Geophys. J. Int. (1998) 132, 654–666
Effects of slight anisotropy on surface waves
Erik W. F. Larson, Jeroen Tromp and Göran Ekström
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA
Accepted 1997 September 7. Received 1997 September 7; in original form 1997 March 26
SU MM A RY
We present a complete ray theory for the calculation of surface-wave observables from
anisotropic phase-velocity maps. Starting with the surface-wave dispersion relation in
an anisotropic earth model, we derive practical dynamical ray-tracing equations. These
equations allow calculation of the observables phase, arrival angle and amplitude in a
ray theoretical framework. Using perturbation theory, we also obtain approximate
expressions for these observables. We assess the accuracy of the first-order approximations by using both theories to make predictions on a sample anisotropic phasevelocity map. A comparison of the two methods illustrates the size and type of errors
which are introduced by perturbation theory. Perturbation theory phase and arrivalangle predictions agree well with the exact calculation, but amplitude predictions are
poor. Many previous studies have modelled surface-wave propagation using only
isotropic structure, not allowing for anisotropy. We present hypothetical examples to
simulate isotropic modelling of surface waves which pass through anisotropic material.
Synthetic data sets of phase and arrival angle are produced by ray tracing with exact
ray theory on anisotropic phase-velocity maps. The isotropic models obtained by
inverting synthetic anisotropic phase data sets produce deceptively high variance
reductions because the effects of anisotropy are mapped into short-wavelength isotropic
structure. Inversion of synthetic arrival-angle data sets for isotropic models results in
poor variance reductions and poor recovery of the isotropic part of the anisotropic
input map. Therefore, successful anisotropic phase-velocity inversions of real data
require the inclusion of both phase and arrival-angle measurements.
Key words: anisotropy, Earth structure, surface waves, tomography.
1
I NTR O D UC TIO N
Lateral variations in phase velocity of surface waves have
been recognized for many years, starting as early as the 1920s
(Tams 1921). Large-scale isotropic variations in surface-wave
phase velocity were mapped in the 1960s, 1970s and 1980s (e.g.
Oliver 1962; Dziewonski 1971; Nakanishi & Anderson 1984;
Wong 1989). In some studies, anisotropic structure also has
been used to model surface waves (e.g. Forsyth 1975; Nataf,
Nakanishi & Anderson 1986; Nishimura & Forsyth 1988;
Montagner & Tanimoto 1990). However, in recent highresolution studies based on phase measurements, only isotropic
structure has been modelled (Zhang, Lay & Ammon 1993;
Trampert & Woodhouse 1995; Ekström, Tromp & Larson
1997). Anisotropy was not included for a number of reasons:
it was assumed to be small, it was more difficult to model, and
it was not clear that phase measurements were very sensitive
to anisotropy on a global scale. Arrival-angle measurements
may increase our sensitivity to anisotropy, but currently only
isotropic structure has been used to model arrival-angle data
(Laske & Masters 1996).
654
Anisotropic surface-wave ray-tracing equations which determine the ray and phase have been presented previously
(Tanimoto 1987; Mochizuki 1990; Tromp 1994). A full practical
ray theory for dealing with anisotropic phase-velocity maps
such as exists for isotropic maps (Woodhouse & Wong 1986;
Wang & Dahlen 1994) is, however, lacking. Equations for
predicting phase, arrival-angle and amplitude based upon exact
ray theory as well as perturbation theory are needed for the
analysis of surface-wave propagation in anisotropic structures.
To analyse large data sets of surface-wave observations, it is
much more efficient to use inversion techniques based upon
linear perturbation theory than exact ray tracing, but the
accuracy of the linear approximation needs to be assessed. We
present practical dynamic ray-tracing equations and perturbation
theory approximations for surface-wave propagation and we
assess the accuracy of these approximations.
The effect of neglecting anisotropy in studies of global phasevelocity heterogeneity has not been previously addressed.
Recent high-resolution isotropic phase-velocity maps which
have been derived from only phase measurements generally
agree at long wavelengths. Good agreement does not, however,
© 1998 RAS
EVects of slight anisotropy on surface waves
preclude a bias of these maps due to neglecting anisotropy,
even though the studies use different data sets. The addition
of arrival angle data results in higher-resolution isotropic phasevelocity maps, but this may be a trade-off with unmodelled
anisotropic effects. In this paper, we assess the effects of
neglecting anisotropy in surface-wave phase-velocity modelling.
We also study the sensitivity of surface-wave observables to
anisotropic structure.
(Tromp & Dahlen 1993; Tromp 1994)
A
S LI GH T A N IS O TR O P Y
In this section we determine the effects of slight anisotropy
on dispersed surface-wave packets. Position vectors defined
relative to the Earth’s centre of mass are denoted by r,
and points on the unit sphere are denoted by r̂=r/r, where
r=drd denotes the radius. We define the wavevector k in
terms of the surface-wave phase y by
k=V y ,
(2.1)
1
where the surface gradient V is related to the gradient V by
1
V=r̂∂ +r−1V . The surface-wave dispersion relation on an
r
1
anisotropic earth model may be written in the form (Tromp
& Dahlen 1993; Tromp 1994)
v2I −kΩI Ωk−kΩI −I =0 ,
(2.2)
1
2
3
4
where v denotes angular frequency. We choose to neglect the
effects of rotation and self-gravitation in this paper. The scalars
I and I , the vector I and the symmetric, second-order tensor
1
4
3
I are defined by
2
a
rsΩs*r2dr ,
(2.3)
I =
1
0
a
sΩCΩs*dr ,
(2.4)
I =
2
0
a
{sΩC : [(∂ s−r−1s)*r̂+r−1(r̂Ωs)*I ]
I =i
r
3
0
−[(∂ s−r−1s)r̂+r−1(r̂Ωs)I ] : CΩs*} rdr ,
(2.5)
r
a
[r̂(∂ s−r−1s)+r−1(r̂Ωs)I ] : C
I =
r
4
0
:[(∂ s−r−1s)*r̂+r−1(r̂Ωs)*I ] r2dr .
(2.6)
r
Here r denotes the Earth’s distribution of density, C denotes
the elastic tensor, a denotes the Earth’s radius, and an asterisk
denotes complex conjugation. The vector s may be defined in
terms of three local radial eigenfunctions U, V and W as
P
P
P
P
s=Ur̂+iV k̂−iW r̂×k̂ .
(2.7)
The unit vector k̂ is given by k/k, where k=(kΩk)1/2 denotes
the magnitude of the wavevector, i.e. the wavenumber. In a
general anisotropic earth model, the local radial eigenfunctions
U, V and W are determined by a system of three coupled
second-order ordinary differential equations (Tromp 1994,
eqs 48–55). The equations are a function of the direction of
the wavevector k̂. Only in the case of a transversely isotropic
earth model do the equations for U and V decouple from the
equations for W . In that case U and V describe the Rayleighwave motion and W describes the Love-wave motion. The
local group velocity in an anisotropic earth model is given by
© 1998 RAS, GJI 132, 654–666
B
1
G=∂v/∂k=(cI )−1 I Ωk̂+ k−1I .
2
3
1
2
(2.8)
In this paper we are interested in the propagation of surface
waves on a slightly anisotropic earth model. Therefore, we
write the elastic tensor in the form
C
2
655
ijkl
A
B
2
= k− m d d +m(d d +d d )+c ,
ij kl
ik jl
il jk
ijkl
3
(2.9)
where k and m describe the bulk and shear moduli in an
isotropic reference model, and c represents the slight anisotropy. The results in this section are also applicable to a
transversely isotropic rather than an isotropic reference model.
As a result of Rayleigh’s principle, correct to first order
in c, the fractional angular frequency of a surface wave with
wavevector k is perturbed by an amount
dv/v=(2v2I )−1(kΩdI Ωk+kΩdI +dI ) ,
(2.10)
1
2
3
4
where dI , dI and dI are given by
2
3
4
a
sΩcΩs*dr ,
(2.11)
dI =
2
0
a
{sΩc : [(∂ s−r−1s)*r̂+r−1(r̂Ωs)*I ]
dI =i
r
3
0
−[(∂ s−r−1s)r̂+r−1(r̂Ωs)I ] : cΩs*} rdr ,
(2.12)
r
a
[r̂(∂ s−r−1s)+r−1(r̂Ωs)I ] : c
dI =
r
4
0
:[(∂ s−r−1s)*r̂+r−1(r̂Ωs)*I ] r2dr ,
(2.13)
r
and s is the local surface-wave displacement in the isotropic
reference model. Thus for Love waves we have s=−iW r̂×k̂
and for Rayleigh waves s=Ur̂+iV k̂, where U, V and W are
the local radial eigenfunctions of the isotropic reference model.
We assume that the unperturbed Love and Rayleigh waves are
normalized such that
P
P
P
cGI =1 ,
(2.14)
1
where G denotes the group velocity in the isotropic reference
model. For Love waves the integral I is given by
1
a
rW 2r2 dr ,
(2.15)
I =
1
b
where b denotes the radius of the core–mantle boundary. For
Rayleigh waves we have instead
P
I =
1
P
a
r(U2+V 2) r2dr .
(2.16)
0
The fractional phase-velocity perturbation e=dc/c is related
to the fractional angular-frequency perturbation (2.10) by
1
e=dc/c=(c/G)(dv/v)= k−2(kΩdI Ωk+kΩdI +dI ) ,
2
3
4
2
(2.17)
as in Backus (1964), where we have used the normalization
(2.14).
At this point is is convenient to introduce spherical coordinates h and w denoting colatitude and longitude, respectively.
Unit vectors ĥ and ŵ are defined in the directions of increasing
656
E. W. F. L arson, J. T romp and G. Ekström
Then we have for Love waves
Figure 1. In an anisotropic earth model, the wavevector k is generally
not aligned with the group velocity G. Energy travels along the ray
in the direction of the group velocity. The angles j and f, measured
counterclockwise from the east, determine the directions of the group
velocity and the wavevector, respectively.
colatitude and longitude. Let us express the wavevector k in
terms of an angle f, measured counterclockwise from due east,
as shown in Fig. 1, such that
k=−k sin fĥ+k cos fŵ .
P
1
e = k−2
1 2
P
1
e = k−2
2 2
P
Ĝ=[k̂+(∂ ln c)r̂×k̂][1+(∂ ln c)2]−1/2 .
(2.19)
f
f
Notice that on an anisotropic earth model the direction of the
group velocity Ĝ generally does not coincide with the direction
of the wavevector k̂, as illustrated in Fig. 1. The direction of
the group velocity is defined by the angle j, which is determined
by
tan(j−f)=ĜΩk̂=∂ ln c .
(2.20)
f
After a significant amount of algebra, eq. (2.17) can be
rewritten in the form
e=e +e cos 2f+e sin 2f+e cos 4f+e sin 4f .
(2.21)
0
1
2
3
4
Smith & Dahlen (1973) and Montagner & Nataf (1986) derived
an equation of the form (2.21) for a flat, laterally homogeneous
reference model, and Romanowicz & Snieder (1988) derived
a similar expression from a normal-mode perspective for a
spherical reference model. The result (2.21) is valid for a
laterally slowly varying, spherical reference model. Let us
define five transversely isotropic elastic parameters A, C, L , N
and F as in Love (1927):
1
1
3
)+ c
+ c
,
A= (c +c
hhhh
wwww
hhww
8
4
2 hwhw
(2.22)
C=c
(2.23)
(2.24)
1
1
1
)− c
+ c
,
N= (c +c
wwww
8 hhhh
4 hhww 2 hwhw
(2.25)
1
F= (c +c ) .
rrww
2 rrhh
(2.26)
a
b
[Nk2r−2W 2+L (∂ W −r−1W )2] r2dr ,
r
a1
(c −c )(∂ W −r−1W )2r2dr ,
rwrw r
2 rhrh
b
a
c
b
PC
1
e =− k−2
4
2
P
(2.27)
(2.28)
(∂ W −r−1W )2r2dr ,
rhrw r
1
e =− k−2
3
2
(2.29)
D
1
1
a 1
(c +c
)− c
− c
k2W 2 dr ,
wwww
8 hhhh
4 hhww 2 hwhw
b
(2.30)
a1
(c −c
)k2W 2 dr .
wwwh
2 hhhw
b
(2.31)
For Rayleigh waves we have instead
1
e = k−2
0 2
(2.18)
In terms of the angle f, the direction of the group velocity
Ĝ=G/dGd is given by (Tromp 1994)
,
rrrr
1
L = (c +c ) ,
rwrw
2 rhrh
1
e = k−2
0 2
P
a
0
[C(∂ U)2+L (∂ V −r−1V +kr−1U)2
r
r
+2Fr−1∂ U(2U−kV )+(A−N)r−2(2U−kV )2
r
+Nr−2k2V 2] r2dr ,
1
e = k−2
1 2
(2.32)
PC
a 1
(c −c )(∂ V −r−1V +kr−1U)2
rhrh r
2 rwrw
0
1
)kr−2V (2U−kV )
+ (c −c
wwww
2 hhhh
+(c
rrhh
1
e = k−2
2 2
+(c
−c
P
+c
rrhw
D
)kr−1V ∂ U r2dr ,
r
(2.33)
(∂ V −r−1V +kr−1U)2
rhrw r
[−c
0
hhhw
+2c
a
rrww
wwwh
)kr−2V (2U−kV )
kr−1V ∂ U] r2dr ,
r
1
e = k−2
3 2
PC
1
e = k−2
4 2
P
(2.34)
D
a 1
1
1
k2V 2dr ,
(c +c
)− c
− c
wwww
8 hhhh
4 hhww 2 hwhw
0
(2.35)
a1
(c −c
)k2V 2 dr .
wwwh
2 hhhw
0
(2.36)
These expressions are in agreement with Romanowicz &
Snieder (1988). The parameters e , e , e , e and e are functions
0 1 2 3
4
of latitude and longitude. Once maps of the fractional
phase-velocity perturbation e have been established over a
range of frequencies, these maps may be subsequently inverted
in terms of 3-D lateral variations in anisotropy based
upon (2.27)–(2.36). It is interesting to note that surface-wave
propagation on a slightly anisotropic earth model is governed
© 1998 RAS, GJI 132, 654–666
EVects of slight anisotropy on surface waves
by 13 elastic parameters rather than the full 21 elements of c
(Montagner & Nataf 1986).
In terms of the generalized spherical harmonics (Phinney &
Burridge 1973)
3
1
1
P0 (cos f)= [3(cos f)2−1]= cos 2f+ ,
20
2
4
4
(2.37)
1
1
P1 (cos f)= √ 6 cos f sin f= √ 6 sin 2f ,
20
2
4
(2.38)
1
P0 (cos f)= [35(cos f)4−30(cos f)2+3]
40
8
=
1
(35 cos 4f+20 cos 2f+9) ,
64
A
(2.39)
B
1
1
1
P3 (cos f)= √ 35 cos f(sin f)3= √ 35 sin 2f− sin 4f .
40
4
16
2
(2.40)
We may express eq. (2.21) in the form
e=e∞ +e∞ P0 +e∞ P1 +e∞ P0 +e∞ P3 ,
0
1 20
2 20
3 40
4 40
(2.41)
1
1
e∞ =e − e − e ,
0
0 3 1 15 3
A
B
given by
A eyr =n̂Ωs ,
r
r
(2.42)
(2.50)
A eyp =(8pk S )−1/2exp i
p
r r
GP
D
0
k[1+(∂ ln c)2]−1/2 dD
f
H
p
p
,
+ −M
4
2
(2.51)
P
D v
dD ,
(2.52)
2GQ
0
A eys =−(iv)−1(M : E* ) .
(2.53)
s
s
The polarization direction of the receiver is denoted by n̂.
Geometrical spreading is represented by the factor S , the
r
angular distance along a ray is denoted by D, the local quality
factor is denoted by Q, and M denotes the Maslov index,
which starts at zero and increases by one upon each passage
through a caustic. Because the local radial eigenfunctions U,
V and W are real, the additional variation in phase (the Berry
phase) reported in Tromp (1994) vanishes. The Fourier transform of the moment-rate tensor is denoted by M, and E is
s
the JWKB strain tensor at the source. For Love waves we
have (Tromp & Dahlen 1992)
A =exp−
a
s =−iW r̂ ×k̂ ,
r
r
r r
where
657
(2.54)
C
D
1
(M −M ) sin 2f −M cos 2f
M : E* =r−1 k W
ww
s
hw
s
s
s s s 2 hh
−i(∂ W −r−1 W )(M cos f +M sin f ) ,
r s
s
s
rh
s
rw
s
whereas for Rayleigh waves
(2.55)
s =U r̂ +iV k̂ ,
r
r r
r r
(2.56)
4
4
e1 = e − e ,
1 3 1 7 3
(2.43)
2
e∞ = √ 6(e +2e ) ,
2 3
2
4
(2.44)
64
e∞ = e ,
3 35 3
(2.45)
1
(M −M ) cos 2f +M sin 2f
+r−1 k V
ww
s
hw
s
s s s 2 hh
32
e∞ =− √ 35e .
4
4
35
(2.46)
−i(∂ V −r−1 V +r−1 k U )(M cos f −M sin f ) .
r s
s s
s s s
rw
s
rh
s
(2.57)
The representation (2.41) may be more convenient for practical
purposes because it enables one to damp the directional
dependence on f with the angular degree of the generalized
spherical harmonics (Trampert & Woodhouse 1996).
Using the notation of Wang & Dahlen (1994) and the
surface-wave JWKB response on an anisotropic earth model
obtained by Tromp (1994, eq. 153), we write the JWKB
response on a slightly anisotropic earth model in the form
u= ∑ Aeiy ,
rays
(2.47)
A=A A A A ,
r p a s
(2.48)
y=y +y +y .
r
p
s
(2.49)
Throughout this paper we use subscripts s, p, a and r to label
source-, path-, attenuation- and receiver-dependent parameters,
respectively. The quantities A , A , A , A , y , y and y are
r p a s r p
s
© 1998 RAS, GJI 132, 654–666
B
C
D
We have ignored the effects of slight anisotropy at the source
and receiver locations on the radial eigenfunctions U, V and W
in (2.54)–(2.57). This cannot be strictly justified theoretically,
and the effects are difficult to calculate (Martin & Thomson
1996). If the earth model exhibits anisotropy at the location
of the receiver, its effect would be to produce a Love-wave
arrival on the vertical and longitudinal components and a
Rayleigh-wave arrival on the transverse component. However,
in the case of isotropic source and receiver locations, eqs
(2.54)–(2.57) are correct.
3
where
A
1
M : E* =∂ U M +r−1 U − k V (M +M )
s 2 s s
hh
ww
s
r s rr
s
AN I SO T R O PI C R AY TH EO R Y
In this section we derive the dynamical ray-tracing equations
for surface-wave propagation on an anisotropic earth model.
The theory in this section is applicable to completely general
anisotropic earth models.
In terms of its covariant components k =∂ y and k =∂ y,
h
h
w
w
the wavevector k may be written in the form
k=V y=k ĥ+k (sin h)−1ŵ .
1
h
w
(3.1)
658
E. W. F. L arson, J. T romp and G. Ekström
Using the angle f defined in (2.18), the covariant components
of the wavevector may be expressed in terms of this angle as
k =−k sin f ,
k =k cos f sin h .
(3.2)
h
w
Conversely, the angle f is determined in terms of colatitude
and the covariant components of the wavevector by
tan f=−sin hk /k .
(3.3)
h w
For any given angular frequency v, the local phase
velocity on an anisotropic earth model can be expressed as
c=c(h, w, f, v), where the dependence on f reflects the fact
that surface waves travel with different phase velocities in
different directions. Following Tromp (1994), the surface-wave
dispersion relation may be written in the form
H=0 ,
(3.4)
where the Hamiltonian, H, is defined by
H(h, w, k , k , v)
h w
1
= [k2 +(sin h)−2k2 −v2c−2(h, w, f(h, k , k ), v)] .
w
h w
2 h
(3.5)
Notice how the explicit dependence of the angle f on colatitude
and the covariant components of the wavevector has been
indicated, as stipulated by (3.3).
The surface-wave ray geometry is determined in terms of a
generating parameter n by considering h(n), w(n), k (n) and
h
k (n). Hamilton’s equations for H are
w
dh
=∂ H=k −k (sin h)−1∂ ln c ,
(3.6)
kh
h
w
f
dn
dw
=∂ H=k (sin h)−2+k (sin h)−1∂ ln c ,
w
h
f
kw
dn
dk
h =−∂ H=−v2c−2∂ ln c+k2 (sin h)−3 cos h
h
h
w
dn
+k k (sin h)−2 cos h∂ ln c ,
h w
f
dk
w =−∂ H=−v2c−2∂ ln c .
w
w
dn
(3.7)
(3.8)
(3.9)
(3.10)
The phase y may be determined from
dy dh
dw
= ∂ y+ ∂ y=k2 .
dn dn h
dn w
dh
=−sin h(tan f+∂ ln c)(1−tan f∂ ln c)−1 ,
f
f
dw
(3.15)
df
=(sin h∂ ln c+tan f∂ ln c−cos h)(1−tan f∂ ln c)−1 ,
h
w
f
dw
(3.16)
We use eqs (3.15) and (3.16) to determine the surface-wave
ray geometry on an anisotropic earth model. Finding the
geometric ray that connects a given source and receiver
constitutes a shooting or bending problem:
h(w )=p/2 ,
h(w )=p/2 .
(3.17)
s
r
For any given surface-wave orbit, the objective is to find the
initial direction of the wavevector, f , for which the ray hits
s
the receiver. Remember that the take-off angle of the ray is
determined by j =f +(∂ ln c) , as illustrated in Fig. 1.
s
s
f
s
In terms of longitude w, the surface-wave phase y is
subsequently determined from
dy
=vc−1 sin h(cos f)−1(1−tan f∂ ln c)−1 .
f
dw
(3.18)
The effects of attenuation may be determined from
d
ln A =−v(2GQ)−1 sin h(cos f)−1(1−tan f∂ ln c)−1 .
a
f
dw
(3.19)
Tromp (1994) demonstrates that the ray-tracing equations
(3.6)–(3.9) are equivalent to those obtained by Tanimoto
(1987). The epicentral distance D is related to the generating
parameter n by (Tromp 1994)
dD
=k[1+(∂ ln c)2]1/2 .
f
dn
We can always rotate our coordinate system such that both
the source and the receiver are located on the equator. It is
then convenient to choose w as the independent parameter,
which reduces the number of ray-tracing equations further
from three to two:
(3.11)
Because the covariant components of the wavevector k and
h
k are related through the wavenumber k, the number of rayw
tracing equations may be reduced from four to three. We obtain
dh
=−vc−1 sin f(1+cot f∂ ln c) ,
f
dn
(3.12)
dw
=vc−1 cos f(sin h)−1(1−tan f∂ ln c) ,
f
dn
(3.13)
df
=vc−1[cos f∂ ln c+sin f(sin h)−1∂ ln c−cos f cot h] .
h
w
dn
(3.14)
The geometrical-spreading factor S defined in (2.51)
r
represents the effects of focusing and defocusing (Tromp &
Dahlen 1993; Tromp 1994). At the receiver, S is determined by
r
∂w
∂h
∂h
S =k−1
= cos f (1−tan f ∂ ln c )
,
r
r
r
r
f
r
∂n
∂f
∂f
s r
s r
r
(3.20)
KA B A B K K
A BK
where we have used the fact that both the source and the
receiver are located on the equator. The partial derivative ∂h/∂f
s
may be obtained by differentiating the ray-tracing equations
(3.15) and (3.16) with respect to the initial direction of the
wavevector f , holding the longitude w fixed, with the result
s
d ∂h
=−{cos h(tan f+∂ ln c)+sin h
f
dw ∂f
s
×[1+(1−tan f∂ ln c)−1
f
×tan f(tan f+∂ ln c)]∂ ∂ ln c}
f
h f
∂h
×(1−tan f∂ ln c)−1
f
∂f
s
−sin h{[(cos f)−2+∂2 ln c]
f
+(tan f+∂ ln c)(1−tan f∂ ln c)−1
f
f
×[(cosf)−2∂ ln c+tan f∂2 ln c]}
f
f
∂f
×(1−tan f∂ ln c)−1
,
(3.21)
f
∂f
s
A B
A B
A B
© 1998 RAS, GJI 132, 654–666
EVects of slight anisotropy on surface waves
A B
d ∂f
=[cos h∂ ln c+sin h∂2 ln c+tan f∂ ∂ ln c+sin h
h
h
h w
dw ∂f
s
+(sin h∂ ln c+tan f∂ ln c−cos h)
h
w
×(1−tan f∂ ln c)−1 tan f∂ ∂ ln c]
f
h f
∂h
+{(cos f)−2∂ ln c
×(1−tan f∂ ln c)−1
w
f
∂f
s
+sin h∂ ∂ ln c+tan f∂ ∂ ln c
f h
f w
+(sin h∂ ln c+tan f∂ ln c−cos h)
h
w
×(1−tan f∂ ln c)−1[(cos f)−2∂ ln c+tan f∂2 ln c]}
f
f
f
∂f
.
(3.22)
×(1−tan f∂ ln c)−1
f
∂f
s
The appropriate initial conditions are
A B
A B
A B
A B
∂h
∂f
(0)=0 ,
(0)=1 .
(3.23)
∂f
∂f
s
s
We use (3.21) and (3.22) to determine the surface-wave
amplitude on an anisotropic earth model.
To conclude this section we demonstrate that on an isotropic
earth model, when ∂ ln c=0, our results reduce to those of
f
Tromp & Dahlen (1993). In that case the ray-tracing equations
(3.15) and (3.16) become
dh
=−sin h tan f ,
dw
(3.24)
df
=sin h∂ ln c+tan f∂ ln c−cos h .
h
w
dw
(3.25)
The phase is determined by
dy
=vc−1 sin h(cos f)−1 ,
dw
(3.26)
and attenuation is determined from
d
ln A =−v(2GQ)−1 sin h(cos f)−1 .
a
dw
(3.27)
The dynamical ray-tracing equations (3.21) and (3.22) reduce
to
A B
A B
A B
∂h
∂f
d ∂h
=−cos h tan f
−sin h(cos f)−2
,
dw ∂f
∂f
∂f
s
s
s
A B
d ∂f
=[cos h∂ ln c+sin h∂2 ln c+tan f∂ ∂ ln c
h
h
h w
dw df
s
∂h
∂f
+(cos f)−2∂ ln c
.
+sin h]
w
∂f
∂f
s
s
A B
A B
for tomographic inversions. The ray-perturbation theory presented here is similar to that discussed by Farra & Madariaga
(1987), Coates & Chapman (1990) and Mochizuki (1990), and
is valid for weakly heterogeneous and slightly anisotropic
earth models.
We assume that lateral variations in phase velocity and
anisotropy are small and write the phase velocity as
c(v)[1+e(h, w, f, v)] .
(4.1)
The laterally homogeneous, isotropic phase velocity in the
spherical reference model is denoted by c, and lateral variations
in velocity and anisotropy are represented by the fractional
phase-velocity perturbation e=dc/c given in (2.21). We make
the following substitutions in (3.15) and (3.16):
h p/2+dh ,
f df ,
(4.2)
where we have used the fact that on a spherically symmetric
earth model the surface-wave ray is determined by h=p/2 and
f=0. The ray direction j, shown in Fig. 1, may be obtained
by making the substitution
j dj=df+∂ e .
(4.3)
f
The perturbation df represents the direction of the wave vector
k̂ and the perturbation dj represents the direction of the group
velocity Ĝ. Surface-wave energy travels with the group velocity,
therefore it is the group arrival angle which is observed. To
first order in the small perturbations we obtain the following
equations for the perturbed ray:
d
dh=−df−∂ e ,
f
dw
(4.4)
d
df=dh+∂ e .
h
dw
(4.5)
Let us assume for simplicity that the source is located on the
equator at 0° longitude, and that the receiver is located on the
equator at longitude D, such that D is the epicentral distance.
We require that the perturbed ray emanates from the same
source as the unperturbed ray, and that the perturbed ray hits
the same receiver as the unperturbed ray. This implies that
dh(0)=0 ,
dh(D)=0 .
(4.6)
Eqs (4.4) and (4.5) may be written in the vector form
dy
=Ay+f ,
dw
(3.28)
(4.7)
where
A B
A
dh
,
A=
0 −1
B
A B
−∂ e
f .
(4.8)
df
1 0
∂ e
h
The solution to (4.7) is most conveniently expressed in terms
of a propagator matrix P(w, w∞) (Aki & Richards 1980), which
is determined by
y=
(3.29)
659
,
f=
4 A NI S O TR O P IC R AY- P ER T UR B ATI O N
T HE O R Y
dP
=AP ,
dw
In this section we demonstrate how take-off and arrival angles
may be estimated based on perturbation theory. Good estimates of the take-off angle substantially decrease the number
of iterations required to ‘hit’ the receiver using exact ray
tracing. The arrival-angle predictions may be used as a basis
where I denotes the 2×2 identity matrix. The propagator
matrix P is given by
© 1998 RAS, GJI 132, 654–666
P(w, w∞)=
P(w, w)=I ,
A
(4.9)
cos(w−w∞)
−sin(w−w∞)
sin(w−w∞)
cos(w−w∞)
B
.
(4.10)
E. W. F. L arson, J. T romp and G. Ekström
660
In terms of the propagator matrix (4.10), the solution to (4.7)
may be written in the form
y(w)=P(w, 0)
CP
w
P−1(w∞, 0)f(w∞) dw∞+y(0)
0
=
P
D
w
P(w, w∞)f(w∞) dw∞+P(w, 0)y(0) .
0
The second equality holds because of the relation
(4.11)
P(w, 0)P−1(w∞, 0)=P(w, 0)P(0, w∞)=P(w, w∞) .
(4.12)
The result (4.11) is most easily verified by direct substitution
in (4.7). Written in vector form, the boundary conditions (4.6)
become
y(0)=
A B
0
df
,
y(D)=
A B
0
df
,
(4.13)
s
r
where df and df denote the perturbed direction of the
s
r
wavevector at the source and receiver, respectively. Using the
boundary conditions (4.13) we obtain from (4.11) the fact that
the perturbed wavevector directions df and df are given by
s
r
D
df =−(sin D)−1
[cos(D−w)∂ e+sin(D−w)∂ e] dw ,
s
f
h
0
(4.14)
P
df =(sin D)−1
r
P
dj =df +(∂ e) ,
(4.16)
s
s
f s
dj =df +(∂ e) .
(4.17)
r
r
f r
Eqs (4.16) and (4.17) are in agreement with eq. 8 of
Mochizuki (1990).
Using (4.14) to determine an estimate of the initial direction
of the wavevector can substantially reduce the number of
iterations required to hit the receiver. Eq. (4.17) may be used
as a basis for anisotropic arrival-angle tomography. Using
(4.11) we determine that the perturbed ray is determined by
P
w
0
[cos(w−w∞)∂ e+sin(w−w∞)∂ e] dw∞−sin w df ,
f
h
s
(4.18)
df(w)=
P
w
0
h p/2+dh ,
f df ,
A B
A B
[−sin(w−w∞)∂ e+cos(w−w∞)∂ e] dw∞+cos w df .
f
h
s
(4.19)
Note that dh(0)=dh(D)=0, that df(0)=df , and that
s
df(D)=df , as required.
r
On a spherically symmetric earth model the partial
derivatives ∂h/∂f and ∂f/∂f are given by −sin w and cos w,
s
s
respectively. To determine the partial derivatives on a laterally
heterogeneous, anisotropic earth model, we make the following
A B
A B
∂h
∂h
−sin w+d
,
∂f
∂f
s
s
(4.20)
∂f
∂f
cos w+d
.
∂f
∂f
s
s
(4.21)
To first order in the small perturbations we obtain the following
equations for the perturbed partial derivatives:
A B A B
A B A B
∂h
∂f
d
d
=−d
+sin w∂ ∂ e−cos w∂2 e ,
h f
f
dw
∂f
∂f
s
s
(4.22)
∂f
∂h
d
d
=d
−sin w∂2 e+cos w(∂ e+∂ ∂ e) .
h
w
f h
dw
∂f
∂f
s
s
(4.23)
The initial conditions for the perturbed partial derivatives are
d
A B
∂h
(0)=0 ,
∂f
s
d
A B
∂f
(0)=0 .
∂f
s
(4.24)
Eqs (4.22) and (4.23) can be solved based on the same
approach as we used for solving (4.4) and (4.5). Since the
matrix A is the same in both cases, so is the propagator matrix
P. In this case
D
(−cos w∂ e+sin w∂ e) dw .
(4.15)
f
h
0
Note that an interchange of the source and receiver, which
amounts to making the substitutions w D−w and D−w w
in (4.14) and (4.15), interchanges df and df , as expected. The
s
r
perturbed take-off and arrival angles may be obtained from
(4.3):
dh(w)=−
substitutions in (3.21) and (3.22):
y=
A
B
d(∂h/∂f )
s ,
d(∂f/∂f )
s
(4.25)
subject to the initial condition y(0)=0, and
f=
A
B
sin w∂ ∂ e−cos w∂2 e
h f
f
.
−sin w∂2 e+cos w(∂ e+∂ ∂ e)
h
w
f h
(4.26)
Using (4.11) we determine that the perturbed partial derivatives
are given by
d
A B
P
d
A B
P
D
∂h
(D)=
{cos(D−w)(sin w∂ ∂ e−cos w∂2 e)−sin(D−w)
h f
f
∂f
s
0
×[−sin w∂2 e+cos w(∂ e+∂ ∂ e)]} dw , (4.27)
h
w
f h
D
∂f
(D)=
{sin(D−w)(sin w∂ ∂ e−cos w∂2 e)+cos(D−w)
h f
f
∂f
s
0
×[−sin w∂2 e+cos w(∂ e+∂ ∂ e)]} dw . (4.28)
h
w
f h
The perturbed phase dy is determined by
p
dy =−vc−1
p
P
D
e dw .
(4.29)
0
The effects of lateral variations in attenuation dQ/Q may be
determined from
d ln A =v(2GQ)−1
a
P
D
dQ/Q dD ,
(4.30)
0
where G is the group velocity in the isotropic reference
model, as in eq. (2.14). Based on (2.51) and (3.20), the relative
perturbation in amplitude due to focusing and defocusing of
© 1998 RAS, GJI 132, 654–666
EVects of slight anisotropy on surface waves
surface-wave energy is given by
C
A BD
CP
1
∂h
e +(sin D)−1d
r
2
∂f
s r
D
1
1
= (e +e )− (sin D)−1
cos(D−2w)e dw
r
2 s
2
0
D
D
− sin(D−w) sin w∂2 e dw+
sin(D−2w)∂ ∂ e dw
h
f h
0
0
D
cos(D−w) cos w∂2 e dw ,
(4.31)
+
f
0
P
P
A
D
B
M : dE*
s ,
M : E*
s
dA /A = Re
s s
A
P
d
P
where the term involving ∂ e has been integrated by parts.
w
Notice that interchanging the source and receiver leaves the
relative amplitude perturbation (4.31) unchanged, as expected.
We have verified that the contribution dS /S is in agreement
r r
with eq. (10) of Mochizuki (1990).
Eq. (4.31) may be used as a basis for surface-wave amplitude
tomography. Another option is to correct the observed amplitudes for the effects of focusing and defocusing, local structure
at the source and receiver, and perturbations in take-off and
arrival angles, and to invert the remaining signal for lateral
variations in intrinsic attenuation based on (4.30).
For the sake of completeness, using the results obtained by
Wang & Dahlen (1994), we note that perturbations in the
receiver phase and amplitude, dy and dA /A , as defined in
r
r r
eq. (2.50), are zero, and that
dy =Im
s
directions df and df . They are given by
s
r
D
sin(D−w)∂ e dw ,
(4.36)
df =−(sin D)−1
h
s
0
D
(4.37)
sin w∂ e dw ,
df =(sin D)−1
h
r
0
and the perturbed partial derivatives are determined by
P
1
dA /A =− (dk /k +dS /S )
r r
p p
2 r r
=
(4.32)
B
M : dE*
s .
M : E*
s
(4.33)
For Love waves we have
M : dE* ={r−1 k W [(M −M ) cos 2f +2M sin 2f ]
s
s s s
hh
ww
s
hw
s
−i(∂ W −r−1 W )(−M sin f +M cos f )}df
r s
s
s
rh
s
rw
s
s
1
+ e r−1 k W [(M −M ) sin 2f −2M cos 2f ] ,
hh
ww
s
hw
s
2 s s s s
(4.34)
and for Rayleigh waves
M : dE* ={r−1 k V [−(M −M ) sin 2f +2M cos 2f ]
s
s s s
hh
ww
s
hw
s
+i(∂ V −r−1 V +r−1 k U )
r s
s s
s s s
×(M sin f +M cos f )}df +e r−1 k V
rw
s
rh
s
s s s s s
1
1
× (M +M )− (M −M ) cos 2f −M sin 2f
ww
ww
s
hw
s
2 hh
2 hh
C
−ie r−1 k U (M sin f −M cos f ) .
s s s s rh
s
rw
s
D
(4.35)
On an isotropic earth model the perturbed take-off and
arrival angles dj and dj are equal to the perturbed wavevector
s
r
© 1998 RAS, GJI 132, 654–666
661
A B
P
∂h
D
sin(D−w)(−sinw∂2 e+cos w∂ e) dw ,
(D)=−
h
w
∂f
s
0
(4.38)
d
A B
P
D
∂f
(D)=
cos(D−w)(−sin w∂2 e+cos w∂ e) dw .
h
w
∂f
s
0
(4.39)
These results are in accordance with the results obtained by
Woodhouse & Wong (1986). The relative perturbation in
amplitude due to focusing and defocusing (4.31) reduces to
1
1
dA /A = (e +e )− (sin D)−1
r
p p 2 s
2
−
P
D
CP
D
cos(D−2w)e dw
0
D
(4.40)
sin(D−w) sin w∂2 e dw ,
h
0
which is in agreement with the result of Wang & Dahlen (1994).
5 AC CUR A C Y O F P ER TU R BATIO N
TH EO R Y
Perturbation theory is preferable to exact ray tracing for largescale inversions because it requires less computational effort.
In order to understand the character and size of the errors
introduced by perturbation theory, we compare predictions of
perturbation theory and exact ray theory. Phase depends
directly on phase velocity c, arrival angle on its first derivatives,
and amplitude on phase velocity and its second derivatives.
Therefore, we expect the errors in the predictions of arrival
angle and amplitude by perturbation theory to increase when
the size of small-scale structure increases.
We have compared predictions of phase, arrival angle and
amplitude from perturbation theory with exact ray theory. The
predictions are for 8905 first-orbit surface waves in the distance
range 30°–150°. The source and receiver locations are at the
vertices of a tessellation of the sphere, resulting in uniform
coverage of the globe. The predictions were made by ray
tracing, using the equations in Section 3, through an anisotropic phase-velocity map (Fig. 2), which was obtained by a
preliminary inversion of phase measurements of Rayleigh
waves at 35 s from an enlarged version of the data set of
Ekström et al. (1997). The anisotropic structure of the map is
expanded as in eq. (2.21), but the 4f anisotropy is neglected
as it is small; only the isotropic term e and the 2f anisotropic
0
terms e and e are included. Each term is expanded in spherical
1
2
harmonics up to degree 16 for the isotropic term e and degree
0
12 for the anisotropic terms e and e for a total of 627 model
1
2
parameters.
As illustrated in Fig. 3, the phase from exact ray theory is
always less than the Fermat phase along the great circle; this
is to be expected, since the true path is the route of minimum
phase or traveltime. Caustics are an exception to this, and our
662
E. W. F. L arson, J. T romp and G. Ekström
Figure 3. Predictions of phase anomalies for 8905 paths on the
anisotropic phase-velocity map shown in Fig. 2 for Rayleigh waves at
a period of 35 s. The values on the x-axis are perturbation-theory
predictions from eq. (4.29). The values on the y-axis are exact
ray-theory predictions from eq. (3.18).
results do not include paths with receivers near caustics, such
as the antipode. The difference between perturbation theory
and exact ray theory is small for phase, even when the absolute
size of the perturbation is large.
The arrival and take-off angles should be viewed together
(Fig. 4), since they are reciprocals of each other. If the source
and receiver are interchanged, the values for the angles are
also interchanged, because the same path would be used in
the reverse direction. For both angles, there is no consistent
pattern of over- or under-prediction by perturbation theory
relative to exact ray tracing. However, as the perturbation
theory prediction increases in magnitude, its error also tends
to increase.
Fig. 5 shows a comparison of amplitude predictions based
on ray theory and on perturbation theory. There is a pattern
in the differences between the perturbation-theory amplitude
and ray-theory amplitude. When perturbation theory predicts
a large amplitude decrease, this is typically an over-prediction;
the absolute amplitude from ray theory is usually larger
than the perturbation-theory amplitude. When perturbation
theory predicts a large amplitude increase, it is typically an
under-prediction, although it is sometimes a very large overprediction. We see the best agreement with exact ray theory
when perturbation theory predicts only a small amplitude
correction, but there is a small bias here as well: the ray-theory
amplitude tends to be smaller. We find that these patterns are
found for isotropic phase-velocity variations also, but it was
not apparent in a previous study because of corrections to the
amplitude for perturbed take-off angles in the calculation of
source excitation (Wang & Dahlen 1994). The main source of
these patterns can be understood by looking at the distribution
of predictions. The predictions from exact ray theory are lognormally distributed, as seen in Fig. 6. This distribution is to
Figure 4. Predictions of take-off angle (top) and arrival angle ( bottom)
for 8905 paths on the anisotropic phase-velocity map shown in Fig. 2.
The x-axis is perturbation-theory predictions from eqs (4.14) and
(4.15), and the y-axis is exact ray-theory predictions determined from
ray tracing with eqs (3.15) and (3.16).
be expected: a fast region causes an amplitude decrease which
is the reciprocal of the amplitude increase caused by a slow
region of equal but opposite phase-velocity perturbation. For
example, a slow region may cause a doubling in amplitude
(100 per cent increase) where an equal but opposite fast region
would cause the reciprocal: halving (only a 50 per cent decrease).
Fig. 7 shows predictions from perturbation theory, which are
normally distributed around zero. Eqs (4.31) and (4.40) predict
© 1998 RAS, GJI 132, 654–666
EVects of slight anisotropy on surface waves
663
Figure 6. Distribution of exact ray-theory amplitude predictions for
8905 paths on the anisotropic phase-velocity map shown in Fig. 2
based on eq. (3.20).
Figure 5. Predictions of amplitude anomalies, in per cent, for 8905
paths on the anisotropic phase-velocity map shown in Fig. 2. The
x-axis is perturbation-theory predictions from eq. (4.31) and the y-axis
is exact ray-theory predictions from eq. (3.20).
this type of distribution since the amplitude perturbation is
dependent on path integrals of the phase velocity and its
second derivative, which both have normal distributions. The
different distribution resulting from perturbation theory causes
the pattern in the correlation with exact ray theory shown
in Fig. 5.
In the above analysis we have assumed that exact ray tracing
can be used as ‘ground truth’, that is that the predictions it
makes are without error. This may not be correct, since ray
theory assumes that the seismic wave can be expressed as a
1-D ray. Wang & Dahlen (1995) have shown by comparison
with mode-coupling calculations that ray theory is a satisfactory method for predicting the phase and arrival angle of
first-orbit surface waves, but it is not adequate for making
amplitude predictions. The study was performed using only isotropic models, but we expect the same results for anisotropic
models.
6
A NI S O TR O P IC I NVE R S IO N S
We have designed an experiment to answer two questions
which have arisen in the study of global surface-wave data.
Most previous global studies have modelled only isotropic
structure, but some studies have found that there is significant
anisotropy in the true Earth (Nishimura & Forsyth 1988;
Montagner & Tanimoto 1990). Therefore, we want to know if
and how true anisotropic structure might be mapped to isotropic
structure. Second, we want to know what surface-wave
observables are sensitive to anisotropic structure.
To answer these questions, we produce synthetic data vectors
of phase and arrival angle from an anisotropic phase-velocity
model and invert for isotropic structure. The input anisotropic
model, shown in Fig. 2, is the same as the model used in the
previous section, except that the anisotropic part is tripled
in size to illustrate the results more clearly. This spherical
© 1998 RAS, GJI 132, 654–666
Figure 7. Distribution of perturbation-theory amplitude predictions
for 8905 paths on the anisotropic phase-velocity map shown in Fig. 2
based on eq. (4.31).
harmonic expansion (to degree 16 for the isotropic part and
degree 12 for the 2f anisotropic part) results in a total of 627
parameters, 289 for the isotropic part and 338 for the anisotropic part. We determine phase and arrival-angle predictions
using exact ray tracing for 8905 paths. As in the previous
section, the paths are chosen to obtain optimal coverage; the
sources and receivers are not actual earthquake and station
locations.
The synthetic anisotropic data sets are inverted for isotropic
phase-velocity maps. The inverted maps are expanded in
spherical harmonics to degree 24, such that we have nearly
the same number of parameters, 625, as the anisotropic input
model, 627. We invert the phase and arrival-angle data sets
both independently and jointly. We note that since we have no
data errors and optimal coverage, no damping or regularization
is necessary. This is the ‘best-case’ example and our results are
not influenced by a subjective choice of damping.
664
E. W. F. L arson, J. T romp and G. Ekström
The phase data can be well explained by a purely isotropic
model. The isotropic part of the anisotropic input model
explains 89.0 per cent of the phase data. The isotropic model
resulting from the inversion of the anisotropic phase data
explains about half of the remaining variance (total variance
reduction is 93.7 per cent). This leaves only a small amount of
signal to be explained. However, the extra variance reduction
gained must be due to ‘incorrect’ isotropic structure; the
‘correct’ answer would have the same variance reduction as
obtained from the isotropic part of the input model. The
nature of this incorrect structure is determined by comparing
the input model, Fig. 2, to the output model, Fig. 8. The largescale features are the same, but there are small-scale features
in the inverted model which do not occur in the input model,
particularly in regions of strong anisotropy in the north Pacific
and along the East Pacific Rise. The correlation of the inverted
model to the isotropic part of the input model is very high
(Fig. 10). However, there is a large misfit to the isotropic part
of the input model in the power at degrees three and four, for
which there is the most anisotropy (see Fig. 9). There is also
noticeable power above degree 16 in the inverted model where
there was no power in the anisotropic input model.
The anisotropic arrival-angle data are not as well explained
by an isotropic map. Only 34.2 per cent of the variance is
explained by the isotropic part of the input anisotropic model.
In the inversion of the anisotropic arrival-angle data for an
isotropic map, about half of the remaining variance is explained
in terms of ‘incorrect’ isotropic structure, but there is still a
significant signal to be modelled (total variance reduction is
64.4 per cent). The improvement in fit is due to a general
reduction of the size of the inverted model (see Figs 11 and 12)
and incorrect low-order structure, particularly at degrees four,
seven and ten (Fig. 10). Inversion of the anisotropic arrivalangle data set does not introduce power above degree 16 as
inversion of the phase data set does: arrival angles do not allow
anisotropy to be modelled as small-scale isotropic structure.
The anisotropic arrival-angle data cannot be satisfied by
any isotropic model, and significant variance remains to be
explained.
Inversion of the anisotropic phase and arrival-angle data
sets together compromises between the two individual data
sets. The map and spectra of the inverted isotropic model from
the combined data set are shown in Figs 13 and 14. Up to
about degree five, the inverted model from the combined data
Figure 10. Correlation of inverted isotropic models obtained from
the anisotropic phase data set (dotted), from the anisotropic arrivalangle data set (dashed), and from the combined data sets (solid) with
the isotropic part of the anisotropic input model. Note that the
minimum value on the y-axis is 0.75.
Figure 12. Power spectrum of the isotropic model obtained by
inversion of the anisotropic arrival-angle data set.
Figure 14. Power spectrum of the isotropic model obtained by
combined inversion of the anisotropic phase and arrival-angle data sets.
Figure 9. Power spectrum of the isotropic model obtained by
inversion of the anisotropic phase data set.
sets is very similar to the isotropic model inverted from the
phase data set (Fig. 9), and above degree ten, it nearly matches
the model inverted from the arrival-angle data set (Fig. 12).
This mimics the sensitivity of the observables: phase is more
sensitive to long-wavelength structure and arrival angle to
© 1998 RAS, GJI 132, 654–666
EVects of slight anisotropy on surface waves
short wavelengths. In fact, the correlation of the isotropic part
of the anisotropic input model with the isotropic model
inverted from phase data is better than the correlation of the
inverted model with the combined data set (Fig. 10). Likewise,
the power of the inverted phase model more closely matches the
power of the isotropic part of the input model than does
the power of the model inverted from the combined data set
(Fig. 15). The one improvement of the inverted isotropic model
from the combined data set over the model inverted from the
phase data set is that there are not false small-scale features.
We have repeated this experiment with several other anisotropic input models. These include models which are derived
from phase measurements (without re-scaling of anisotropy, as
we did in the experiments discussed above). All input models
produce the same pattern of results, and the magnitude of the
effects is larger when the size of the anisotropy is larger. The
variance reductions for each experiment are summarized in
Table 1. We note that as the anisotropy becomes smaller,
Table 1. Variance reductions of inverted isotropic models to anisotropic data sets. Each row is for a different anisotropic input model
from which data sets of phase and arrival angle were created: row 1
is an anisotropic phase-velocity model inverted from Rayleigh-wave
phase measurements at 150 s; row 2 is the same model as row 1, but
with the anisotropic part multiplied by 3; row 3 is an anisotropic
phase-velocity model inverted from Rayleigh-wave phase measurements at 35 s; row 4 is the same model as row 3, but with the
anisotropic part multiplied by 3. The columns represent models for
which the variance reductions of the data sets were calculated:
column A is the isotropic part of the anisotropic input model; column B
is the model inverted from the anisotropic phase data set; column C
is the model inverted from the anisotropic arrival-angle data set; and
column D is the model inverted from the combined anisotropic phase
and arrival-angle data set. The numbers in each cell are the variance
reductions to the anisotropic phase data set (y) and the arrival-angle
data set (j), respectively.
A
y, j
1
2
3
4
92.7,
61.6,
96.9,
89.0,
52.3
39.6
38.6
34.2
B
y, j
95.7,
78.0,
98.2,
93.4,
51.6
10.3
44.5
30.2
C
y, j
75.4,
57.2,
62.6,
55.3,
70.5
67.6
65.7
64.4
D
y, j
93.7,
74.1,
96.6,
91.5,
69.3
65.9
63.7
62.1
665
phase is less able to resolve it, but arrival angles remain
quite sensitive.
7
DI S C US S IO N
We have presented an exact ray theory and a perturbation
theory for surface-wave propagation on an anisotropic earth
model. This theory can be used in inversions of surface-wave
phase, arrival angle and amplitude measurements for anisotropic phase-velocity structure and for forward modelling. For
perturbation theory, phase is sensitive directly to the phase
velocity, arrival angle is sensitive to the first derivative of phase
velocity, and amplitude to the phase velocity and its second
derivatives. This implies that arrival angles are more sensitive
to short-wavelength phase-velocity structure than phase, and
amplitudes depend even more on short-wavelength structure.
Ray perturbation theory is an adequate approximation to
exact ray theory for phase and arrival angles for first-orbit
surface waves. The approximation for arrival angles becomes
worse as the perturbation increases in magnitude, therefore
caution must be used when interpreting large arrival-angle
measurements. In most cases, the linear approximation is quite
sufficient. Significant problems arise in the linear approximation for amplitude, and it seems unlikely that a linear theory
is sufficiently accurate to interpret amplitude observations.
In summary, we note the following.
(1) Isotropic maps inverted from phase data have more
power above degree 12 than isotropic maps from arrivalangle data. However, anisotropic structure may be the cause
of the reduced power in isotropic inversions which include
arrival-angle data.
(2) Long-wavelength anisotropic structure is mapped to
small-scale isotropic structure by phase data.
(3) The variance reduction attained by inversion of either
anisotropic phase data or anisotropic arrival-angle data for
isotropic structure is deceptively high.
(4) Anisotropic arrival-angle data cannot be fit with isotropic maps and therefore are highly sensitive to anisotropic
structure.
A CKN O W LE DG M ENT S
We thank Zheng Wang and Gabi Laske for assistance in
bench-marking the ray-tracing code. This material is based
on work supported under a National Science Foundation
Graduate Fellowship, by the NSF under Grant EAR-9219239,
and by the David and Lucile Packard Foundation.
R EF ER EN C ES
Figure 15. Power spectra of isotropic models obtained from the
anisotropic phase data set (dotted) and from the combined anisotropic
phase and arrival-angle data sets (dashed), compared to the power
spectrum of the isotropic part of the anisotropic input model (solid).
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