Geophys. J. Int. (1997) 131, 253-266
P-SHconversions in a flat-layered medium with anisotropy of
arbitrary orientation
Vadim Levin and Jeffrey Park
Department ofGeology and Geophysics, Yale University, PO Box 208109, New Haven, CT 06520-8109, USA. E-mail: [email protected]
Accepted 1997 June 8. Received 1997 February 2; in original form 1996 June 6
SUMMARY
P-SH conversion is commonly observed in teleseismic P waves, and is often attributed
to dipping interfaces beneath the receiver. Our modelling suggests an alternative
explanation in terms of flat-layered anisotropy. We use reflectivity techniques to
compute three-component synthetic seismograms in a 1-D anisotropic layered medium.
For each layer of the medium, we prescribe values of seismic velocities and hexagonally
symmetric anisotropy about a common symmetry axis of arbitrary orientation. A
compressional wave in an anisotropic velocity structure suffers conversion to both SVand SH-polarized shear waves, unless the axis of symmetry is everywhere vertical or the
wave travels parallel to all symmetry axes. The P-SV conversion forms the basis of the
widely used ‘receiver function’ technique. The P-SH conversion occurs at interfaces
where one or both layers are anisotropic. A tilted axis of symmetry and a dipping
interface in isotropic media produce similar amplitudes of both direct ( P ) and converted
(Ps) phases, leaving the backazimuth variation of the P-Ps delay as the main discriminant. Seismic anisotropy with a tilted symmetry axis leads to complex synthetic
seismograms in velocity models composed of just a few flat homogeneous layers. It
is possible therefore to model observations of P coda with prominent transverse
components with relatively simple 1-D velocity structures. Successful retrieval of
salient model characteristics appears possible using multiple realizations of a geneticalgorithm (GA) inversion of P coda from several backazimuths. Using GA inversion,
we determine that six P coda recorded at station ARU in central Russia are consistent
with models that possess strong (> 10 per cent) anisotropy in the top 5 km and between
30 and 43 km depth. The symmetry axes are tilted, and appear aligned with the seismic
anisotropy orientation in the mantle under ARU suggested by SKS splitting.
Key words: anisotropy, crustal structure, inverse problem, layered media, seismic
modelling, synthetic seismograms.
INTRODUCTION
The occurrence of P-SH conversion is common in teleseismic
P waves at periods of 0.5-5 s, and has been observed in a
variety of tectonic environments. Besides misaligned seismometers, explanations typically offered include the following:
(1) ray divergence from the great-circle path due to lateral
velocity heterogeneities (e.g. Hu & Menke 1992); (2) the
presence of dipping interfaces beneath the receiver (Langston
1977); (3) scattering from topography on the surface and/or
on velocity interfaces under the station (e.g. Clouser &
Langston 1995); (4) birefringence of P-S converted phases
(e.g. McNamara & Owens 1993). All but the last of these
scenarios entrain an implicit assumption that the near-receiver
region is isotropic. There is mounting evidence that seismic
0 1997 RAS
isotropy may actually be a rarity rather than the rule in the
shallow Earth. The majority of minerals and rocks that
form the crust and upper mantle display seismic anisotropy
in laboratory measurements (Babuska & Cara 1991). Bulk
anisotropy in the oceanic crust and lithosphere was established
by marine refraction experiments over two decades ago
(e.g. Raitt ef al. 1969). Shear-wave splitting in broadband
seismic data suggests that the continental lithosphere also has
significant elastic anisotropy (e.g. Vinnik et al. 1992; Silver
1996).
Of the isotropic causes for P-SH conversion, by far the most
popular is option (2). A dipping interface predicts a systematic
azimuthal variation in P-SV conversions, complementary
to the azimuthal variation of the P-SH conversions. I n
the context of the receiver-function technique (Burdick &
253
254
V: LevinandJ. Park
Langston 1977) properties of this mechanism for P-SH conversion have been explored by Owens & Crosson (1988)
and Cassidy (1992). The application of this technique to
observed data sets is often plagued by the low amplitude of
predicted transverse signals (e.g. Zhang & Langston 1995). As
a consequence, proposed velocity models sometimes contain
interfaces with dips in excess of 15" (e.g. Zhu et al. 1995).
To the best of our knowledge, all earlier efforts to constrain
seismic anisotropy using P-SH conversions in teleseismic P
coda employed models with horizontal axes of symmetry.
Kosarev, Makeyeva & Vinnik (1984) and, more recently,
Vinnik & Montagner (1996) used long-period records containing P-SH conversions to infer anisotropic layers in the
mantle under Northern and Central Europe. A complementary
scenario would be an S-P conversion at an interface in an
anisotropic velocity model, which is considered by Farra et al.
(1991) as a possible tool to study upper-mantle interfaces.
Birefringence in a P-S conversion at the base of the crust was
used in a number of studies (e.g. McNamara & Owens 1993;
Herquel, Wittlinger & Guilbert 1995) to constrain the total
crustal anisotropy. These models also assumed a horizontal
orientation of the symmetry axis.
Implications of an inclined anisotropic symmetry axis
were explored by Babuska, Plomerova & Sileny (1993) and
Plomerova, Sileny & Babuska (1996) as means to reconcile
observations of SKS splitting and traveltime delays of teleseismic P waves in Europe and the western US. Similarly, joint
analysis of SKS splitting and P traveltimes and polarizations
led Levin, Menke & Lerner-Lam (1996) to conclude that
tilted-axis anisotropy is likely in the upper mantle under the
northeastern US. Silver & Savage (1994) mentioned seismic
anisotropy with an inclined axis as a possible source of azimuthal variation in SKS splitting measurements. An inclined
anisotropy axis is also favoured in a tomographic inversion of
teleseismic traveltimes for the Pyrenees by Gressilaud & Cara
(1996). In the context of local earthquakes and upper-crustal
structure, seismic anisotropy with a dipping symmetry axis was
explored (and rejected) by Booth & Crampin (1985) in their
study of North Anatolian fault seismicity. Their computations
were used in a later study of microseismicity near Long Valley,
California, which concluded that microcracks in the vicinity of
one station dip by as much as 15" (Savage, Peppin & Vetter
1990).
Early investigations of anisotropic wave propagation (Keith
& Crampin 1977a,b) express the reflectivity solution in terms of
the full elastic tensor. Although its 21 independent components
offer the most general representation of elastic stress-strain
relations, this large number of free parameters is not likely to
be constrained by seismic data, at least not very soon. Even in
forward calculations, researchers typically resort to elastic
tensors based on lab measurements of single minerals or else
make simplifying assumptions of hexagonal or orthorhombic
symmetry in order to gain intuition into the basic physics. In
this paper we allow the velocity model space to include any
orientation of hexagonally symmetric seismic anisotropy. The
elastic tensors in this model space have only seven free parameters (five elastic parameters and two angles to specify the
axis of symmetry). We consider it more feasible, at least in the
current state of our understanding, to estimate a restricted set
of parameters from seismic data if we wish to interpret anisotropy in a geological context. More complex symmetry can be
justified if hexagonal symmetry fails. We use a representation
of hexagonally symmetric elastic tensors that expresses the
coupling of compressional and shear motion directly in terms
of seven free parameters, rather than the general elastic tensor.
This has the small advantage of identifying terms that are nonzero only when the axis of symmetry fi is tilted between vertical
and horizontal. Developed by Park (1993, 1996) in connection
with surface-wave coupling, it adapts easily to the case of
subvertical body waves.
The goal of this paper is to demonstrate that P-SH
conversions resulting from reverberations of a plane wave in
a stack of flat anisotropic layers can account for various
common features in broadband records of teleseismic P waves.
We present a technique for computing the transmission
response of a flat-layered medium with arbitrarily oriented
hexagonally symmetric anisotropy, and describe the main
features of the resulting synthetic seismograms. We show
that the observable effects of seismic anisotropy are similar
to those of an inclined velocity interface in an isotropic
medium. We describe experiments, with both synthetics and
data, with a genetic-algorithm (GA) search engine to automate the forward-modelling procedure. We compute multiple
realizations of the GA model search with different random
seeds to assess the non-uniqueness of P-coda modelling. As an
example of the potential applications, we estimate anisotropic
velocity profiles via a GA inversion for a set of teleseismic
records with prominent P-SH conversions from station ARU
in central Russia.
REVERBERATIONS I N ANISOTROPIC
LAYERS
We consider homogeneous flat layers over a homogeneous
half-space. The half-space is isotropic, and the layers may
possess seismic anisotropy with an axis of symmetry +. The
axis of symmetry can vary among the layers. Our seismograms
are computed with a reflectivity algorithm originally based
on the algorithm of Chen (1993), but the propagator-matrix
derivation does not differ in an essential manner from the
algorithm developed by Keith & Crampin (1977a,b). A subvertical compressional wave is assumed to propagate upwards
from the half-space into the layered part of the model, where it
undergoes refraction and conversion. The combination of
pulses arriving at the free surface is the 'transmission response'
of the media. Once computed, this transmission response can
be convolved with the pulse of the original compressional
wave, yielding a synthetic seismogram. A compressional plane
wave in an anisotropic 1-D flat-layered structure suffers
conversion to both vertically ( S V ) and horizontally (SH)
polarized shear waves, with three exceptions: (1) no P-SH
conversion occurs if the axis of symmetry is everywhere
vertical, or (2) if the ray direction coincides with that of the
symmetry axis in each layer, or (3) if the ray azimuth is
perpendicular to the azimuth of a horizontal symmetry axis.
We express elastic properties as a function of depth as A(z),
where Alrkfis 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 fi, according to
formulae similar to the head-wave formulae of Backus (1965):
pa2(5) = A + s c o s
pp2(()=
25 + c c o s 45,
D + E cos 25.
(1)
0 1997 RAS, GJI 131,253-266
P-SH conversions in aflat-layered medium
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 isotropicmedium, B = C = E = O
and A = ,
I
+ 2 p and D = p, where I”,p are the Lame parameters.
Park (1993, 1996) showed how these azimuthal relations
generalize to other orientations of 6.It is possible to form a
linear combination of anisotropic deviations from an isotropic
reference model, each deviation with its own axis of symmetry
9.This would be useful for media with both oriented cracks
and oriented minerals, if the orientations differ.
We assume a flat earth, z=O at the free surface, and z
increasing downwards. We assume a plane-wave solution of the
form U(x, t )= u(x) exp [i(k.x - wt)]. Phase velocity for P and S
waves in hexagonally symmetric media can be represented
by smooth surfaces symmetric about the axis in 3-space defined
by (Fig. 1). If B , E > 0, defines the ‘fast’ axis for wave
propagation, leading to phase-velocity surfaces that resemble
tilted watermelons. If B, E < 0, +defines the ‘slow’axis for wave
propagation, leading to phase-velocity surfaces that resemble
tilted pumpkins. The cos45 coefficient C, small in most estimates from data, would lead to modest alterations of these
ellipsoidal shapes. The axis of symmetry+ is parametrized with
polar angles that describe tilt (0) and counterclockwise
in the locally Cartesian reference frame
(CCW) azimuth
of the reflectivity calculation. In our choice of geographic
+
+
(r)
Anisotropy Parameterization
isotropic layer
‘+ t
I”
’
,
Vertical axis of symmetry
B<O E<O
255
coordinates, 2 points downwards, S points south and 9 points
describe the same symmetry axis,
west. Because and
the tilt and azimuth angles 0 , l can also be interpreted as a
symmetry axis tilted Q from the vertical, with strike 4 clockwise
(CW) from north.
In a layer with constant anisotropic elastic properties, one
can calculate three upgoing and three downgoing plane-wave
solutions to the equations of motion, with vertical wavenumbers and polarizations determined by the eigenvectors
of a 6 x 6 matrix eigenvalue problem (Keith & Crampin
1977a; Garmany 1983; Fryer & Frazer 1984; Park 1996).
Assume K layers over an isotropic half-space, with interfaces
at z l , z2, . . . ,ZK. We compute the generalized transmission
response of the layer stack to determine the particle motion at
the free surface zo = 0 due to an upgoing input wavelet with
horizontal phase velocity c (slownessp= 1/ c ) at the base of the
stack. We restrict attention to phase velocities c for which both
P and S waves in the half-space are oscillatory, thus bypassing
the problem of leaky-mode reverberation. As long as the halfspace velocities exceed those of the layers above, we also avoid
the defective matrices that can impede the calculation of
evanescent S waves (Park 1996).
We use the notation of Chen (1993), extended to anisotropic
layers. Let cik),cT) be 3-vectors containing the plane-wave
amplitudes of the three upgoing and three downgoing waves in
the kth layer. Plane-wave amplitudes (that is phase and/or
exponential decay accumulates from) are referenced to the
interface that the wave departs, so that c r ) is referenced to
z = zk- 1, and cik)is referenced to z = z k . As an exception to this
rule, we prescribe the coefficients
of the upgoing waves
in the half-space at the interface z = ZK.At the kth interface, the
coefficients of waves that leave the interface are related to
waves that approach the interface by a matrix of reflection and
transmission coefficients:
+
-+
where Tf), etc., are 3 x 3 submatrices. These relations at the K
interfaces are manipulated to calculate generalized reflection
-(k)
(k)
and transmission coefficient submatrices T, ,Rdu, so that the
amplitudes of waves that leave an interface can be expressed
solely in terms of upgoing waves in the layer below:
-
Horizontal axis of symmetry
B>O E>O
-(k) -(k)
The calculation of T,
,Rdu follows Kennett (1983) closely,
and readers who wish a more complete treatment of the
reverberation problem can consult this book. The generalized
coefficients are determined recursively from the top layer
downwards through the stack, so it is sufficient to outline the
solution for k = 1, the interface at the base of the surface layer.
The coefficient vectors for waves that depart the interface at
Z = Z ~ are expressed in terms of the coefficient vectors for waves
that impinge the interface:
isotropic halfspace
Figure 1. Schematic diagram illustrating possible shapes of velocity
distribution for various choices of anisotropic parameters and axis
orientations. Diagrams are not scaled with B, E values; rather they
represent the type of velocity distribution in the vertical plane that
contains the symmetry axis.
0 1997 RAS, GJI 131,253-266
The free-surface reflection matrix (see Park 1996, eq. 30; Chen
1993, eq. 27) also relates the upgoing and downgoing waves in
the surface layer:
(5)
256
V. Levin and J. Park
Substituting ( 5 ) into (4) results in formulae for the
generalized reflection and transmission coefficients at z =z1:
c!1) = ( I - R ~ ~ . R 4 ) - 1 . T
( 1u) . =T(').c(~),
~ ( 2u) u
u
cf) = (TS:).R:((.T.'," + R:j).cLz) =R:J.ckz)
effects in this procedure are minimized by padding the input
wavelet with zeroes in the time domain to interpolate the
spectrum.
(6)
.
S Y N T H E T I C EXAMPLES
The generalized reflection and transmission coefficients for the
deeper layers are calculated recursively, using Rfi to calculate
- ( a,Rud,
-aand so on. The plane-wave coefficients in the surface
T,
layer are computed from the upgoing waves at the base of the
stack by
=T:).Tf'. . , , .Tu
-(fa.c,( K + l )
(7)
and (5). The particle motion at the surface is a linear combination of these waves. Since the reflection and transmission
submatrices in (2) have phase-propagation factors hidden
within them that depend on frequency w, the transmission
response of the medium must be calculated at the evenly spaced
frequency values of the fast Fourier Transform of a chosen
input wavelet. Particle motion at the surface is obtained by an
inverse Fourier Transform (Fig. 2a). Non-causal 'wraparound'
This section is devoted to the description of the main
properties of the P-SH conversion mechanism. An extensive
exploration of the model space is presented elsewhere (Levin &
Park 1997a), while here we focus on the most significant
aspects. All simulations discussed in this section are performed
in a model that consists of an isotropic half-space beneath an
anisotropic layer 30 km thick, with velocity values and anisotropic parameters described in Table 1 and illustrated in
Fig. 2(b). The plane wave used in simulations has an incidence
angle of 25", typical of P waves from sources 70"-75" away.
Fig. 2(c) shows ray diagrams for the phase we analyse.
Transmission responses are convolved with the spectrum of a
tapered cosine pulse prior to their transformation into the time
domain. Azimuthal variation of P-SH conversions for two
different positions of fi is illustrated in Fig. 3. If the symmetry
velocity, km/s
3.0
0
4.0
5.0
6.0
7.0
8.0
tilt = 45
E
O
10
Y
c 20
U
30
Vs
R
0.38
0
VP
I
Psrns
,
5
I
10
15
20
25
times (see)
Figure 2. P-SH conversion in a flat-layered anisotropic medium. (a) A sample synthetic seismogram computed in a simple anisotropic model
(Table 1). Ray backazimuth is 45", incidence angle is 25". A number at the start of a horizontal trace indicates the maximum amplitude of that trace
scaled relative to the vertical component of P.(b) Velocity model used to compute synthetic seimograms. Parameters of seismic anisotropy in the layer
are shown, along with the schematic diagram illustrating symmetry axis orientation, and the distribution of velocity in the axis plane. (c) A schematic
ray diagram for phases marked on the seismogram. Solid line indicates a P wave, dashed line indicates an S wave. Shaded triangle signifies the seismic
station.
0 1997 RAS, GJI 131,253-266
P-SH conversions in ajlat-layered medium
impedance contrast across the interface, the level of seismic
anisotropy and the relative amount of P and S anisotropy.
S-velocity anisotropy has maximum effect for a horizontal
symmetry axis, while the influence of the P-velocity anisotropy
is strongest when the axis is tilted by 45".
The polarity and amplitude dependence of converted phases
is encouraging, but also challenging, since there are so many
free parameters. Fig. 5(a) illustrates the effect of varying one
anisotropic parameter (symmetry axis tilt) in two layers of a
three-layer model (Fig. 5b).
To summarize, the presence of arbitrarily oriented seismic
anisotropy in a flat-layered medium results in P-SH conversions in all but exceptional cases. The timing of converted
phases is primarily a function of vertical velocity structure, but
their amplitude and particle motion are governed by the
orientation and strength of elastic anisotropy. The pattern of
azimuthal variation combines variations proportional to sin 4
and sin25. Overall, P-SH conversions stemming from 1-D
anisotropic velocity structures exhibit azimuthal variations
that may easily be misinterpreted as evidence of 2-D or 3-D
lateral heterogeneity.
Table 1. Velocity model used for synthetic simulations. The
parameters B and E scale peak-to-peak variations of compressional
and shear velocity, respectively, each with cos 25 azimuthal dependence.
B=E=0.05 corresponds to 5 per cent peak-to-peak variation in
respective velocities.
layer bottom, km
I V p , km sC1
Vs, km s-'
p, g cm-3
3.6
2.7
3.3
4.6
a,
I
B
E
0.05
0.05
axis is tilted (Fig. 3, left column), amplitudes of SH-polarized
phases (e.g. P, Ps) may exhibit two-lobed (sin 5) patterns as the
ray backazimuth varies. These patterns are four-lobed (sin 29)
for the commonly used case of a horizontal symmetry axis
(Fig. 3, right column). The most general description of the
azimuthal amplitude pattern would be Wl sin 5 + Wz sin 29, a
weighted sum of the two, where phase-specific weights Wl and
Wz depend on velocity profile, anisotropy parameters and ray
geometry. Regardless of the pattern, little energy is transferred
to the transverse component if the azimuth of the ray is close to
that of +, or, in the case of a horizontal axis, if the ray is nearly
normal to the ii. direction. Two rays forming the same angle on
opposite sides of the symmetry axis lead to converted waveforms that are mirror images of each other.
influences the waveforms strongly, with no
The tilt of
P-SH conversion in the case of a vertical axis of symmetry.
An example of P-SH amplitude variation with axis tilt is
shown in Fig. 4. For a given orientation of the ray and the
symmetry axis, converted-phase amplitudes scale with the
axis tilted 45"
P Ps
75'
ISOTROPIC INTERFACE D I P OR
ANISOTROPIC A X I S TILT?
If a plane wave refracts as it passes through a dipping
velocity interface, both P and SV motion are deflected out
of the vertical plane. Viewed in a coordinate system defined by
the source-receiver path (radial-transverse-vertical), these
deflected phases will possess a transverse component. This
horizontal axis
P Ps
Psrns
Psrns
-J.-++
::::$----;p
North
'@'-+--++-
a.
::::;-;-;p
255'
257
II
I1
%--+-
0
5
10
15
20
time (sec)
anisotropy
\
axis
velocity
surface
East
B,E>O
I1
V
25
'
0
5
10
15
20
25
time (sec)
Figure 3. Azimuthal dependence of P-SH conversions in simple anisotropic models. Transverse components of synthetic seismograms generated
for different ray directions (see diagram on the right) in a simple model (Fig. 2b andTable 1). Left column depicts tracescomputed in a model with the
symmetry axis tilted at 45", right column depicts traces computed in a model with a horizontal axis. Traces within each column are plotted to the same
scale and labelled with ray backazimuth. Amplitudes of P-SH converted phases vary as sin5 in the left column, and as sin25 in the right column,
where 5 is the clockwise angle between the axis and the ray.
0 1997 RAS, GJI 131,253-266
258
K Levin and J. Park
solid -
P
BAZ 284",Anisotropy Axis 320"
thick grey - Ps
6
U
3
.-
c,
e
"1
4
0
s
baz45'north
8
I
0
s
s
inc. ang. 25'
0
0
8
( _ _ _
0
. . . . .r . . . .
,.
10
_ _ . . . . , . _ . _ _ _ _,.!
..,
30 0
20
10
Time (sec)
cd
20
30
Time (sec)
s
s
L
velocity, km/s
2.5 3.5 4.5 5.5 6.5 7.5 8.5
0
I
I
I
I
I
1
30
60
90
120
150
180
axis tilt from vertical,
O
B I -0.08
tilt = 20 *
tilt = 60
Figure 4. Amplitude of transverse phases in P coda as a function of
+(anisotropic symmetry axis) tilt from the vertical. Solid line: P wave;
thick grey line: Ps;dashed line: Psms. Schematic ray-path diagrams are
shown in Fig. 2(c). A simple anisotropic model (Fig. 2b and Table 1)
with iv tilted 45" to the north is used. All computations are performed
for the compressional wave with an incidence angle of 25" arriving
from a backazimuth 45".
mechanism is often invoked to explain transverse motions in
teleseismic P coda that exhibit systematic variations of both
amplitude and polarity with backazimuth (e.g. Zhu, Owens &
Randall 1995; Zhang & Langston 1995).
In this section we compare the azimuthal behaviour of
SH-polarized phases in the anisotropic model discussed above
and a simple isotropic model with one inclined interface. The
isotropic model consists of a half-space beneath a layer, with
the boundary between them dipping to the north. Depth to the
layer-half-space interface directly below the receiver is 30 km,
and velocity and density values are identical to the anisotropic
model. Computations for a plane-wave interaction with an
inclined interface are performed following Langston (1977).
Fig. 6 depicts amplitudes of direct (P)and converted (Ps)
phases and Ps-P delay as functions of ray backazimuth. The
time of the Ps arrival is measured from the radial component,
where it has a significant amplitude and the same polarity for
all ray directions. Three cases are shown: an isotropic model
with the interface dipping 10" to the north, an anisotropic
model with the symmetry axis tilted 45" to the south, and an
anisotropic model with a horizontal symmetry axis aligned
north-south.
When the symmetry axis is horizontal, amplitude patterns
resulting from anisotropy are four-lobed (cos 25 on the radial,
sin 25 on the transverse), and can easily be distinguished from
the two-lobed (cost on the radial, sin5 on the transverse)
pattern due to the dipping interface. In the case of an axis tilted
at 45" this distinction is not as obvious, since the pattern of
50'
I
I
Figure 5. Influence of anisotropy on synthetic seimograms in 1-D
structures. The upper part of the plot shows synthetic seismograms
computed for two velocity models that differ only in the orientation of
anisotropy. The lower part of the plot depicts a 1-D velocity model
composed of three layers over an isotropic half-space. Two of the layers
are anisotropic, with anisotropy parameters shown below the corresponding waveforms (a and b). In the upper anisotropic layer B, E < 0,
resulting in a pumpkin-shaped velocity distribution. In the lower layer
B, E > 0, yielding a melon-shaped velocity distribution. Anisotropy in
the upper layer may be caused by aligned cracks or layers of alternating
velocity, while that in the lower layer may be caused by alignment
of mineral crystals due to deformation. The incident P wave has a
phase velocity of 16.5 km sec-' and arrives from a backazimuth of
284". In two simulations the orientations of the symmetry axes
are varied as illustrated by ellipsoidal diagrams (a) and (b). The transverse components of the resulting synthetic seismograms are quite
different, although their maximum amplitudes (indicated by the
number at the start of the trace) are similar. Radial components are
less affected by changes in anisotropy directions. The most significant
difference in radial waveforms between models (a) and (b) is in the size
of the arrival marked by an arrow.
amplitude variation due to anisotropy is also nearly two-lobed.
The 'phase' relationship of the radial and transverse amplitude
patterns may serve as a discriminant. For example, the dipping
interface model, as well as the tilted anisotropy model, impose
0 1997 RAS, GJI 131,253-266
P-SH conversions in a flat-layered medium
259
4-
solid - interface dipping north 10'
dotted - axis tilted 45' south
dashed - horizontal N<->S axis
v)
-4
0
90
180
270
360
0
I
I
90
180
ray baz '
i
270
360
270
360
ray baz
0
8
IS
0
v)
a
a
I
?t
3.4I
,
0
90
180
270
360
0
90
ray baz '
180
ray baz
Figure 6. Azimuthal patterns of amplitudes and relative traveltimes imposed by dipping interfaces and tilted-axis anisotropy on P and Ps phases.
A solid line indicates the effects of a dipping interface model. Dashed and dotted lines show the effects of anisotropic models with axes oriented
horizontally and at a 45" angle, respectively. In all cases the vertical velocity profile under the receiver is that of Fig. 2(b). The dip of the interface is 10"
to the north; the anisotropic axis points south. All amplitudes are expressed as a percentage of the maximum on the vertical component of a respective
three-component seismogram, and expressed in per cent.
cos (- 5) variation on the radial component of P (smallest at
5 =O", largest at 5 = 180").The transverse P follows sin 5 for the
dipping interface model, and sin (- 5 ) for the tilted anisotropy
model. However, an isotropic model with a velocity inversion
across the interface dipping to the south would yield a sin5
azimuthal pattern for the transverse component of P, while
preserving the cos (- 5) variation of the radial component, and
thus mimicking the anisotropic model. Examples of such a
pattern were found at a number of stations in Tibet by Zhu et al.
(1995), and interpreted as evidence for a strong mid-crustal
velocity inversion under an interface dipping 15"-20".
A better discriminant between anisotropy and interface
dip is the relative timing of P and Ps pulses. For an equivalent
SH amplitude, the variations in ray-path length for the
inclined interface lead to larger variations in the Ps-P differential traveltime. Also, the Ps-P delay exhibits a clear pattern
following cos 5. Anisotropic models yield more complex
patterns of Ps-P delays, and much smaller delay variations.
Interpretation will be more difficult, however, if both tilted
anisotropy and dipping interfaces are present.
Multiple reverberations like Psms may attain significant
amplitude in anisotropic models (see Figs 2 and 3 ) , as well as in
models with dipping interfaces. For these phases the problem
of misinterpretation is less severe. In anisotropic models, the
0 1997 RAS, GJI 131,253-266
transverse amplitude patterns of multiple reverberations
generally follow sin25. The transition from a four-lobed azimuthal pattern such as this to a two-lobed one is mainly controlled by the symmetry-axis tilt ( Levin & Park 1997a). For
multiples such as Psms, two-lobed patterns are observed only
when the symmetry axis is nearly vertical, in which case the
amplitude of P-SH conversion is weak (Fig. 4). In dippinginterface models the amplitude variation of multiple reverberations follows sin 5 (e.g. Owens & Crosson 1988). Thus, if a
reverberation such as Psms is indeed identified in the data, the
number of polarity changes (two or four) in a complete circle
will indicate its origin.
In summary, the azimuthal behaviour of SH-polarized conversions in simple tilted-axis anisotropic models is similar to
that of P-SH conversions induced through refraction at a
dipping interface in an isotropic medium. The observation of
systematically varying transversely polarized motion in the
coda of teleseismic P waves thus has two possible interpretations. While the introduction of anisotropy increases the
number of unknowns in potential models, considering it
appears necessary. It may also prove advantageous, because
simple 1-D anisotropic structures may turn out to be more
geologically meaningful than their more complex isotropic
alternatives.
260
I.: Levin and J. Park
GENETIC MODELLING O F TELESEISMIC
RECORDS
Forward-modelling experiments indicate that the reverberation inverse problem for 1-D anisotropic structure is
quite non-linear. To aid the search for an acceptable model
(or models) for P-SH conversions within body-wave codas,
we adapted a genetic-algorithm (GA) model search ( Sen &
Stoffa 1992; Sambridge & Drijkoningen 1992; Zhou, Tajima
& Stoffa 1995). GA algorithms mimic the processes by which
Darwinian selection in nature is thought to increase the
adaptive ‘fitness’ of organisms to their environment. One
starts with a randomly generated population of models, whose
‘fitness scores’ are calculated from waveform misfit and
possibly other desirable model attributes, such as increasing
seismic velocity with depth. Succeeding generations of model
populations are obtained by splicing seismic velocities, anisotropic parameters and layer thicknesses from pairs of ‘parent’
models. The probability of ‘mating’ for a given model is
determined according to its ‘fitness’ score, so that desirable
model attributes, such as a velocity discontinuity that generates
a particular Ps converted phase, tend to survive, and undesirable model attributes tend to perish.
In our implementation of GA inversion, we advanced a
population of 100 10-layer crustal models through
1000
generations to approach a satisfactory waveform fit. We
specified several ‘mutations’ during the model ‘reproduction’
process, in order to encourage final models with minimal
structure, and to introduce small perturbations into partially
successful models. These included (1) averaging the properties
of adjacent layers (to discourage unnecessary interfaces),
(2) switching the properties of adjacent layers, and ( 3 ) altering
the thickness, velocities and/or anisotropies within an isolated
layer. Mutations of the anisotropic properties were specified to
vary the coefficients B and E of the cos 25 P and S anisotropies
independently, subject to the constraint that they have the
same sign. The cos45 term C was set to zero to simplify the
interpretation. Experiments showed that, for single records or
multiple records with similar P-wave phase velocities, the GA
algorithm trades-off crustal thickness with average crustal
velocity. This trade-off is well known in receiver-function
inversion, as the delay time of a Ps Moho-converted phase
can be fit by both slower-and-thinner and thicker-and-faster
crustal models. We therefore fixed the crustal thickness in our
tests, anticipating outside constraints on Moho depth, for
example from active-source seismic studies.
If R ( f ) ,T ( f ) ,V ( f ) are the spectra of the radial, transverse and vertical components of the P coda, respectively,
the receiver-function impulse responses ZR(f ) ,IT( f ) for the
radial and transverse motion can be found by spectral
division:
-
IR(f)=R(f)/V(f),
IT(f)=T(f)/V(f).
(8)
Spectral division is unstable, although the calculation can be
damped to stabilize instances where V ( f ) = O . For genetic
inversion, however, we need not calculate the receiver function
directly, but rather attempt to match the ratios R(f)/V ( f )
and T ( f ) / V ( f ) using the dot-product of the complex
3-vectors
The model vector + ( f ) is formed from the frequencydependent response of a 1-D structure to a plane wave
impinging from below at a prescribed phase velocity (i.e.
slowness). The data and model vectors are parallel if
If the model and data vectors are parallel for all frequencies,
there exists an upgoing source wavelet that, impinging the
model from below, generates the data.
by summing these vector dotWe compute a fitness score, 9,
products over a chosen frequency interval. In this study, we
specify 0.2 2 f 20.8 Hz, corresponding to teleseismic periods
1.25 2 T 5 5 s. Using 512-point (25.6 s) P-coda records from
broad-band (20 sps) channels, there are 16 discrete frequencies
that we use in the fitness-score sum:
The modulus of the data spectrum weights the sum to avoid
overweighting frequencies with little signal.
In order to check the success of a particular model, the data
spectra can be projected onto the model response spectra to
synthesize seismic motion:
The inverse FT of this projection is compared with data.
Because this procedure is not a convolution, non-causal
waveform artefacts can occur if the model fit is inaccurate.
In this approach, no specific input wavelet is assumed.
In fact, one retrieves the complex-valued spectrum of ‘the’
upgoing wavelet from the coefficients of the frequencydependent dot-product above. Although this seems an ideal
method for retrieving both structure and source time function,
there is a trade-off between wavelet and 1-D structure.
Regional-distance seismograms typically show intense P-SV
reverberation, with scattering by 3-D structure a probable
contributor. A 1-D GA inversion of such waveforms, unable to
model 3-D scattering, would interpret most of this waveform
complexity as a long, complex source time function.
The number of free parameters in this experiment is quite
large, so it is unlikely that genetic inversion would converge
to the input model exactly. It is instructive to note how often
GA converges to a model far from the input model. We performed synthetic experiments with the crustal model shown
in Fig. 7, which has tilted-axis anisotropy in the top 2 km
( B = E = -0.10) and bottom 13 km (B=E=0.08) of the crust.
In the inversions we specified the depth of the crust, and let
other model parameters vary for 500 generations. Multiple
realizations of the GA procedure, using a single record, produced a ‘cloud’of models that differ greatly in detail. Inversions
of records from several backazimuths were necessary for
successful retrieval of the salient characteristics of the input
model, for example the orientation of the symmetry axis ~.
Fig. 8 shows the symmetry axes for anisotropic layers in the
shallow, middle and deep crust. These models typically earned
fitness scores between 0.85 and 0.925, suggesting the difficulty
of converging precisely to the theoretical (and in this case
known) best-fit model (Fig. 9). Even within this set of GA
0 1997 RAS, GJI 131, 253-266
P-SH conversions in uflut-layered medium
Composite Crustal Model
8000
%
a
r
roo0
ci 6000
a,
4
rd
a,
k
8JJ
5000
-
2 4000
4
3000
r-
-
- density (kglm"3)
I
0
I
I
10
I
I
I
20
Depth (km)
I
I
30
Figure 7. Velocity-density profile of the crustal model used in the
synthetic test of the genetic-algorithm P-coda inversion. Maximum
and minimum velocities in the anisotropic layers are shown. The units
are m s-l for velocities a and fc, and kg m-3 for density p . Numerical
values of all parameters are listed in Table 2.
inversions, there are trade-offs that appear endemic to the
procedure. For instance, similar Ps converted phases can be
generated by models with B, E > 0 and B, E < 0 for axes of
symmetry 6 that differ in strike by 90". As a result, plots of the
scalar parameters B, E as a function of depth appear poorly
constrained for a GA 'inversion set'. This may occur if the data
records are sensitive to only two of the three principal axes of
the ellipsoidal velocity surface in anisotropic layers.
Although Fig. 8 is superficially inconclusive, the GA
'inversion set' appears to identify a 'fast' and a 'slow' direction
consistently and accurately. To confirm this, it is useful
to estimate an average, or composite, of a family of G A
inversions. Since the models are anisotropic with variable
orientation, a simple average of model parameters, such as B,
would be misleading. Instead, we compute a composite model
from a weighted sum of velocity ellipsoids. For P velocity,
define a symmetric matrix
g i a k [ (1 f Bk /2) 6'k @6'k
v p=
+ (1
- Bk
/2)(1 -6'k
@@,)I
k
layer bottom, km
2
20
33
co
0 1997 RAS, GJI 131,253-266
Vp, km s
4.70
6.00
6.60
8.0
Vs, km s
2.70
3.50
3.85
4.6
261
computed as a function of depth, where the Ek is the P velocity
in the kth layer and the sum extends over k = 1 , . . . ,K GA
realizations. We use the cube of the fitness score B as a
weighting parameter. The eigenvectors and eigenvalues of V p
define three principal P velocities and their orientations. A
similar formula computes a composite anisotropy for the S
velocity, with three principal shear polarizations. If anisotropy
is poorly constrained, one expects the composite velocity
ellipse to be nearly spherical, with little net anisotropy and
rapid variations in principal-component orientations. If a fast
or slow symmetry axis in a depth range is clearly evident from
the inversions, two of the principal velocities should be nearly
equal. If not, three distinct principal velocities will be evident
in the composite model. Although it would be tempting to
interpret such behaviour as indicating anisotropy with more
complex symmetry, our synthetic tests indicated that the
underlying model need not possess this. Moreover, the composite model defined by (13) does not typically fit the data
records optimally, but rather represents features shared by the
set of models that do.
The composite model for 14 GA inversions of synthetic data
from six backazimuths shows general agreement with the depth
extent and orientation of anisotropy in the input model
(Fig. 10). In particular, the set of GA inversions predicts sharp
velocity jumps (1) beneath a shallow low-velocity layer with
tilted anisotropy, and (2) at 20 km atop a deep-crustal layer
with 8 per cent peak-to-peak tilted anisotropy. There is better
agreement for the fast-axis orientation in the deep crust than
there is for the slow-axis orientation in the shallow layer, and
the intermediate principal velocity for this shallow layer is
midway between the extreme velocities. Anisotropy with low
amplitude is 'found' in the upper 5-10 km of the composite
model, as well as a gentle velocity inversion in the middle crust.
Since neither of these spurious features would generate a Ps
converted phase, their existence is probably an artefact of the
GA procedure.
We applied GA inversion to P coda recorded at IRIS GSN
station ARU (Arti Settlement, Russia) from six teleseismic
events (Fig. 11). We used inversion parameters identical to the
synthetic tests, running the GA for 2000 generations. Many P
coda at ARU display a prominent P-SH conversion roughly
5 s after direct P,indicating a deep-crustal conversion, as well
as a polarization anomaly in direct P, suggesting anisotropy in
a shallow layer (Levin & Park 1997b). We fixed the crustal
thickness at 42 km, consistent with active-source transects
of the Ural Mountains suture south of ARU, reported by
Thouvenot et al. (1995) and Berzin et ul. (1996). We also
fixed the velocities in the mantle as a=8.0 km s - ' and
p , g cm
2.3
2.8
2.9
3.2
~
B
E
-0.1
-0.1
0.08
0.08
0,
cr,
30
30
75
120
262
-10%
K Levin and J. Park
-1%
1%
a 4
0
6-Record GA Inversion Test
10%
0
,
8
Synthetic for best-fit model
Fitness=O.921
2.0
P anisotropy
S anisotropy
0"
0'
90" 2
90"
n
L
180"
3.0 -
-
180"
O->5km
0"
15
20
phase velocity
0
5
Time (sec) -->
25
10
19.00
"Observed" input record
0'
.
o
2
0
H
270"
0.0
0
5
Time (sec) -->
25
20
phase velocity
19.00
Figure 9. Synthetic test of GA waveform inversion for a P wave
arriving with a phase velocity of 19 km s-' at a backazimuth of 60".
The direct-P input pulse is taken from an earthquake record. It suffers
polarity reversal in the Ppmp reverberation that follows direct P by
11 s. Note how the GA inversion attempts to fit the transverse motion
coeval with direct P, as well as the Ps converted phase that trails it
b y 4 s.
180'
10 -> 15 km
0'
15
10
0'
90"
Composite Crustal Model
r
8000
180"
180"
25 -> 30 km
Figure 8. Anisotropy within the uppermost, middle and deep crust
among models obtained by GA inversion of synthetic P coda computed
for the model shown in Fig. 7. Six synthetic records were used in the
inversions, with backazimuths from 0" to 150", spaced at 30" intervals.
Different realizations of the genetic algorithm are obtained by using
different random-number seeds. Within each depth range, we plot
parameters for each layer with thickness greater than 0.5 km. The
orientation angles 0, 5 are plotted in polar coordinates, equivalent to a
map view of an upward-pointing 6.Symbol size indicates the peak-topeak velocity anisotropy. Squares indicate anisotropy with a fast
symmetry axis (B, E > 0). Triangles indicate anisotropy with a slow
symmetry axis ( B , E < 0). The anisotropy of the input model, non-zero
only in the shallow and deep crust, is indicated by the shaded symbols. In
the shallow and deep crust, fast and slow symmetry axes from different
GA inversions tend to cluster at an angular distance of 90" from each
other, suggesting the ability of P coda to distinguish between them.
p=4.6 km s-'. Since the data are real, not synthetic, waveform fitness scores are much lower, typically 0.55 < 9 < 0.65,
probably due to departures from 1-D geometry. Nevertheless,
the GA inversions produce 10-layer models that reproduce
salient features on the transverse component (Fig. 12).
%
a
roo0
-
6 6000 e,
& 5000 -
4
e,
k
QO
2
4000 3000
S-velocity ( d s e c )
1
density (kg/m^,33'
4
2000'
I
0
I
I
10
I
I
20
1
I
I
30
Depth (km)
Figure 10. Composite of 14 10-layer anisotropic models, calculated
according to weighted sum (13). The models were obtained with GA
waveform inversion of synthetic P coda. Maximum (solid line), minimum (solid line) and intermediate (dashed line) velocities in the
anisotropic layers are shown. Though computed from hexagonally
symmetric crustal models, the composite model's anisotropy does not
typically possess an axis of symmetry, nor does it necessarily achieve an
optimal waveform fit. Nevertheless, model features shared by many
GA inversions, such as a sharp transition from isotropic to anisotropic
velocity at 20 km depth, are highlighted by the composite model. The
units are m SKI
for velocities a and B, and kg m-3 for density p .
0 1997 RAS, GJI 131,253-266
P-SH conversions in ajat-layered medium
263
Events Analysed for P-Reverberation
40'
60'
80'
100'
im'
148'
...
160'
.
.
Event #1 - Model/Data Comparison
___
i8n'
6-record Genetic Inverse - Fitness=0.642
600
$
60'
60'
pI
400
-9
8
40'
10°
200
2
: o
U
0
u)'
20'
0'
al
60'
80'
loo'
120'
140'
160'
180'
Figure 11. Locations of six events used in the GA inversion of P
coda at station ARU (Arti, Russia), for anisotropic crustal structure.
Hypocentral parameters of these events are listed in Table 3.
PI
Table 3. Earthquakes recorded at the seismic station ARU (58.2"N;
#6
13. D
date
13/05/93
15/11/94
31/03/95
10/16/94
24/06/95
23/04/95
latitude
14.4"N
5.6"s
38.2"N
45.7"N
3.9"s
51.3"N
400
200 L
2
0
0
5
Time (sec)
10
-->
15
20
phase velocity
25
13.90
Event #2 - Model/Data Comparison
6-record Genetic Inverse - Fitness=0.642
$
16000
a 12000
Synthetic
8000
fl
2 4000
0
G
*
O t l
I
0
5
10
15
20
Time (sec) -->
phase velocity
I
25
19.20
,
P)
pI
I
16000
12000
8o
Observed
li
2 4000
8
0
0
5
Time (sec) -->
15
20
phase velocity
10
25
19.20
Figure 12. GA waveform inversion for P waves arriving at station
56.4"E) in central Russia.
#5
25
I
8
U
#4
20
4
U
A composite of 11 six-record GA inversions (Fig. 13) shows
a clear preference for anisotropy at 0-5 km and 3 2 4 2 km
depth. There is a tendency for near-mantle velocities in the
lower 2 km of the crust, suggesting that our prescribed crustal
thickness is too great. Note also the velocity gradient in the
lower, anisotropic portion of the composite crustal model. No
single GA inverse model replicates this behaviour precisely, but
many possess a 5-10 km anisotropic layer of intermediate
velocity in the deep crust. A layer of this type suggests the
imbrication of mantle and crustal rocks, similar to the mixture
of ultramafic and metapelite rocks exposed in the Ivrea region
of Northern Italy (Quick, Sinigoi & Mayer 1995).
Although anisotropy in the composite model for ARU peaks
at roughly 8 per cent in the shallow and deep crust, the 10-layer
models obtained by GA inversion typically possess layers with
anisotropy in excess of 10 per cent. We enforced an upper limit
of 20 per cent anisotropy in the GA model search, and this level
was met in some layers. The orientation of anisotropy in the
six-record 'inversion set' for ARU shows consistent 'fast' and
'slow' orientations in the upper and lower crust (Fig. 14). In
both depth ranges, the values of fi correlate with the SKS fastaxis strike of 68" reported by Helffrich, Silver & Given (1994),
and confirmed by our own estimates. In the shallow crust,
models tend to exhibit either a subhorizontal slow axis with
strike subparallel to the SKS fast axis, or tilted axes, either fast
or slow, that lie in an equatorial 'belt' perpendicular to the SKS
#2
#3
15
phase velocity
$ 600
B
2
#1
10
-->
D'
40'
event
5
Time (sec)
longitude
40.2"E
110.2"E
135.O"E
149.1"E
153.9"E
179.7"E
01997 RAS, GJI 131,253-266
depth, km
19
561
354
117
386
17
A,
44
75
52
54
96
62
backazimuth, '
206
126
76
60
88
37
ARU from (a) event #1 at backazimuth 206" and (b) event #2 at
backazimuth 126". Note how the GA inversion attempts to fit the
transverse-motion reverberation evident in both events, particularly
the Ps converted phase that trails direct P by 5 s in event #2. The
'projection' (9) of the data spectra [ V ( f ) , R ( f )T, ( f ) ] onto the
spectral response [ k ( f )T, ( f ) , P ( f ) ] of the GA velocity model can
lead to non-causal waveforms, as evident for event # I .
264
V. Levin and J. Park
Composite Crustal Model
8000
I
arei 7000
d
-10%
P-velocity (dsec)
Qa
5000
al
1Yo
a .
D
P anisotropy
6000
10%
0
S anisotropy
A'
al
4
-1%
J
k
4000
4
3000
density (kglm"3)
I
I
0
10
.
I
I
I
20
30
Depth (km)
I
40
Figure 13. Composite of 11 10-layer anisotropic models for the crust
beneath ARU, calculated according to weighted sum (13). The models
were obtained with GA waveform inversion of P coda from six events.
Maximum (solid line), minimum (solid line) and intermediate (dashed
line) velocities in the anisotropic layers are shown. Although computed
from hexagonally symmetric crustal models, the composite model's
anisotropy does not typically possess an axis of symmetry, nor does
it necessarily achieve an optimal waveform fit. Nevertheless, model
features shared by many GA inversions, such as anisotropic velocity in
the top few kilometres and in the 3 2 4 2 km depth range, are highlighted by the composite model. The units are m s-' for velocities o! and
/3, and kg m-3 for density p.
fast axis. In the deep crust, the distribution of fi also favours
strikes parallel and normal to the SKS fast axis, with a
cluster of slow axes with 15-75" tilt near 240" azimuth. This
anisotropy would contribute to SKS splitting, but its effect
is small, approximately 0.25 s. The results of this experiment
are encouraging, and suggest a reconnaissance method for
detecting crustal anisotropy. The use of a handful of singleevent records in the GA inversion would be useful, say, for data
from short deployments of portable broad-band seismometers.
For permanent stations, however, inverting composite, or
stacked, receiver functions for narrow backazimuth sectors
would probably be a more efficient use of larger data sets (e.g.
Abers, Hu & Sykes 1995; Savage 1997; Levin & Park 1997b).
However, GA inversion cannot converge on crustal models
which lie outside its range, for example models with dipping
interfaces. This type of discrimination requires more careful
analysis of azimuthal variation, as suggested above; see also
Savage (1997). Levin & Park (1997b) show that the Ps amplitude at ARU has a mixture of two- and four-lobed azimuthal
dependence, which argues for anisotropy.
0"
*O"
0"
10.0 -> 15.0 km
180'
32.0 -> 38.0 km
180'
0"
"O'
Figure 14. Anisotropy within the uppermost, middle and deep crust
in models obtained by GA inversion of six P coda observed at
ARU. Different realizations of the genetic algorithm are obtained by
using different random-number seeds. Within each representative
depth range, we plot parameters for layers exceeding 0.5 km in thickness. The orientation angles 0, 5 are plotted in polar coordinates,
equivalent to a map view of an upward-pointing fi. Symbol size
indicates the peak-to-peak velocity anisotropy. Squares indicate
anisotropy with a fast symmetry axis ( B , E > 0). Triangles indicate
anisotropy with a slow symmetry axis (B, E < 0). Solid arrows on the
bottom left plot depict the fast direction from SKS splitting measurements (Helffrich et d.1994). The orientation of anisotropy in the crust
suggested by the clustering of symmetry axes from multiple geneticinversion realizations appears to be consistent with the direction of the
fast shear velocity in the mantle.
CONCLUSIONS
Synthetic seismograms computed in a stack of horizontal
layers with constant anisotropic velocity and arbitrary orientation of an axis of symmetry contain P-SH conversions.
These converted phases are similar to transverse phases
commonly observed in teleseismic P codas. In addition to the
impedance contrast, the amplitudes of these P-SH converted
phases are controlled by the intensity and relative proportions
of P-and S-velocity anisotropy, and the mutual orientation of
the ray path and the axis of symmetry. Anisotropy in a 1-D
velocity structure also affects the amplitudes of P-SV conversion. Azimuthal amplitude patterns of P-S converted phases
in anisotropic structures are proportional to sin 5 and sin 25 for
SH-polarized phases, and cos 5 and cos 25 for SV-polarized
0 1991 RAS, GJI 131, 253-266
P-SH conversions in uflat-layered medium
phases. This last feature of anisotropically imposed P-SH
conversions presents a problem for techniques that seek to
enhance specific phases via stacking with a predetermined
azimuthal pattern, usually cos 25. If observations from different backazimuths are stacked to detect anisotropy, it seems
necessary to consider patterns of the more general form, e.g.
W’l sin 5
Wz sin 25.
Relatively simple crustal models that include anisotropy
may yield reverberations of high complexity. P-SH conversions stemming from 1-D anisotropic velocity structures
exhibit azimuthal variations that may easily be misinterpreted
as evidence of 2-D or 3-D lateral heterogeneity. Relatively
simple anisotropic models can satisfy P-wave observations
with prominent P-SH conversions, promising an expanded
spectrum of options for studies of the lithosphere.
The effects of dipping interfaces and seismic anisotropy with
a tilted axis of symmetry are not formally distinguishable if
only single records are considered. Discrimination may be
hard even when multi-azimuth observations are available, as
certain geometries yield virtually identical amplitude patterns.
Azimuthal variation of relative P-Ps phase delay may provide
the best clues to the origin of observed P-SH conversion.
Alternatively, one may use nearby refraction studies and nonseismological information, such as considerations of the local
geology.
Using a genetic-algorithm model search, we have shown that
salient model characteristics, such as the depth range of
anisotropy and ‘slow’ and ‘fast’ orientations, can be retrieved
from as few as six P-coda from events at a variety of backazimuths. The non-linearity of waveform fitting and the
large number of model parameters lead to substantial nonuniqueness among models. This appears to be best overcome
by forming a composite model from multiple realizations of
the GA inversion scheme, and by plotting a stereoscopic plot of
‘fast’ and ‘slow’axis orientations for all models.
+
ACKNOWLEDGMENTS
This work was supported by the US Air Force Office of
Scientific Research contract F49620-94-1-0043. Computational
firepower was supported by NSF grant EAR-9528484. Zachary
Grinspan contributed to the genetic-algorithm code development. This work benefited from the stimulating atmosphere
of the 7th International Conference on Seismic Anisotropy,
held in Miami, Florida, in March 1996. Comments and
suggestions from M. K. Savage, G. Abers, S. Ward and an
anonymous reviewer helped us to improve the manuscript.
GMT graphical software (Wessel & Smith 1991) was used in
figure preparation.
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