (1992) define
'quasi-Love' mantle waves in long-period data to describe
Love-to-Rayleigh scattered waves that arrive on the vertical
component in the Love-wave arrival window. In this context,
we use the alternative definition of a quasi-Love wave as an
eigenmode of the equation of motion with dominantly Love
polarization-e.g. Maupin ( 1989).] The fundamental dispersion branches couple only weakly for these models. For an
overtone quasi-Love surface wave, the SH amplitude fraction
is close to unity in most cases. The SH amplitude of an
0 1996 RAS, G J I 126, 173-183
0.0
0.1
0.2
0.3
0.4
Frequency (Hz)
0.5
Figure 3. Dispersion characteristics of surface waves for anisotropic
crustal model B, obtained by specifying B/A = 0.04, C/A = 0.00,
E/D = 0.02 in layers 1 and 2 of the isotropic model in Table 1. The
axis of symmetry \ir is horizontal and oriented 45" counterclockwise
from % in layer 2, and tilted 30" above the horizontal at the same
azimuth in layer 1. The fundamental and five overtone branches are
present at f = 0.5 Hz for both Rayleigh- and Love-polarized motion.
(a) Phase velocity c = w/k, versus frequency f . (b) Group velocity
U,= dw/dk, versus frequency f .The misalignment of group and phase
velocities for these modes, represented by non-zero U , = dw/dk,, is no
more than 1.5", and typically much smaller.
SH Amplitude: Lowermost Crustal Anisotropy
5 1.0
Ti
0.8
0.6
0.4
0.2
4
0.0
a
0.0
0.2
0.4
Frequency (Hz)
Figure 4. For anisotropic Model A, the relative size of the SH
component of motion for the surface-wave dispersion branches graphed
in Fig. 2, expressed as the amplitude ratio at the free surface
lu,(O)l/lu(O)l. This ratio is close to unity for Love-dominant, or 'quasiLove' modes, and hovers near 0.2 for Rayleigh-dominated overtone
modes. The dashed lines correspond to the fundamental dispersion
branches, which couple weakly.
180
a
a
J . Park
SH Amplitude: Mid and Upper Crustal Anisotropy
where c,, U , , k, and I?) are the phase velocity, group velocity,
horizontal wavenumber and kinetic energy functional of the
nth surface-wave mode, respectively. The sum extends over all
surface-wave modes at frequency w. The complex conjugate of
u,(h) is used as an ‘adjoint’ particle motion to transform a
point-force vector F to a basis set of eigensolution particle
motions.
For a point force F(w), seismic excitation at frequency w is
F ( w ) G(x, hL, w). Particle motion in the time domain can be
obtained by performing an inverse Fourier transform. For a
moment-tensor point source M(w) at depth h, r = 0, the
expression for particle motion can be obtained from the formula
M(w):V’G(x, hL, w). where V‘ denotes a gradient taken with
respect to source coordinates (x’,y’, 2’). The partial derivatives
of r with respect to source coordinates are drlax’ = - 1, and
&lay‘ = arpz‘ = 0. Neglecting the derivative of (2/nk,r)’/’ in
the far-field limit,
-
,,-...___
________ ____ ____ ____ ____________.-----....................... ----___
I
~
I
a
0.0
~
I
I
I
0.2
I
0.4
Frequency (Hz)
Figure5. For anisotropic Model B, the relative size of the S H
component of motion for the surface-wave dispersion branches graphed
in Fig. 3, expressed as the amplitude ratio at the free surface
lu,(O)l/lu(O)/.This ratio is close to unity for Love-dominant, or ‘quasiLove’ modes, and hovers near 0.2 for Rayleigh-dominated overtone
modes. The dashed lines correspond to the fundamental dispersion
branches, which couple less strongly.
overtone quasi-Rayleigh wave is typically 15-25 per cent for
Models A and B, and much larger for selected branches in
selected frequency intervals. Because a similar hybridization
occurs at the depth of source excitation, substantial LoveRayleigh mixing can be expected in synthetic seismograms.
@ u,( 0)
(
L)l’’
nk,r
exp [i ( k , r
+ :)
].
( 34)
The z-derivative of u,*(h)can be assembled as a sum of planewave terms. In order to express the seismograms in terms of
radial, transverse and vertical components, we switch the signs
of the y and z components of
and u,(O) in (34).
An important consequence of Love-Rayleigh coupling in an
anisotropic layered structure is the presence of either Love or
Rayleigh waves in unexpected sourcereceiver geometries.
Fig. 6 graphs a three-component synthetic seismogram at the
Rayleigh radiation node of a strike-slip source at 15 km depth,
using Model A with anisotropy in the lowermost crust. In the
isotropic reference model, the motion is Love-polarization
SYNTHETIC SEISMOGRAMS
We make several approximations to derive a compact computational algorithm for synthetic seismograms from summed
surface-wave resonances. To accommodate point momenttensor sources, we switch from Cartesian (x,y, z ) to cylindrical
(r, 4, z ) coordinates. To keep the formulas compact, we treat only
the case where q5 = 0, so that f = 2 and 4 = 9. More general
sourcereceiver geometries can be obtained by coordinate
rotations. We assume weak anisotropy, so that synthetics can
be generated with reasonable accuracy with a single slowness
integration in the radial coordinate (Nolte et al. 1992). We
seek far-field asymptotic forms of the particle motion, in
which the Bessel-function expressions of the exact solution
are replaced by a decaying phase-advanced cylindrical wave.
Neglecting terms that involve larger powers of ( l l r ) , the
surface-mode wavetrain at frequency w can be expressed in
terms of a Green’s function G(x, x‘, w ) for motion at the surface
z = 0 for a point-force vector F at r’ = 0, z’ = h (Aki & Richards
1980 Kennett 1983). We generalize the expression to allow
complex-valued displacement in the nth surface-wave mode
u,(z), with the understanding that the imaginary parts
correspond to phase lags in the time domain:
Anisotropic Model A: 15-km Strike Slip Source
vertical
80
0
-32 60
v
Y
E
40
8a
-& 20
.r.
n
0
80
120
160
200
Time (seconds)
Figure6. Surface-wave particle motion for Model A for a station
500 km distant from a strike-slip source at 15 km depth with strike
parallel to the source-receiver path, i.e. at the nominal Rayleigh
radiation node. Note the particle motion on the vertical component,
which arrives in the Love-wave window, but is poorly coherent with the
transverse-component motion. The moment tensor is M , = 10’’ N m,
all other M,, = 0. Particle motion is expressed in microns.
0 1996 RAS, G J I 126, 173-183
Surface waves in layered anisotropic media
only, restricted to the transverse component. In the anisotropic
model, motion occurs on the radial and vertical components
as well, with amplitude 5-10 per cent that of the main Love
wave. Note as well that the motion on the vertical and
transverse components is only weakly coherent, which suggests
two distinct surface-wave arrivals, rather than a perturbation
to the Love polarization. At periods T > 100 s, Park & Yu
(19931 and Yu & Park (1994) attributed similar behaviour
tc fundamental-mode Love-to-Rayleigh scattering caused by
lateral gradients of upper-mantle anisotropy associated with
regional deformation. At short periods (< 10 s), coupling occurs
without lateral structure, primarily among overtone-branch
surface waves with near-equal dispersion characteristics.
Because the dispersions of the Love and Rayleigh fundamental
surface waves differ greatly for the crust, their coupling is weak
in the absence of lateral structure.
For Model B with anisotropy through most of the crust and
a tilted axis of symmetry in the upper 4 km, polarization
distortion of surface waves is evident. Fig. 7 shows synthetics
for a shallow ( 3 km) 45" thrust, with slip motion parallel to the
sourcereceiver path: M,, = 1, M,, = - 1, M,, = 0 otherwise.
In addition to the Love wave that precedes the fundamental
Rayleigh wave on the transverse component, the anisotropy
also generates a transverse component to the Rayleigh fundamental that coheres with the vertical component, tilting the
particle-motion ellipse from the vertical.
Love waves can be observed from explosive sources, in the
absence of identifiable 3-D scatterers. Figs 8 and 9 show the
response of Models A and B to a shallow (1 km) explosive
source at A = 800 km. The motion is dominated by the fundamental Rayleigh wave, carrying energy primarily trapped in
the upper crustal layer. Aside from timing shifts associated
with the azimuthal velocity variations, the Rayleigh waveforms
are visually identical among the isotropic and anisotropic
Anisotropic Model B: 3-km Thrust Source
120 t
h
'a
I
Y
8
E
E
V
40
.
I
Y
20
E
a
1
transverse
rn
s
- 10
I
I
160
I
I
200
I
240
I
I
280
I
I
320
Time (seconds)
Figure 8. Surface-wave particle motion for anisotropic Model A for
a station 800 km distant from an explosive source at 1 km depth, for
which the nominal Love-wave radiation is zero. The moment tensor
is M,, = M y , = M,, = lot7N m, all other May= 0. Particle motion is
expressed in microns.
Anisotropic Model B: 1-km Explosive Source
I
3
radial
160
V
W
m
a - 10
A I\ II
vertical
Anisotropic Model A: 1-km Explosive Source
Q)
100
80 -
181
60
200
240
280
320
Time (seconds)
40 -
8m 20 B o-
Figure 9. Surface-wave particle motion for anisotropic Model B for
a station 800 km distant from an explosive source at 1 km depth, for
which the nominal Love-wave radiation is zero. Note the particle
motion on the transverse component, which comprises both a
'scattered Love wave and the expression of a tilted Rayleigh
ellipse, coherent with the vertical component. The moment tensor is
M,, = M,,, = M,, = lOI7 N m, all other May= 0. Particle motion is
expressed in microns.
transverse
CI
s -20 80
radial
120
160
200
Time (seconds)
Figure7. Surface-wave particle motion for Model B for a station
500 km distant from a 45" thrust source at 3 km depth, with strike
perpendicular to the source-receiver path and slip parallel to it, i.e. at
the nominal Love radiation node. Note the particle motion on the
transverse component, which comprises both a 'scattered Love-wave
and the expression of a tilted Rayleigh ellipse, coherent with the
vertical component. The moment tensor is M,, = -M,, = 10'' N m,
all other M,, = 0. Particle motion is expressed in microns.
0 1996 RAS, GJI 126, 173-183
layered models. However, a significant Love wave is generated
on the transverse component, even for anisotropy restricted to
the base of the crust. Its arrival window does not match either
the Rayleigh fundamental or overtone arrivals. The amplitude
of an explosion-generated Love wave depends on the strike of
the axis of symmetry relative to the propagation path (Fig. lo),
with minimal Love-Rayleigh coupling for \;v parallel (0" in the
figure) and perpendicular (*90") to the path. Coupling for
f90" arises from the tilt of \;v out of the horizontal plane in
J. Park
182
Model B: Coupling vs. Propagation Azimuth
6o
1
I
90"
67.5"
a
0
a
a
45"
20
0
20"
0"
-20"
II
$ -20
-45"
k
-60
cause the Love-Rayleigh scattering to resemble a spurious
addition to the source mechanism.
We have developed and implemented a reflectivity-based
algorithm to calculate surface-wave motion in a layered anisotropic structure, in which the anisotropy can be expressed as
a deviation from isotropy with a particular axis of symmetry,
or a linear combination of such deviations. In some cases
where both upgoing (and/or both downgoing) quasi-shear plane
waves are evanescent within a layer, their particle-motion vectors
can become nearly parallel, leading to numerical instability.
This can sabotage automatic root-finding algorithms for the
surface-wave dispersion curves. An alternative solution for
propagating waves within the layer can be used which removes
this instability.
-90"
I
160
I
I
I
I
I
I
200
240
280
Time (seconds)
I
I
320
Figure 10. Transverse-component particle motion for Model B for an
isotropic source at 1 km depth, at a range of propagation azimuths,
ranging from -90" to 90" relative to the projection of \iv on the
horizontal plane. Note the azimuthal variation of the 'scattered' Lovewave amplitude, which resembles that of a thrust earthquake.
Model B-note that the transverse-component motion in the
fundamental-Rayleigh arrival window is greatest for this case.
Park (1993) showed that the dependence of coupling on
propagation azimuth differed for the anisotropic parameters
B, C and E , as well as for the tilt of $+, analogous to the
azimuthal velocity dependence for plane waves. Despite these
complications, the azimuthal minima of coupling intensity
would make waveforms from an explosion appear as though
the source possessed a small thrust component-note that the
Love waveform polarity toggles with the sign of the azimuth
of $+. For this choice of parameters, the Love/Rayleigh amplitude ratio does not change appreciably for source-receiver
distances from 200 to 1000 km, which would strengthen an
(erroneous) identification of the Love wave with the source
and not the structure. This ambiguity is likely to worsen if
larger anisotropy is assumed in the top kilometre of the crust.
This could arise from parallel crack distributions in shallow
basement rocks. Simple stochastic models of short-wavelength
3-D isotropic structure would not show this azimuthal variation, unless the structures possessed geometric anisotropy,
for example a large aspect ratio and preferred orientation of
velocity anomalies.
CONCLUSION
Because the dispersion characteristics of higher-mode Love
and Rayleigh surface-wave modes are quite similar, they can
suffer strong coupling in a layered anisotropic structure. With
plausible levels of crustal anisotropy, Rayleigh-dominant
surface-wave overtones can exhibit S H amplitudes at the
surface that are 15-25 per cent of the total displacement. The
fundamental-mode branches have different dispersion relations,
and so are more weakly coupled. Synthetic seismograms for
crustal models demonstrate that significant Love-to-Rayleigh
and Rayleigh-to-Love 'scattering' can occur in the absence of
lateral variation in seismic properties. The dependence of this
scattering on azimuth relative to the axis of symmetry $+ can
ACKNOWLEDGMENTS
E-mail exchanges with Robert Odom and Jan Garmany were
very helpful during the period in which the author could not
distinguish computer bugs from genuine anisotropic effects.
This work was supported by Air Force Office of Scientific
Research contract F49620-94-1-0043.
REFERENCES
Aki, K. & Richards, P.G., 1980. Quantitatiue Seismology, W.H.
Freeman, San Francisco, CA.
Babuska, V. & Cara, M., 1991. Seismic Anisotropy in the Earth, Kluwer
Academic, Dordrecht.
Backus, G.E., 1962. Long-wave elastic anisotropy produced by horizontal layering, J. geophys. Res., 67, 4427-4440.
Backus, G.E., 1965. Possible forms of seismic anisotropy of the
uppermost mantle under oceans, J. geophys. Res., 70, 3429-3439.
Booth, D.C. & Crampin, S., 1983a. The anisotropic reflectivity technique: theory, Geophys. J. R. astr. SOC.,72, 755-766.
Booth, D.C. & Crampin, S., 1983b. The anisotropic reflectivity technique: anomalous reflected arrivals from an anisotropic upper
mantle, Geophys. J. R. astr. SOC.,72, 767-782.
Chapman, C.H., 1994. Reflection/transmission coefficient reciprocities
in anisotropic media, Geophys. J. Int., 116, 498-501.
Chen, X., 1993. A systematic and efficient method of computing normal
modes for multilayered half-space, Geophys. J . Int., 115, 391-409.
Crampin, S., 1970. The dispersion of surface waves in multilayered
anisotropic media, Geophys. J. R . astr. SOC.,21, 387-402.
Crampin, S., 1984. Effective elastic constants for wave propagation
through cracked solids, Geophys. J. R. astr. SOC., 76, 135-145.
Crampin, S. & Taylor, D.B., 1971. The propagation of surface waves
in anisotropic media, Geophys. J. R . astr. SOC.,25, 71-87.
Finney, R.L. & Ostberg, D.R., 1976. Elementary Differential Equations
with Linear Algebra, Addison Welsey, Reading, MA.
Fisk, M.D. & McCartor, G.D., 1993. Fluctuation analysis of elastic
waves in random media via phase screen simulations, J. geophys.
Res., 98, 685-698.
Frazer, L.N. & Fryer, G.J., 1989. Useful properties of the system
matrix for a homogeneous anisotropic visco-elastic solid, Geophys. J.,
97, 173-178.
Fryer, G.J. & Frazer, L.N., 1984. Seismic waves in stratified anisotropic
media, Geophys. J. R. astr. SOC., 78, 691-710.
Fryer, G.J. & Frazer, L.N., 1987. Seismic waves in stratified anisotropic
media-11. Elastodynamic eigensolutions for some anisotropic systems, Geophys. J. R . astr. SOC.,91, 73-102.
Garmany, J., 1983. Some properties of elastodynamic eigensolutions
in stratified media, Geophys. J. R. astr. SOC.,75, 565-570.
Goff, J.A., Hollinger, K. & Levander, A,, 1994. Modal fields: a new
method for characterization of random seismic velocity heterogeneity, Geophys. Res. Lett., 21, 493-496.
0 1996 RAS, GJI 126, 173-183
Surface waves in layered anisotropic media
Kanamori, H. & Hadley, D., 1975. Crustal structure and temporal
velocity change in Southern California, Pageoph, 113, 257-280.
Kennett, B.L.N., 1983. Seismic Wave Propagation in Stratged Media,
Cambridge University Press, Cambridge.
Levander, A.R. & Hollinger, K., 1992. Small-scale heterogeneity and
large-scale velocity structure of the continental crust, J. geophys.
Res., 97, 8797-8804.
Levin, V., Lerner-Lam, A.L. & Menke, W., 1995. Anomalous mantle
structure at the Proterozoic-Paleozoic boundary in northeastern
US, Geophys. Res. Lett., 22, 121-124.
Levin, V., Menke, W. & Lerner-Lam, A.L., 1996. Seismic anisotropy
in northeastern US as a source of significant teleseismic P traveltime
anomalies, Geophys. J. Int., in press.
Lou, M. & Crampin, S., 1991. Dispersion of guided waves in thin
anisotropic media, Geophys. J. Int., 107, 545-555.
Mallick, S. & Frazer, L.N., 1991. Reflection/transmission coefficients
and azimuthal anisotropy in marine seismic studies, Geophys. J . Int.,
105, 241-252.
Maupin, V., 1989. Surface waves in weakly anisotropic structures: on
the use of ordinary or quasi-degenerate perturbation methods,
Geophys. J., 98, 553-563.
Maupin, V., 1990. Modeling of three-component L, waves in anisotropic crustal models, Bull. seism. SOC.Am., 80, 1311-1325.
Q 1996 RAS, GJI 126, 173-183
183
Nolte, B., Frazer, L.N. & Mallick, S., 1992. Seismogram synthesis for
azimuthally anisotropic media with a single slowness integration,
Geophys. J. Int., 111, 127-140.
Park, J., 1993. The sensitivity of seismic free oscillations to upper
mantle anisotropy I. Zonal structure, J. geophys. Res., 98,
19 933-19 949.
Park, J. & Yu, Y., 1992. Anisotropy and coupled free oscillations:
Simplified models and surface wave observations, Geophys. J. Int.,
110, 401-420.
Park, J. & Yu, Y., 1993. Seismic determination of elastic anisotropy
and mantle flow, Science, 261, 1159-1162.
Shearer, P.M. & Orcutt, J.A., 1986. Compressional and shear-wave
anisotropy in the oceanic lithosphere-the Ngendie seismic refraction experiment, Geophys. J. R. astr. Soc, 87, 967-1003.
Thomson, C., Clarke, T. & Garmany, G., 1986. Observations on
seismic wave equations and reflection coefficient symmetries in
stratified media, Geophys. J . R. astr. Soc., 86, 675-686.
Wu, R . 3 , 1994. Wide-angle elastic wave one-way propagation in
heterogeneous media and an elastic wave complex screen method,
J. geophys. Res., 99, 751-766.
Yu, Y. & Park, J., 1994. Hunting for azimuthal anisotropy beneath
the Pacific Ocean region, J . geophys. Res., 99, 15 399-15 421.