Download New images of the Earth`s upper mantle from measurements of

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B4, 2059, 10.1029/2000JB000059, 2002
New images of the Earth’s upper mantle from measurements
of surface wave phase velocity anomalies
Lapo Boschi and Göran Ekström
Department of Earth and Planetary Sciences, Harvard University, Cambridge,
Massachusetts, USA
Received 14 November 2000; revised 24 September 2001; accepted 29 September 2001; published 11 April 2002.
[1] The sensitivity of a surface wave to the lateral structure of the Earth’s mantle depends on the
local structure and thickness of the crust; particularly at short periods, the strong lateral
heterogeneity of the crust has a significant nonlinear effect on the surface wave ‘‘sensitivity
functions.’’ Thus far, this effect has not been quantified, and tomographic studies of the upper
mantle based on surface wave observations have not taken it into account. On the basis of a
Jeffreys-Wentzel-Kramers-Brillouin (JWKB) description of surface wave propagation we find and
discuss the lateral variations of the sensitivity of Love and Rayleigh waves to mantle seismic
anomalies, associated with a realistic model of the crust [Mooney et al., 1998]. We then use the
resulting laterally varying sensitivity kernels to determine a new tomographic image of horizontally
and vertically polarized shear velocity in the upper mantle from Love and Rayleigh wave phase
velocity measurements (periods from 35 to 300 s). In terms of large-scale patterns of high and low
velocity our model is similar to previously published ones, confirming, among other features, the
anomalous anisotropy underlying the central Pacific [Ekström and Dziewonski, 1998]; we find,
however, that the lateral dependence of the surface wave sensitivity kernels has a significant effect
on the absolute amplitude of the anomalies, which should not be neglected as we attempt to obtain
higher-resolution images of the upper mantle.
INDEX TERMS: 7218 Seismology: Lithosphere and
upper mantle; 7255 Seismology: Surface waves and free oscillations; 8180 Tectonophysics:
Tomography; KEYWORDS: Tomography, surface waves, JWKB theory, anisotropy, continental roots,
upper mantle
1. Introduction
[2] Measurements of Love and Rayleigh waves at short and
intermediate periods provide the principal seismological constraint,
on a global scale, on the properties of the Earth’s crust and upper
mantle [e.g., Ekström, 2000]. Today, the crust has been characterized successfully by local seismic reflection and refraction
studies, and accurate laterally varying global models of its structure
and thickness are becoming available [e.g., Mooney et al., 1998].
The next step, then, is to make use of surface wave measurements
to achieve an equally accurate understanding of the upper mantle
three-dimensional (3-D) structure.
[3] In practice, the local perturbations to surface wave velocity
specifically due to the crust can be estimated a priori by a
theoretical calculation [e.g., Woodhouse and Dziewonski, 1984],
based on existing 3-D crustal models like CRUST-5.1 [Mooney et
al., 1998]. These perturbations are then subtracted from the
original measurements, and the resulting, ‘‘corrected’’ velocity
anomalies should be the result of anomalous mantle structure only.
A set of sensitivity functions that relate surface wave measurements to the underlying heterogeneities of the parameters that we
wish to constrain (horizontal and vertical shear and compressional
velocity, density, etc.) is then used to set up a linear inverse
problem, whose solution is a 3-D model of the upper mantle
[e.g., Montagner and Tanimoto, 1991; Masters et al., 1996;
Ekström and Dziewonski, 1997, 1998; Ritsema et al., 1999;
Ekström, 2000; Ritsema and van Heijst, 2000].
[4] Because the Earth’s crust is characterized by strong lateral
heterogeneities, the sensitivity of surface waves to the underlying structure must also vary with latitude and longitude:
Copyright 2002 by the American Geophysical Union.
0148-0227/02/2000JB000059$09.00
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depending on the local values of thickness, density, and seismic
velocities of the crust, the propagation of a surface wave along
the surface might be affected in different ways by a mantle
anomaly located at a given depth. Nevertheless, in all global
tomographic studies of the upper mantle thus far [e.g., Montagner and Tanimoto, 1991; Masters et al., 1996; Ekström and
Dziewonski, 1997, 1998; Ritsema and van Heijst, 2000; Ekström,
2000], lateral variations of the sensitivity functions have been
neglected. The partitioned waveform inversion algorithm of
Nolet [1990] (later employed, for example, by van der Lee
and Nolet [1997] and Simons et al. [1999]) allows for a 3-D
reference model but requires that the reference model (and thus
the sensitivity functions) be averaged along each source-station
path, when the second step of the inversion is carried out to
determine the structure of the mantle; since a teleseismic surface
wave ray path usually spans a portion of the Earth large enough
to include significant variations of crustal structure, the problem
that we address remains.
[ 5 ] In the Jeffreys-Wentzel-Kramers-Brillouin (JWKB)
approximation [e.g., Tromp and Dahlen, 1992a, 1992b] the
propagation of surface waves through a 3-D Earth is governed
by ‘‘local’’ normal-mode eigenfunctions, coinciding, at any given
location, with those of a 1-D Earth model whose properties are
everywhere equal to those of the 3-D Earth at the same location
[Tromp and Dahlen, 1992a, section 2.4]. After finding the local
eigenfunctions of a 3-D starting model that includes an accurate
map of the crust, we can use them to determine, everywhere on
the Earth’s surface, the local sensitivity functions that are
naturally associated with them; the resulting set of laterally
varying kernels can then be used, in a linear inverse problem,
to constrain the laterally varying structure of the upper mantle
[Wang and Dahlen, 1994]. This way, the lateral variations in
sensitivity associated with the crust are taken into account, within
1-1
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BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
the accuracy of the JWKB approximation. Application of this
technique to our set of high-quality surface wave measurements
has resulted in a new 3-D, transversely isotropic image of shear
velocity in the upper mantle; we discuss this model in comparison with previous results, focusing on the specific effect of our
new treatment of sensitivity.
2. Tomographic Inversions of Observed
Surface Wave Phase Velocity Anomalies
2.1. Phase Velocity Maps
[6] Given a spherically symmetric reference Earth model, a
theoretical Rayleigh or Love wave seismogram for any sourcestation geometry can be calculated and written, as a function of
frequency w, in the form
iw
;
uðwÞ ¼ AðwÞ exp
cðwÞ
ð1Þ
where denotes the epicentral distance, A(w) denotes the wave
amplitude, and c(w) is the surface wave phase velocity at a
frequency w.
[7] We can use a perturbation approach to write a mathematical
expression for a seismogram associated with a 3-D Earth [e.g.,
Ekström et al., 1997],
iw
þ dðwÞ ;
uðwÞ þ duðwÞ ¼ ½AðwÞ þ dAðwÞ exp
cðwÞ
ð2Þ
where dA(w) and d(w) are small perturbations to the amplitude
and phase of the seismogram, respectively. In particular, the phase
anomaly can be written
Z
parameters are density, horizontally polarized compressional velocity vPH, vertically polarized compressional velocity vPV, horizontally polarized shear velocity vSH, vertically polarized shear
velocity vSV, the nondimensional parameter h (described, for
example, by Dziewonski and Anderson [1981]), and the attenuation
factors qK and qm [e.g., Dziewonski and Anderson, 1981].
[10] We denote pi(r) + dpi(r, q, f) the values of the same
structural parameters in a 3-D Earth. Then, assuming that the phase
slowness p is a function of the local radial structure only [Jordan,
1978], dp(q, f; w) must depend on the dpi via
Z
I
X
a
dpðq; f; wÞ ¼
0
ð4Þ
where a is the Earth’s radius and the functions Ki usually are
referred to as ‘‘sensitivity kernels’’ or simply ‘‘partial derivatives’’
[Anderson and Dziewonski, 1982]. The functions Ki are determined
numerically on the basis of normal mode theory [e.g., Takeuchi and
Saito, 1972].
[11] Let us substitute (4) into (3). We find a relation between the
phase anomaly and the heterogeneities dpi,
Z
Z
a
dðwÞ ¼ w
0
I
X
0
Ki ðr; wÞdpi ½r; qðsÞ; fðsÞdrds:
ð5Þ
i¼1
Given our set of phase anomaly measurements, (5) can be used to
constrain dpi, thereby obtaining a 3-D model of the upper mantle:
first, the dpi need to be expanded over a set of basis functions
fk(r, q, f) (k = 1, 2,. . ., K ),
dpi ðr; q; fÞ ¼
K
X
xik fk ðr; q; fÞ;
ð6Þ
k¼1
dðwÞ ¼ w
Ki ðr; wÞdpi ðr; q; fÞdr;
i¼1
dpðqðsÞ; fðsÞ; wÞds
ð3Þ
0
where dp(q, f; w) is the difference between the surface wave phase
slownesses p = 1/c associated with the 3-D Earth and the initial,
one-dimensional (1-D), reference model. The letters r, q, f denote
radius, colatitude and longitude; the equations q = q (s) and f = f(s)
describe the geometry of the ray path; here and in the following, as
in most of the literature on surface wave tomography, the ray path is
assumed to follow the great circle connecting source and receiver,
and the lateral width of the ray is neglected.
[8] Let the difference between the phase of a measured
seismogram and that predicted by the 1-D reference model be
denoted d(w). A large set of such phase anomaly measurements,
collected according to the procedure devised by Ekström et al.
[1997], is available. We then treat dp(q, f; w) as the unknown
solution to the inverse problem implicitly defined by (3) and
expand it over a set of basis functions (of q and f only); the
resulting linear system of equations is then solved to obtain twodimensional phase velocity maps of Earth [e.g., Montagner and
Tanimoto, 1990; Trampert and Woodhouse, 1995; Laske and
Masters, 1996; Zhang and Lay, 1996; Ekström et al., 1997;
Boschi and Dziewonski, 1999].
2.2. Three-Dimensional Heterogeneities With Respect to
a Spherical Reference Model
[9] The phase velocity of surface waves is a function of the
velocity and density structure of the top few hundred kilometers of
the Earth. Consequently, measurements of surface wave velocity
(or phase) anomalies can provide information about the crust and
upper mantle. Let us denote pi(r) (i = 1, 2,. . ., I ) the set of I
parameters that define our reference 1-D Earth model; in a transversely isotropic (radially anisotropic) Earth, I = 8, and the eight
which holds for i = 1, 2,. . ., I. Equation (6) can then be substituted
into (5),
dj ðwÞ ¼ w
I X
K
X
i¼1 k¼1
Z
j
Z
a
Ki ðr; wÞfk ðr; q; fÞdrds;
xik
0
ð7Þ
0
where the additional subscript j = 1, 2,. . ., M is simply used to
distinguish each of the M measurements available in our data set.
[12] We simplify the notation by establishing a one-to-one
correspondence between the couples of indexes i, k and a single
index l. Let L be the total number of combinations i, k. We
subsequently call Ajl the double integral in (7), which can now
be calculated numerically for each j and l. Equation (7) collapses to
dj ðwÞ ¼ w
L
X
Ajl xl
ð8Þ
l¼1
( j = 1, 2,. . ., M ), a linear system that can be solved for the
unknown vector x, and therefore the model vector dpi(r, q, f), in a
regularized least squares sense. As discussed below, (8) is usually
corrected to account a priori for the 3-D structure of the crust. An
alternative approach is to treat the properties of the crust, and in
particular its thickness, as free parameters of the least squares
inversion [Li and Romanowicz, 1996]; in this case, no further
manipulation of (8) is necessary.
2.3. Three-Dimensional Heterogeneities With Respect to
a 3-D Reference Model
[13] If the reference model is chosen to be 3-D, (1) has no
meaning (the phase velocity c now being dependent on the location
BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
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Earth’s crust by applying a crustal correction, as in (9), but then
neglecting the inherent lateral variations of the sensitivity kernels
Ki, which are still assumed to be constant with respect to q and f
[e.g., Woodhouse and Dziewonski, 1984; Nataf et al., 1986;
Montagner and Tanimoto, 1991; Masters et al., 1996; Ekström
and Dziewonski, 1997, 1998; Ekström, 2000]. Here, instead, we
replace the 1-D kernels Ki(r) by the appropriate JWKB 3-D
kernels, which we shall denote Ki0(r, q, f; w).
[16] In the JWKB approximation the local radial eigenfunctions
(and dispersion relation) at (q, f) are precisely those of a 1-D Earth
model whose properties are everywhere equal to those of our 3-D
model at (q, f); we can therefore find Ki0(r, q, f; w) with the same
procedure used in the entirely 1-D case (section 2.1), repeated at all
the nodes of a fine (q, f) grid, depending on the lateral resolution
of the 3-D reference model, using at each node the appropriate set
of local eigenfunctions. This approach is computationally intense,
but feasible. Equation (7) is replaced by
d00j ðwÞ ¼ w
I X
K
X
i¼1 k¼1
Figure 1. (a) Sensitivity of 300-s Rayleigh waves to vSV (solid
lines) and vSH (dashed lines) at two locations: Tibet (solid lines)
and a region of 25-km-thick crust in Antarctica (shaded lines).
(b) Sensitivity of 300-s Rayleigh waves to vPV (solid lines) and
vPH, at the same two locations as in Figure 1a. (c, d) Same as
Figures 1a and 1b, respectively, but for 35-s Rayleigh waves.
Following Anderson and Dziewonski [1982], we denote the
sensitivity dT/T: We define it as the relative perturbation in the
surface wave period, produced by a uniform 1% perturbation in
velocity over a 1-km-thick shell, whose depth is plotted on the
horizontal axis. Here and in Figure 2, the sensitivity curve only
extends to the local Moho depth: the crustal structure, in fact, is not
treated as a free parameter in our least squares inversions (see
section 2.3).
q, f), and the above treatment does not hold. Remaining in the
framework of geometrical optics ray theory, we shall then follow
the description of Wang and Dahlen [1994], based upon the JWKB
approximation [Tromp and Dahlen, 1992a, 1992b]. In this
approach, the propagation of surface waves is governed by local
normal-mode eigenfunctions (changing as functions of q and f),
naturally giving rise to local sensitivity kernels, when the theory is
used to formulate a tomographic inverse problem.
[14] Once a 3-D reference model pi(r) + dpi(r, q, f) (i = 1, 2,. . ., I )
is chosen, we can calculate the associated local surface wave
slowness p(q, f; w) via normal mode theory, finding at each block
the appropriate local eigenfunctions [e.g., Tromp and Dahlen,
1992a], and subsequently determine, with a line integral along every
path j, the phase anomaly with respect to the initial, 1-D reference;
we shall denote it dj0(w). Each datum dj(w), which is also a phase
anomaly measured with respect to the 1-D reference, is then replaced
by
d00j ðwÞ ¼ dj ðwÞ d0j ðwÞ;
ð9Þ
with j = 1, 2,. . ., M.
[15] In the existing literature on upper mantle tomography, there
are attempts to account for the laterally varying properties of the
Z
j
Z
xik
0
0
a
Ki0 ðr; q; f; wÞfk ðr; q; fÞdrds; ð10Þ
which can also be rewritten in the form of (8), to define the linear
inverse problem associated with the chosen 3-D reference model.
[17] The modeling technique described in this section represents
a theoretical improvement with respect to the previous attempts at
accounting for the strong lateral anomalies of a 3-D reference
model that includes a realistic crustal structure; the ‘‘partitioned
waveform inversion’’ method of Nolet [1990] and the ‘‘path
average’’ approximation of Woodhouse and Dziewonski [1984]
both involve averaging of the reference model along each sourcestation path, while within the limits of the JWKB approximation
we treat our reference model strictly as 3-D. A further step forward
could consist in quantifying the effects of ray curvature, already
treated by Wang and Dahlen [1994] but, for simplicity, neglected
here.
3. Sensitivity Kernels Associated With a 3-D
Reference Earth Model
[18] In this study we determine tomographic models of the
upper mantle, defined in terms of perturbations to a 3-D reference
model. Our reference model, which we shall refer to as model A, is
spherically symmetric everywhere below the local depth of the
Moho discontinuity. It coincides with preliminary reference Earth
model (PREM) at depths >220 km; above this depth, all parameters
vary linearly with slopes equal to those between 400 and 220 km
depth: in contrast to PREM, model A is entirely isotropic. The
surface topography, crustal structure, and depth of the Moho
discontinuity of model A change laterally as in model CRUST5.1 of Mooney et al. [1998].
[19] Once the reference model is given, we can compute the
corresponding sensitivity kernels Ki0(r, q, f; w), as described in
section 2.3. CRUST-5.1 is defined in terms of a discrete, 5 5
block parameterization. In order to determine Ki0(r, q, f; w) unambiguously it is therefore sufficient to perform the computation once
for each one of these blocks. By construction, the sensitivity kernels
will be zero within the crust, whose effect has been taken into
account with the correction (9); this choice rests on the assumption
that CRUST-5.1 be an accurate image of the crust at spatial wavelengths of 5 5 or larger. In Figure 1 we show the sensitivity of
fundamental mode Rayleigh waves at periods 300 s (Figures 1a
and 1b) and 35 s (Figures 1c and 1d) to horizontally (dashed lines)
and vertically (solid lines) polarized, shear (Figures 1a and 1c) and
compressional (Figures 1b and 1d) velocities within the Earth’s
mantle, as a function of depth. As a general rule, we know that
surface waves of longer period are sensitive to deeper structure; it
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BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
Figure 2. Sensitivity of (left) Rayleigh and (right) Love waves, at periods of (top) 35 s and (bottom) 300 s, to
perturbations in vertical and horizontal shear velocities, respectively, at a depth of 70 km. The quantity plotted, dT/T,
is the same as in Figures 1 and 3. See color version of this figure at back of this issue.
is evident from Figure 1 that 35-s Rayleigh waves are only
sensitive to very shallow structure, while 300-s Rayleigh waves
sample the entire upper mantle.
[20] Figure 1 includes a set of sensitivity kernels for two
locations on the Earth, one characterized by a particularly thick
continental crust (solid lines; Tibet/Himalaya, 90E, 30N) and the
other characterized by a thinner (
25 km) crust (shaded lines;
260E, 75S); the differences in crustal structure and thickness
significantly affect the values of Ki0(r, q, f; w) within the upper
mantle, at least at shallow depths. This effect is also seen in Figure
2 (left), where the sensitivity of 35- and 300-s Rayleigh waves to
vSV is shown as a function of longitude and latitude, at a fixed
depth of 70 km below the Earth’s mean surface; the lateral
variations of Ki0(r, q, f; w) are significant at both short and long
periods. The most prominent feature of the Rayleigh wave sensitivity maps of Figure 2 is the systematic contrast between the
values of Ki0(r, q, f; w) in continents and in oceans; the ratios
between the highest and lowest values of sensitivity of 35- and
300-s Rayleigh waves to vSV are 2 and 1.5, respectively.
[21] Love waves (Figure 3) are more sensitive to shallow
structure than Rayleigh waves, and model A’s crustal heterogeneities thus result in large lateral variations of the corresponding
sensitivity functions. As to be expected, this effect is particularly
evident at short periods (Figure 3b). Figure 2 (right), showing the
kernels Ki0(r, q, f; w) associated with vSH for Love waves at periods
of 35 and 300 s, as functions of q and f and at a fixed depth of 70
km, confirms that the lateral variations of the Love wave sensitivity
kernels are more significant than those of the Rayleigh wave ones.
For example, the ratio between the highest and lowest values of
sensitivity of 35-s Love waves to vSH is 5; at this depth and
period the value of Ki0(r, q, f; w) is highest in areas underlying the
oceanic crust, away from mid-ocean ridges and therefore characterized by a thick water layer and lowest in regions with thick
sedimentary layers.
[22] Neglecting the lateral dependence of Ki0(r, q, f; w) is equivalent to assuming, for inversion purposes, a constant crustal
thickness, even if the correction (9) is derived from a reference
model of variable crustal thickness. A consequence of this approx-
imation is that in regions of thick crust the sensitivity to mantle
structure is assumed to be nonzero at depths that fall within the
crust and, likewise, in regions of thin crust the sensitivity is
assumed to be zero (unless crustal structure is inverted for)
throughout the shallowest mantle. Figures 1 and 3 indicate that
this effect can be significant even at long periods.
4. New Tomographic Images of the Earth’s
Upper Mantle
4.1. Three-Dimensional Parameterization
and Phase Anomaly Measurements
[23] The 3-D tomographic images presented in this study are
defined as linear combinations of all the products between the
surface cells of Figure 4a and the radial cubic splines [e.g.,
Lancaster and Šalkauskas, 1986] of Figure 4b (i.e., these products
are the basis functions fk(r, q, f) introduced in section 2.2). The
choice of a block parameterization of the Earth’s surface allows us
to treat CRUST-5.1, which is given in terms of 5 5 blocks, as a
reference model, without any intermediate step. Naturally, this
requires that the boundaries between neighboring blocks in our
parameterization always coincide with the boundaries of CRUST5.1, even though the size of our blocks is occasionally larger
(Figure 4a).
[24] The choice of a radial parameterization, on the other hand,
must be related to the capability of the available data to resolve
structure at different depths. Our data set is an expanded version of
the one assembled by Ekström et al. [1997], with approximately
twice as many observations [Larson, 2000]. It consists mostly of
phase anomaly measurements associated with Love and Rayleigh
waves of 35- to 150-s period, very sensitive, as discussed in section
3, to crustal and shallow mantle structure. We therefore expect to
have good resolution of shallow mantle structure and choose for
this region a fine radial parameterization, with five spline functions
centered every 60 km, from 0 to 240 km depth (Figure 4b).
[25] In order to achieve a better resolution of the deeper portion
of the upper mantle we have also collected minor arc and major arc
BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
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within the upper mantle, by accounting for the different properties
of phase anomaly measurements associated with different types
and periods of surface waves. In fact, after a number of preliminary
inversions we found that the quality of the fit strongly depends on
the period and type of the measured surface wave. Our goal is to
obtain a solution that explains equally well the data associated with
all the different periods of Love and Rayleigh waves included in
our data set; we achieve it by assigning different weights to
different subsets of data. In practice, we multiply all the rows of
the linear system resulting from (10), corresponding to a certain
period and type of wave, by
2P
6
weightðwÞ ¼ 4
M ð wÞ
j
h
i2 312
d00j ðwÞ d00 ðwÞ 7
5 :
M ðwÞ
ð11Þ
This quantity is a measure of the deviation of a set of M phase
anomalies, observed at a frequency w, from their mean value d00 ðwÞ.
4.2. Inversion Procedure
[27] The 3-D tomographic images of the upper mantle that we
obtain are solutions to the inverse problem implied by (10),
simplified by the assumption that the sensitivity of our data to
Figure 3. (a) Sensitivity of 300-s Love waves to vSV (solid lines)
and vSH (dashed lines) at the same two locations (solid and shaded
lines) as Figure 1. (b) Same as Figure 3a, but for 35-s Love waves.
The propagation of Love waves is mostly affected by heterogeneities in vSH; their sensitivity to vSV is, in comparison,
negligible, and their sensitivity to compressional velocities is
exactly zero.
phase anomaly measurements for longer-period (200 – 300 s) surface waves and included them in the tomographic inverse problem.
However, short- and intermediate-period data are far more numerous than long-period ones, and our data set is most likely unable to
resolve small-scale radial variations of the Earth’s structure at
depths >300 km; there we make use of a coarser parameterization,
with splines of larger radial extent, centered at depths of 450, 600,
and 750 km.
[26] In addition to choosing an adequate parameterization we
attempt to optimize our resolution of lateral structure, at all depths
Figure 4. (a) The grid of approximately equal-area blocks (5 5 at the equator) that we employ as the surface parameterization of
our model. (b) The set of nine radial splines that we use to describe
its radial dependence. These splines are part of a whole mantle
parameterization, normalized so that their summed values be equal
to 1 at any depth; in this study, we have excluded the splines
centered at depths >800 km.
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Figure 5. Model B percent lateral perturbations (relative to the 3-D reference model A) in (left) vertical and
(middle) horizontal S velocity and (right) their difference, at four depths in the upper mantle (50 to 200 km as
indicated, top to bottom). The average perturbation of each map has been subtracted from it in order to focus on
lateral variations; it is plotted separately, as a function of depth, in Figure 6. See color version of this figure at back of
this issue.
lateral variations of parameters other than vPH, vPV, vSH, and vSV can
be neglected.
[28] In order to limit the amount of computer memory needed to
find a least squares solution to (10), we also take into account
laboratory observations of compressional and shear velocity anisotropy found in upper mantle rocks [Peselnick and Nicolas, 1978;
Christensen and Salisbury, 1979; Anderson, 1989]; although a clear
correlation exists between the signs of compressional and shear
velocity anisotropy (vPH > vPV is always associated with vSH > vSV,
and vice versa), the available data are not enough to determine a
scaling factor relating the two quantities. We simply assume
dvSH/vSH = dvPH/vPH, and, likewise, dvSV/vSV = dvPV/vPV, everywhere in the upper mantle. (We have also solved the inverse
problem in the assumption of no sensitivity to compressional
velocity. The resulting solution differed only marginally from the
models discussed below.) Using these equations in the treatment
of section 2.3 results in a reduction, by a factor 2, in the
number of free parameters. In practice, we are left with only two
unknown velocity distributions, dvSH/vSH and dvSV/vSV, and the
corresponding sensitivity kernels, K20 + K40 and K30 + K50.
[29] We further reduce the size of the inverse problem by
grouping the original measurements into summary data: Given
an equal-area (2 2 at the equator) grid of indexed blocks
spanning the entire surface of the Earth, we identify each datum by
the block indexes of the corresponding seismic source and station
(or vice versa); we then replace the subset of data corresponding to
each combination of source-block and station-block with their
mean value; the summary source and station associated with a
summary datum are placed in the mean locations of the actual
sources and stations.
[30] The crustal structure of our 3-D reference model coincides
with the crustal model CRUST-5.1, derived from a variety of data
whose resolution of crustal properties is higher than that achievable
from surface wave observations [Mooney et al., 1998]. We therefore rely on the accuracy of CRUST-5.1 and constrain the solutions
to our inversions to equal the reference model exactly within the
crust. As mentioned in section 3, this is achieved by defining the
sensitivity kernels Ki0(r, q, f; w) to be zero anywhere at values of r
larger than the local radius (at q, f) of the Moho discontinuity. In
Figure 6. Model B globally averaged values of vSV (thick dashed
line) and vSH (thick solid line), as functions of depth, compared
with PREM and with the isotropic model A.
BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
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Figure 7. Local corrections to the constant value of 35-s Love wave phase velocity associated with PREM, due to
(a) model A, including crustal heterogeneities from CRUST-5.1, (b) the lateral upper mantle anomalies of Figure 5
(model B), and (c) their combined effect. (d) Phase velocity map resulting from a direct inversion of 35-s Love wave
phase anomalies. The similarity of Figures 7c and 7d demonstrates the internal consistency of our procedure. See
color version of this figure at back of this issue.
this approach, crustal heterogeneities not accounted for by CRUST5.1 are likely to map into the solution; this effect could be made
more severe by lateral variations in the sensitivity of surface waves
to crustal structure: If a spatial resolution higher than that of
CRUST-5.1 is to be achieved, the reference model should include
a finer image of the crust.
[31] We use the LSQR algorithm of Paige and Saunders [1982]
to find a least squares solution; the mixed-determined linear system
is regularized by means of norm and roughness minimization [e.g.,
Boschi and Dziewonski, 1999, 2000]. We define the norm and
roughness of each parameter of the model (dvSH/vSH, dvSV/vSV) as
the integrals, over the entire upper mantle, of the squared norm and
Figure 8. Same as Figure 7, but for 150-s Rayleigh waves. See color version of this figure at back of this issue.
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BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
anisotropic anomaly that Ekström and Dziewonski [1998] found to
characterize the Pacific region around the depths of 50 and 150 km.
Model B is also anisotropic, with vSH larger than vSV, under eastern
Asia and northeastern America, at 100 and 150 km depth, and under
the East Pacific Rise, at 200 km; in the same areas, the maps of
Ekström and Dziewonski [1998] show a less significant anisotropy.
[33] Figure 6 shows the globally averaged profiles of vSH and
vSV associated with model B. Even though the reference model A is
isotropic and includes no mantle discontinuities shallower than 400
km depth, model B is characterized, on average, by a significant
radial anisotropy similar to that of PREM, with the discontinuity at
220 km (the so-called Lehmann discontinuity [see Revenaugh and
Jordan, 1991]) replaced by a very rapid, but smooth change in both
vSH and vSV.
4.4. Phase Velocity Maps
[34] Once a 3-D model is found, equation (4) gives the
corresponding phase velocity of a Love or Rayleigh wave at
Figure 9. Model B average perturbations to (a) Rayleigh and (b)
Love waves phase velocities with respect to their PREM constant
values for all the periods considered here. The values obtained
from independent direct inversions of the phase anomaly
measurements (dot-dashed line, pluses) are compared with those
computed based on the 3-D model discussed in section 4.3 (solid
line, circles); the latter values are the sum of two contributions: that
of the mantle heterogeneities of model B (short-dashed line,
triangles) and that of the reference model A (long-dashed line,
squares), including CRUST-5.1 and the 1-D upper mantle model
described in section 3.
squared horizontal gradient of the parameter itself; the vertical
gradient, therefore, is not minimized.
4.3. A New Radially Anisotropic Image
of the Earth’s Upper Mantle
[32] In Figures 5 and 6 we show a new tomographic image of the
Earth’s upper mantle, which we refer to as model B. Although
resulting from a different tomographic procedure, model B is
consistent, at least in terms of its large-scale pattern, with recent
images of the upper mantle. Ekström and Dziewonski [1998] derived
their anisotropic upper mantle model S20A from a different reference model (PREM, with a uniform crust and a discontinuity at 220
km depth), with a coarser parameterization and a different set of
measurements. Nevertheless, our model and S20A share the same
long-wavelength features. Figure 5, in fact, shows the robust
Figure 10. Variance reduction, with respect to the constant
values of phase velocity associated with PREM, achieved by our
(a) Rayleigh wave and (b) Love wave phase velocity maps. The fit
to the data achieved by theoretical maps based on our 3-D model
(solid line and circles for minor arc data; long-dashed line and
diamonds for major arc data) is compared with that achieved by
phase velocity maps obtained through independent direct inversions of the phase anomaly data (short-dashed line, pluses). At
each period where both minor arc and major arc measurements
were available, they were used together in a single inversion to
obtain a single-phase velocity map.
BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
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Figure 11. Model C, shown in the same fashion as model B in Figure 5, in terms of percent perturbations relative to
model A (the average perturbation, previously subtracted from these plots, is shown separately in Figure 12). This
new image results from a second iteration of our procedure, with model B as the starting point. See color version of
this figure at back of this issue.
any period, as a function of longitude and latitude. In Figures 7
and 8 we compare the resulting (‘‘two-step’’) 2-D maps, for 35-s
Love and 150-s Rayleigh waves, respectively, with the corresponding maps obtained by an independent, direct inversion
(‘‘one-step’’) of the phase anomaly data dj(w) as described in
section 2.1. In this exercise the phase anomaly data are not
corrected to account for the 3-D reference model; therefore, for
every type and period of surface waves the phase velocity maps
are defined as relative perturbation with respect to the constant
phase velocity of PREM.
[35] In order to find the phase velocity anomalies, with respect to
PREM, predicted by our final 3-D model, the quantities dpi(r, q, f)
in (4) must now be defined as perturbations with respect to PREM.
The phase velocity maps are then obtained by summing two
contributions: the anomalies in phase velocity arising from the
differences between PREM and our 3-D reference model A and
those due to the lateral heterogeneities of the upper mantle shown in
Figure 5. At each value of q and f we calculate these two
components of the total phase velocity anomaly and plot them in
Figures 7a and 7b (or Figures 8a and 8b), respectively.
[36] In practice, since the differences between PREM and
model A are purely one-dimensional below the Moho discontinuity, all the lateral structure seen in Figures 7a and 8a originates
from the 3-D heterogeneities of CRUST-5.1. As is to be expected,
this ‘‘crustal’’ component of the total velocity anomaly dominates
short-period Love wave velocity, which is most sensitive to
shallow structure, while the effect of mantle anomalies clearly
becomes more important for Rayleigh waves, especially at relatively long periods. The two-step phase velocity maps, shown in
Figures 7c and 8c, are simply the sum of maps in Figures 7a, 7b,
8a, and 8b. Their consistency with the one-step maps (Figures 7d
and 8d) is an important measure of the stability of our model.
[37] For both Love and Rayleigh waves and at all the periods
considered in this study, we repeat systematically the analysis of
Figures 7 and 8. Figure 9 shows the mean phase velocity anomalies
associated with all the phase velocity maps that we computed. The
agreement between the one-step and two-step images is confirmed
at all periods.
[38] We also calculate the variance reduction, with respect to
PREM, achieved by each of the two-step phase velocity maps;
in Figure 10 we compare it with the value of variance reduction
Figure 12. Globally averaged values of vSV (dashed lines) and
vSH (solid lines) for model B (section 4.3, thin lines) and model C
(section 5, thick lines).
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BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
We thus compute the new sensitivity kernels, associated with
model B; in principle, they should account more accurately for
the actual dependence of the surface wave signal to the underlying
structure. Then we carry out a new least squares inversion,
regularized in the same fashion as the previous one.
[40] The solution model (hereinafter model C) is shown in
Figure 11 in terms of lateral perturbations with respect to
model A. The mean values of vSH (solid lines) and vSV (dashed
lines) in model C are shown as functions of depth in Figure 12
(thick lines), where they are directly compared with those of model
B (thin lines); model B appears to be slightly more isotropic
between 50 and 100 km depth and more anisotropic around 150
km depth. However, these discrepancies are minor, and, as we can
infer from a comparison of Figures 5 and 11, probably smaller than
the actual precision of our models; we have also verified that these
fluctuations keep diminishing as more iterations are performed.
[41] As in section 4.4, we then find from (4) the perturbations
dc(q, f, w) = dp(q, f,w)/p2(w) to the phase velocity c of Love and
Rayleigh waves, at all the values of frequency w considered here.
Their lateral variations (not shown here) and average values
(Figure 13) are substantially similar to those obtained from model
B (Figures 7 and 8); minor differences, visible in Figure 13, result
from the approximate character of (4): as new sensitivity kernels
associated with model B are computed, the same calculation is
carried out, again, in an exact fashion, and the resulting discrep-
Figure 13. Model C average perturbations to the constant PREM
values of (a) Rayleigh and (b) Love waves phase velocities for all
the periods considered here. As in Figure 9, the specific
contributions of the reference model B (long-dashed lines, squares)
and of the mantle perturbations of model C (dot-dashed lines,
triangles) are shown separately.
obtained in the corresponding direct inversion for each period
and type of surface waves. As a general rule, we find that
phase velocity maps directly obtained from the data explain
them better than those calculated from the 3-D model. This
result was to be expected: While the 3-D model is obtained in
one simultaneous inversion of all the measurements, the direct
phase-velocity inversions are independent from each other, each
being allowed to fit the phase anomalies with a solution model
not necessarily consistent with the others. This effect is particularly significant at long periods, corresponding to deeper
structure, which is more difficult to constrain. However, a
visual comparison of Figures 7c and 8c and Figures 7d and
8d, and of analogous maps at different periods, shows a high
correlation.
5. Iterative Inversion Procedure
[39] The technique that we have developed leaves us free to
choose any convenient 3-D model as reference. A reasonable
choice of a new reference is model B, which naturally explains
our surface wave measurements better than the initial one, model A.
Figure 14. Variance reduction achieved by model C (thick lines),
with respect to PREM, plotted as a function of period for both (a)
Rayleigh and (b) Love wave data. For comparison, the curves of
Figure 10, obtained from model B, are also shown (thin lines).
BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
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Figure 15. Radial profiles of vSV (thick dashed lines) and vSH (thick solid lines), as a function of depth, averaged
over four different regions: (a) central Pacific Ocean, (b) Canadian shield, (c) western United States, and (d) Western
Australia. Model C (black lines) is compared to S20A [Ekström and Dziewonski, 1998] (blue) and to the isotropic vS
model S20RTS of Ritsema and van Heijst [2000] (orange), derived from measurements mostly sensitive, within the
upper mantle, to dvSV. Model A is shown as a thin black solid line. See color version of this figure at back of this issue.
ancy is an indication of the slight inadequacy of the sensitivity
kernels used in the first iteration.
[42] Figure 14 shows the values of variance reduction, with
respect to PREM, achieved by models B (thin lines) and C (thick
lines) at different periods for both Rayleigh (Figure 14a) and Love
(Figure 14b) waves. At short periods the two models explain the
data equally well; in fact, short-period measurements are mostly
sensitive to the crust, whose structure has remained, by definition,
Figure 16. Globally averaged profiles of vSV (dashed lines) and vSH (solid lines) for models B (section 4.3, thin
lines) and B0 (section 6, thick lines).
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BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
Figure 17. The lateral perturbations associated with model B, minus those of model B0. We have found model B0
after suppressing the lateral dependence of the surface wave sensitivity kernels. See color version of this figure at
back of this issue.
constant. At intermediate periods, model C achieves a slightly
better variance reduction than model B; the opposite is true at
periods >200 s.
5.1. A Discussion of Our Model in Relation to
Published Images of the Upper Mantle
[43] In Figure 15 we show the mean values of vSH (solid lines)
and vSV (dashed lines) in model C, as functions of depth, computed
within four different regions; for comparison, we include the mean
vSH and vSV, in the same regions and depth range, from two
published global images of the upper mantle: the radially anisotropic S20A and the primarily vSV model S20RTS of Ritsema and
van Heijst [2000]. The radial profiles derived from model C are
everywhere smooth at 220 km depth: the sharp discontinuity
included in PREM, and thereby present in S20A and S20RTS (both
derived with PREM as a reference model), at that depth does not
appear to be required by observations.
[44] The central Pacific Ocean (Figure 15a; longitude 180E to
215E, latitude 0N to 30N), where model C is remarkably
consistent with S20A and S20RTS, is characterized by a large
anisotropic anomaly, reaching its maximum (with vSH > vSV) at a
depth of 150 km. This very anomalous feature was first identified
by Ekström and Dziewonski [1998], who showed that in this region
the observed anisotropic lateral variations are as large as the
corresponding variations in isotropic shear velocity vS [Ekström
and Dziewonski, 1998, Figure 3]. Our study clearly demonstrates
that this result cannot be simply explained as a fictitious effect of
crustal heterogeneities, or neglected lateral variations of the sensitivity kernels.
[45] Beneath the Canadian shield (Figure 15b; longitude 250E
to 280E, latitude 45N to 60N), model C is characterized by a a
very large positive anomaly, both in vSH and vSV, as to be expected
in continental cratons [e.g., Gee and Jordan, 1988]. Although not
as prominent in S20A and S20RTS, this feature is shared by model
SNA [Grand and Helmberger, 1984] and, within approximately
the same region, the vSV model NA95 [van der Lee and Nolet,
1997], with values of vSV locally as large as 4.8 km s1. The
radial profiles of model C have a peak at a depth of 60 km, while
those of NA95 are smoother, S velocity being maximum between
100 and 150 km.
[46] A comparison with the results of van der Lee and Nolet
[1997] can be attempted also for the western United States region
(Figure 15c; longitude 236E to 255E, latitude 33N to 55N). The
overall slow character of NA95 in this region is seen also in model
C, between 100 and 200 km depth; the slow anomaly is strongest at
a depth of 150 km according to our model and 100 km
according to NA95. While NA95, a regional model with nominal
resolution 1, includes values of vSV as small as 3.9 km s1, in
the same region, S20A and S20RTS (global images of long spatial
wavelength) do not differ substantially from PREM; the slow
anomaly of model C (whose nominal resolution is, perhaps,
1000 km; the cumulative lateral extent of two neighboring blocks
in our parameterization), higher than that of S20A and S20RTS, is
only slightly more pronounced. As a general rule, global tomographic inversions often fail to resolve lateral anomalies of strong
amplitude but limited spatial extent, which are best imaged by
regional studies: a problem that should always be taken into account
as tomographic maps are used to test geological hypotheses.
[47] Last, the radial profile of model C underneath Australia
(Figure 15d; longitude 110E to 145E, latitude 40S to 10S)
supports the observation by Simons et al. [1999] of a relatively thin
continental root in this region; both models have their highest
velocity at a depth of 80 km, where Simons et al. [1999] observe
a local maximum vSV of 4.8 km s1. The slowest regions of
Simons et al.’s [1999] model (vSV locally down to 4.3 km s1) are
located between 140 and 300 km depth, which is also in agreement
with Figure 15.
BOSCHI AND EKSTRÖM: SURFACE WAVE UPPER MANTLE TOMOGRAPHY
6. Lateral Variations of Surface Wave Sensitivity
Kernels: Practical Implications
for Global Tomography
[48] Although we have thoroughly discussed, from a theoretical
standpoint, the implications of lateral changes in the sensitivity of
surface waves to the properties of the Earth’s mantle, a quantitative
evaluation of the consequences of this effect on global tomographic models is still necessary. A simple comparison between
our results and those already published would not be sufficient;
other authors have derived their maps from different sets of data,
with different reference models, different parameterizations, and
different regularization schemes: It would be very difficult to
isolate the net effect of our novel treatment of the sensitivity
kernels. For this reason, we determine one more tomographic
image of the upper mantle, applying, to the same data set, exactly
the same procedure described above, including the regularization
scheme that led to the determination of model B. The only
difference is that we assume that the sensitivity of surface waves
to the Earth’s structure is, at all longitudes and latitudes, equal to
that computed for block 3352 of CRUST-5.1 (randomly chosen
among all blocks characterized by a crustal thickness of 25 km,
as in PREM; see shaded lines in Figures 1 and 3). Following
section 2.2, we impose the crustal correction (equation (9)) on the
measurements, although our reference Earth is now, strictly speaking, 1-D.
[49] The general character of the solution model, which we shall
refer to as model B0, is qualitatively consistent with that of model
B, fast and slow anomalies being distributed throughout the mantle
according to a similar large-scale pattern. Discrepancies between
the two models, however, are significant enough to possibly affect
a geophysical interpretation of the images. This is evident, in the
first place, from Figure 16, where the radial dependence of globally
averaged S velocity perturbation is shown; model B0 is, on average,
most anisotropic at a depth of 180 km, larger than the 150 km
of model B. The average lateral anomalies, both in vSH and vSV, are,
in absolute value, smaller at shallow depths (
200 km and above)
and larger as the depth increases. The magnitude of these discrepancies can be as large as 0.1 km s1 (
2%).
[50] We subtract the lateral S velocity anomalies of model B0
from those of model B and in Figure 17 show the resulting maps of
the upper mantle; they are dominated by a coherent, long-wavelength pattern, resembling, to some extent and mostly at shallow
depths, the distribution of oceans and continents. As expected, the
most significant discrepancies, as large as 2% (in percent
perturbation with respect to model A, minus the average correction
of Figure 16) are localized at various depths in areas where the
crust is particularly anomalous with respect to its globally averaged
properties. The Tibetan Plateau and the Canadian and Eurasian
shields, for example, are characterized by a relatively large crustal
thickness; if the depth of the Moho is treated as constant (model
B0), in these areas, fictitious structure can be projected into the
mantle immediately underlying it. Regions of anomalously thin
crust give rise to a similar but opposite problem. Anomalous
topography also has an effect on the sensitivity of surface waves
to mantle structure; in the southeast Pacific mid-oceanic ridge the
water layer, to which Rayleigh waves are sensitive, is thinner than
in the surrounding ocean, while the seafloor topography is elevated
(which affects the propagation of Love waves).
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heterogeneities strongly affect the propagation of surface waves; an
effect that has therefore to be somehow ‘‘removed’’ if the underlying mantle is to be imaged. They have done so in an approximate
fashion, described in section 2.2, subtracting from the data the
contribution of crustal heterogeneities, as predicted by realistic
crustal models. However, they have not attempted to account for
the effect of such crustal heterogeneities on the sensitivity of
surface waves to mantle structure.
[52] We have shown here, making use of the JWKB treatment
of Tromp and Dahlen [1992a, 1992b] and Wang and Dahlen
[1994], how lateral changes in the properties of the crust and,
most importantly, in the depth of the Moho discontinuity result in a
different shape of the kernels that relate the propagation of surface
waves to the heterogeneity of the mantle. We have then accounted
for this effect in the determination of a new tomographic model of
the upper mantle. We have compared our new model to an image
that we obtained from the same data with the same regularization
and parameterization but with the above mentioned approximate
treatment of crustal anomalies, and we have found significant
discrepancies between the two models (section 6). We therefore
suggest that as tomographers attempt to improve their resolution of
the shallow mantle, it will become necessary to also take into
account the lateral variations of the sensitivity kernels. This next
step should be accompanied by the development of more refined
maps of the crust. The tomographic inverse problem could be
further complicated by consideration of ray curvature [Wang and
Dahlen, 1994; Laske, 1995; Larson, 2000], which we have
neglected thus far.
7.2. Current Understanding of the Upper Mantle
Lateral Structure
[53] Numerous large-scale features that have characterized
previous images of the Earth’s upper mantle are confirmed by
ours (section 5.1); in particular, the important anisotropic anomaly of the central Pacific Ocean first identified (with the ‘‘traditional’’ approach) by Ekström and Dziewonski [1998]. Our
images (Figure 5) still include a radially anisotropic Pacific upper
mantle, with dvSV > dvSH in the top portion of the upper mantle
(50 km depth and above) and dvSV < dvSH below; most other
features of model S20A are also present in our new images, in
spite of the many differences between our technique and that of
Ekström and Dziewonski [1998].
[54] An important difference between the approach of Ekström
and Dziewonski [1998] and ours lies in the choice of the reference
model. In the first iteration of our procedure we formulate the
inverse problem with respect to the completely isotropic model A,
which is continuous at 220 km depth; Ekström and Dziewonski
[1998], instead, refer to the radially anisotropic PREM, including
the 220-km discontinuity. Here we have found shear velocity
models whose vertical profiles (Figures 6, 12, and 15) at those
depths are smooth and monotonous: an indication (within the
resolution limits of surface-wave measurements) that the presence
of a sharp discontinuity at 220 km is not required to explain our data.
[55] Acknowledgments. We thank Adam Dziewonski, Jeroen Tromp,
and Guy Masters for many helpful discussions; Jeroen Ritsema for making
his Earth model easily available; and Suzan van der Lee, Don Forsyth,
Barbara Romanowicz, and an anonymous reviewer. Most figures were
prepared with GMT [Wessel and Smith, 1991]. This research has been
supported by National Science Foundation grant EAR-98-05172.
7. Conclusions
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L. Boschi and G. Ekström, Department of Earth and Planetary Sciences,
Harvard University, 20 Oxford Street, Cambridge, MA 02138, USA.
([email protected])
Rayleigh, 35 s, VSV
Love, 35 s, VSH
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B4, 10.1029/2000JB000059, 2002
-0.0150
-0.0079
-0.0100
Rayleigh, 300 s, VSV
-0.00062
-0.0021
Love, 300 s, VSH
-0.00042
-0.0041
-0.0031
Figure 2. Sensitivity of (left) Rayleigh and (right) Love waves, at periods of (top) 35 s and (bottom) 300 s, to
perturbations in vertical and horizontal shear velocities, respectively, at a depth of 70 km. The quantity plotted, dT/T,
is the same as in Figures 1 and 3.
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Figure 5. Model B percent lateral perturbations (relative to the 3-D reference model A) in (left) vertical and
(middle) horizontal S velocity and (right) their difference, at four depths in the upper mantle (50 to 200 km as
indicated, top to bottom). The average perturbation of each map has been subtracted from it in order to focus on
lateral variations; it is plotted separately, as a function of depth, in Figure 6.
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Figure 7. Local corrections to the constant value of 35-s Love wave phase velocity associated with PREM, due to
(a) model A, including crustal heterogeneities from CRUST-5.1, (b) the lateral upper mantle anomalies of Figure 5
(model B), and (c) their combined effect. (d) Phase velocity map resulting from a direct inversion of 35-s Love wave
phase anomalies. The similarity of Figures 7c and 7d demonstrates the internal consistency of our procedure.
Figure 8.
Same as Figure 7, but for 150-s Rayleigh waves.
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Figure 11. Model C, shown in the same fashion as model B in Figure 5, in terms of percent perturbations relative to
model A (the average perturbation, previously subtracted from these plots, is shown separately in Figure 12). This
new image results from a second iteration of our procedure, with model B as the starting point.
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Figure 15. Radial profiles of vSV (thick dashed lines) and vSH (thick solid lines), as a function of depth, averaged
over four different regions: (a) central Pacific Ocean, (b) Canadian shield, (c) western United States, and (d) Western
Australia. Model C (black lines) is compared to S20A [Ekström and Dziewonski, 1998] (blue) and to the isotropic vS
model S20RTS of Ritsema and van Heijst [2000] (orange), derived from measurements mostly sensitive, within the
upper mantle, to dvSV. Model A is shown as a thin black solid line.
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Figure 17. The lateral perturbations associated with model B, minus those of model B0. We have found model B0
after suppressing the lateral dependence of the surface wave sensitivity kernels.
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