Physics of the Earth and Planetary Interiors 146 (2004) 113–124
Constraining large-scale mantle heterogeneity using mantle
and inner-core sensitive normal modes
Miaki Ishii a,∗,1 , Jeroen Tromp b
a
Department of Earth & Planetary Sciences, Harvard University, Cambridge, MA 02138 USA
b Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125 USA
Received 2 September 2002; received in revised form 5 February 2003; accepted 18 June 2003
Abstract
Attempts to resolve density heterogeneity within the mantle using normal-mode data revealed an unexpected feature near
the core-mantle boundary: regions of strong sheer velocity reduction, known as superplumes, are characterized by heavier
than average material [Science 285 (1999) 1231]. Thus far, free-oscillation studies of mantle structure have relied upon modes
which sample only the mantle (e.g., [J. Geophys. Res. 96 (1991) 551; Science 285 (1999) 1231; J. Geophys. Res. 104 (1999a)
993; Geophys. J. Int. 143 (2000b) 478; Geophys. J. Int. 150 (2002) 162]). In order to better constrain mantle heterogeneity,
we add inner-core sensitive modes to our data set, and invert for mantle structure and inner-core anisotropy simultaneously.
The additional modes do not alter the pattern of the density anomalies relative to model SPRD6 [Geophys. J. Int. 145 (2001)
77]. However, they help constrain the amplitude of lateral variations in density. In a previous study, obtaining a density model
with reasonable amplitudes required supplemental gravity data, which are not included in the current study. Nonetheless, the
new density model is generally weaker, especially at the bottom of the mantle. The root-mean square (RMS) amplitude of
the model exhibits two maxima around 600 and 2300 km depth. Experiments indicate that these are robust components of
the model and neither artifacts of the choice of radial basis functions nor the damping scheme. Comparisons of the density
model with shear- and compressional-velocity models show negative or nearly zero correlation in the transition zone and
near the base of the mantle, where the root-mean square density amplitude is high. Finally, the anomalous features near the
core-mantle boundary are confirmed: strong anti-correlation between shear and bulk-sound velocities, and increased density
at the locations of superplumes.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Seismic tomography; Normal modes; Seismic velocities; Density
1. Introduction
Body-wave travel times, the most extensively used
data in seismic tomography, have revealed large
∗ Corresponding author. Tel.: +1-858-534-4643.
E-mail address: [email protected] (M. Ishii).
1 Present address: Institute of Geophysics and Planetary Physics,
Scripps Institution of Oceanography, University of California, La
Jolla, CA 92093, USA.
low-velocity structures near the core-mantle boundary
(Dziewoński, 1984). Although images of these velocity anomalies have been refined using larger data sets
and improved techniques (e.g., Inoue et al., 1990; Li
and Romanowicz, 1996; Dziewoński et al., 1997; van
der Hilst et al., 1997; Masters et al., 2000c), mantle
density anomalies have remained elusive. This is because high-frequency body-wave data are insensitive
to density variations. In contrast, the gravitational
restoring force is important for low-frequency normal
0031-9201/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.pepi.2003.06.012
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modes, and hence they possess sensitivity to lateral
variations in density. Knowledge of the density distribution within the mantle would constitute a substantial
step toward understanding its dynamics and provides
important constraints on separating effects due to temperature and chemistry (Forte and Mitrovica, 2001).
Although controversial (Resovsky and Ritzwoller,
1999b; Masters et al., 2000b; Romanowicz, 2001), a
previous inversion of normal-mode data resulted in
density variations with an unexpected pattern near
the core-mantle boundary (Ishii and Tromp, 1999).
Underneath the Pacific and Africa, where seismic
velocities are anomalously low (features identified
as superplumes), the density is found to be higher
than average. This observation is consistent with convection simulations where chemically distinct, and
heavier, material accumulates under regions of active
upwellings with higher temperatures and lower shear
velocities (Christensen, 1984; Davies and Gurnis,
1986; Hansen and Yuen, 1988; Tackley, 1998). The
disadvantage of normal-mode tomography is that only
large-scale even-degree structure can be resolved.
Nonetheless, the mantle is generally dominated by
large-scale features, and there are structures with a
dominant even-degree component, such as the superplumes. Therefore, a comparison between velocity
and density heterogeneity should give us insight into
the petrology and dynamics of these structures.
Thus far, modeling of density variations within the
mantle has relied upon mantle sensitive modes (Ishii
and Tromp, 1999; Masters et al., 2000b; Kuo and
Romanowicz, 2002). Despite their sensitivity to mantle structure, inner-core sensitive modes have been ignored in studies of the mantle, because data from these
modes exhibit a strong inner-core signature. To overcome this dilemma, we invert mantle and inner-core
sensitive modes simultaneously for mantle heterogeneity and inner-core anisotropy. The results for the
inner core are discussed elsewhere (Ishii et al., 2002),
and we focus on models of the mantle in this paper.
2. Theory
In a seismic spectrum, peaks associated with the
normal modes of the Earth can be easily observed.
Some of them split visibly due to three factors: rotation, excess ellipticity, and non-spherical material
properties (e.g., lateral heterogeneity and anisotropy).
The first two effects can be determined accurately using the precisely known rotation rate and ellipticity of
the Earth (Woodhouse and Dahlen, 1978), facilitating
the isolation of splitting due to internal structure. This
process provides “splitting-function coefficients” for
every mode which combine to produce the “splitting
function”
σ(r̂) =
s
cst Yst (r̂),
s=0 t=−s
where r̂ denotes a point on the unit sphere, cst is the
splitting-function coefficient at spherical degree s and
order t, and Yst represents fully-normalized spherical
harmonics (Edmonds, 1960). This splitting function
represents a radial average of the structure beneath the
point r̂ as uniquely sampled by a given mode. It is
closely related to surface-wave phase-velocity maps,
with the important difference that the splitting function
of an isolated mode is limited to even degrees. This is
because waves traveling in opposite directions destructively interfere to cancel out the odd-degree signal.
Because a splitting function is a radial average of
three-dimensional heterogeneity, its coefficients are
related to internal properties δm and topography δd by
a
cst =
δmKm
δdst Ksd ,
(1)
s dr +
0
m
d
where Ks denotes the degree-dependent sensitivity
kernel of a given mode. The first term on the right-hand
side of Eq. (1) is the volumetric contribution with an
integration over radius from the center to the surface
(a) of the Earth. Both δm and Ksm are dependent on radius and the summation over m represents a sum over
material properties, such as lateral variations in velocity or anisotropy. The second term on the right-hand
side of Eq. (1) is a contribution from topography δd
on various boundaries represented by a sum over d,
e.g., the core-mantle boundary and Moho. The sensitivity kernel for such undulations, Ksd , tends to be
small, hence this term is often neglected. Because the
trade-off between density and topography is not significant (Ishii and Tromp, 2001), we also choose to
ignore this term in this study.
Previous inferences of density heterogeneity within
the mantle are based upon mantle sensitive modes
(Ishii and Tromp, 1999; Masters et al., 2000b; Kuo and
M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124
Romanowicz, 2002), i.e., modes for which the sensitivity kernels Ksm are zero in the inner core (Fig. 1). In
this case, the volumetric term in Eq. (1) is integrated
over the mantle, with density δρ, shear and compressional velocities δβ and δα, respectively, as material
properties δm in Eq. (1):
a
β
ρ
cst =
(δβst Ks + δαst Ksα + δρst Ks ) dr,
b
where b is the radius at the core-mantle boundary.
In contrast, inner-core sensitive modes have non-zero
sensitivity in the inner core (Fig. 1), and they introduce an additional volumetric term associated with
inner-core anisotropy,
a
β
ρ
cst =
(δβst Ks + δαst Ksα + δρst Ks ) dr
b
c
γ
+
δγKst dr,
(2)
0
γ
where c is the radius of the inner-core boundary, and
δγ represents parameters which describe inner-core
anisotropy (Tromp, 1995). Eq. (2) defines the linear
problem
dm = Gm m,
(3)
115
where dm is the data vector containing splitting-function
coefficients from various modes, Gm a matrix containing sensitivity kernels, and m the model vector
with mantle and inner-core components, i.e., m =
(δβstk δαkst δρstk δγ)T for different values of s and t (T
denotes the transpose). The additional superscript k
is the index of the radial basis functions within the
mantle. In most of this paper, we use Chebyshev
polynomials (Su, 1992) up to order 13 (kmax = 13)
as the radial basis function. Using both mantle and
inner-core modes, we can simultaneously invert for
mantle and inner-core structure.
In order to better constrain inner-core anisotropy,
we include PKP travel times in our data set. The relationship between the anisotropic parameters δγ and
travel times is also linear, i.e.,
dt = Gt m,
(4)
where the data vector dt contains travel times and Gt
describes their sensitivity to anisotropy in the inner
core (see Ishii et al., 2002 for a detailed discussion).
Combining Eqs. (3) and (4), we have
d = Gm,
where d = (dTm dTt )T , and G = (GTm GTt )T . Since this
is generally an under-determined inverse problem,
Fig. 1. Comparison of sensitivity kernels. Sensitivity kernels (K2 ) for isotropic variations in shear and compressional velocities and density
for mantle and inner-core sensitive modes. The kernels are for modes 1 S4 , a mantle sensitive mode (left), and 6 S3 , an inner-core sensitive
mode (right). Mode 1 S4 is insensitive to structure within the inner core, but is strongly sensitive to lateral variations in shear velocity
(dashed curve) in the lower mantle, compressional velocity (dotted curve) near the surface, and density (solid curve) in the upper mantle.
On the other hand, 6 S3 has a non-zero kernel within the inner core as well as significant sensitivity to heterogeneity within the mantle,
especially to density.
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weighting and damping are introduced in the inversions, a detailed description of which can be found
in Ishii and Tromp (2001) and Ishii et al. (2002).
Inner-core sensitive modes are strongly sensitive to
the compressional-velocity structure of the mantle,
and we are forced to damp model parameters for
compressional velocity twice as hard as shear velocity or density. Otherwise the compressional-velocity
model in the mantle acquires a strong zonal pattern
which we know to be erroneous based upon other
tomographic models.
3. Data
In the past several years, various groups have measured mode splitting using a multitude of methods
(e.g., Tromp and Zanzerkia, 1995; Resovsky and
Ritzwoller, 1998; Masters et al., 2000a). The currently
available data set includes isolated spheroidal and
toroidal modes, and efforts have been made to determine the splitting of coupled modes (Resovsky and
Ritzwoller, 1995), which provide valuable information about the odd-degree structure of the Earth’s interior. The maximum angular degree, s, of the splitting
function has been extended to degree 12 (Ritzwoller
and Resovsky, 1995), a three-fold improvement
compared to earlier studies (e.g., Giardini et al.,
1988). From this wealth of data we select splitting
coefficients of isolated modes at degrees 2, 4, and
6. We considered higher-degree and coupled-mode
measurements to be insufficient at this time for deriving reliable mantle models. These selection criteria
reduce the number of modes in our data set to 123
spheroidal and 42 toroidal modes, of which 55 are
inner-core sensitive. For some modes, measurements
of splitting functions are available from independent
research groups. Rather than excluding data, we keep
multiple measurements of splitting functions for the
same mode by different groups in our data set, much
the same way as data for the same ray path are used
in body-wave tomography.
The data set for this study differs from that of our
previous study (Ishii and Tromp, 2001) in that it incorporates measurements from the study of Masters et al.
(2000b), which includes surface-wave equivalent as
well as other mantle sensitive modes, and inner-core
sensitive modes from He and Tromp (1996), Resovsky
and Ritzwoller (1998), and Durek and Romanowicz
(1999). Before inversion, these data are corrected for
crustal structure using model Crust5.1 (Mooney et al.,
1998). Because the data set includes splitting coefficients of isolated modes up to and including spherical
harmonic degree 6, corresponding mantle models will
consist only of the even degrees 2, 4, and 6.
The travel-time data included in this study only constrain inner-core anisotropy, hence we omit a discussion of this data set. A detailed description of the data
processing applied to the body-wave data can be found
in Ishii et al. (2002).
4. Results
Statistical results of inversions with different parameterizations within the mantle are presented in
Table 1
Mode and travel-time inversions
Model
χ2 /(N − M)
χ2 /(N − M7 )
VR (DF)
VR (BC)
VR (AB)
VR (modes)
VR (IC)
S
SP
SPR
3.3
3.0
2.7
3.4
3.0
2.7
84.4
83.2
84.4
71.3
72.2
70.5
74.7
75.0
74.2
89.6
91.1
92.0
76.5
79.9
82.6
Table summarizing the fit for different model parameterizations within the mantle. Inner-core anisotropy is assumed to be uniform. χ2 -tests
for the overall fit are given by χ2 /(N − M) and χ2 /(N − M7 ), where N denotes the number of data, and M and M7 are the number of
model parameters with kmax = 13 and 7, respectively. VR (DF) is the variance reduction for PKPDF data, VR (BC) the variance reduction
for PKPBC − PKPDF data, and VR (AB) the variance reduction for PKPAB − PKPDF data. VR (modes) is the fit to the entire normal-mode
data set; VR (IC) the fit to inner-core sensitive modes. The “S” model involves an inversion for mantle and inner-core structure assuming
that compressional velocity and density are related to shear velocity through scaling. The scaling values are δ ln ρ = 0.2 δ ln β and
δ ln α = 0.55 δ ln β, where ρ is the density, β the shear velocity, and α the compressional velocity. “SP” indicates that data are inverted
for shear- and compressional-velocity structure, but density is a scaled shear-velocity model. Finally, the “SPR” model has shear-velocity,
compressional-velocity, and density variations.
M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124
117
Fig. 2. Models of shear velocity, compressional velocity, density, and bulk-sound velocity within the mantle. Three-dimensional models
of the mantle plotted in map view at six discrete depths. Blue indicates regions where the relative perturbation is higher than average and
red indicates values that are lower than average. The scale is fixed at a saturation level of ±1% for all maps. These maps are plotted
using even spherical harmonic degrees 2, 4, and 6, hence some of the common features seen in tomographic maps such as the contrast
between oceans and continents at shallow depths is not obvious. (Shear Velocity) Comparison of the shear-velocity model derived from
mantle and inner-core sensitive modes (left) and SKS12WM13 (right). (Compressional Velocity) Comparison of the compressional-velocity
model derived from mantle and inner-core sensitive modes (left) and P16B30 (right). (Density) Comparison of density models based upon
mantle and inner-core sensitive modes (left) and SPRD6 (Ishii and Tromp, 2001) which is constrained only by mantle sensitive modes
(right). (Bulk Sound Velocity) Comparison of bulk-sound velocity models derived using shear- and compressional-velocity models from
this study (left) and from SPRD6 (right).
Table 1. An increase in the number of parameters
from a shear-velocity only (S) to a shear-velocity,
compressional-velocity, and density (SPR) inversion
is supported by a systematic decrease in χ2 /(N − M),
where N is the number of data and M is the number
of model parameters. In a previous study, this decrease in χ2 /(N − M) is obtained when the maximum
radial parameterization (kmax ) is 7 (Ishii and Tromp,
2001), but the improved modal database supports a
decrease in χ2 /(N − M) even when kmax = 13. The
fit to the body-wave data does not change more than
a couple of percent, suggesting that additional parameters in the mantle do not trade-off significantly
with inner-core anisotropy. On the other hand, the
fit to inner-core sensitive modes improves with an
increased number of mantle parameters, confirming
that mantle structure makes a substantial contribution
to the splitting coefficients of these modes.
In Fig. 2, velocity and density models of the mantle are shown at discrete depths. These models are
obtained using the following set of starting models: shear-velocity model SKS12WM13 (Dziewoński
et al., 1997), compressional-velocity model P16B30
(Bolton, 1996), and a zero density model. Corresponding statistical results appear in Table 1 under
the model name SPR. Fig. 2 also includes plots of
bulk-sound velocity obtained using the shear- and
compressional-velocity models from the inversion and
reference model PREM (Dziewoński and Anderson,
1981). Unlike compressional velocity, bulk-sound
velocity has no dependence on shear velocity, which
tends to mask anomalies associated with incompressibility in compressional-velocity maps. The models
obtained from our inversions are not readily comparable to existing mantle models because they consist
only of the even degrees 2, 4, and 6. For example,
the ocean-continent distribution observed in all tomographic models near the surface is difficult to see
with only even degrees. We therefore compare our
models with degrees 2, 4, and 6 of SKS12WM13,
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P16B30, the density model of SPRD6 (Ishii and
Tromp, 2001), and a bulk-sound velocity model based
upon the shear- and compressional-velocity models of SPRD6. In general, these models are similar
to SPRD6, with average correlation coefficients between SPRD6 and the new set of models of 0.9, 0.8,
and 0.7 for shear velocity, compressional velocity,
and density, respectively. Even though the pattern of
the density model obtained in this study is similar
to that of SPRD6, there are two regions within the
middle mantle, at around 1000 and 1800 km depth,
where the correlation is relatively low (0.5 and 0.6,
respectively). These are noticable in Fig. 2 as slight
differences in the density distribution pattern in the
maps at depths of 1300 and 1800 km. Furthermore,
strong anti-correlation of bulk-sound velocity with
shear velocity at the base of the mantle is observed, as
well as the regional anti-correlation of shear-velocity
and density underneath the central Pacific and Africa.
In a previous study based upon mantle-sensitive
modes, the amplitude of density was highly uncertain and the addition of gravity data was required to
Fig. 4. Correlation coefficients between models of shear and compressional velocities (solid), shear velocity and density (dotted),
and compressional velocity and density (dashed) as a function of
depth. For the number of free parameters in these models, the
correlation is statistically significant at the 90% confidence level
if the correlation coefficient is greater than 0.25.
Fig. 3. Comparison of the RMS amplitude of the density models
from SPRD6 (Ishii and Tromp, 2001; dashed curve), and this study
(solid curve).
constrain it to a reasonable value (Ishii and Tromp,
2001). Here, we do not use gravity data, yet the amplitude of the density model is comparable to or smaller
than that of the compressional-velocity model. In particular, unlike the density model of SPRD6, the new
model has smaller root-mean square (RMS) amplitude
near the core-mantle boundary. As a consequence, the
RMS profile as a function of depth contains two peaks:
one within the transition zone and another at around
2300 km depth (Fig. 3). As we demonstrate in the next
section, these characteristics are independent of the
radial basis functions used in the inversion.
As shown in Fig. 4, the correlation between velocities and density is rather poor. Furthermore, some
trends observed previously in SPRD6 are strengthened, such as the decorrelation of density and velocities in the transition zone. In fact, the correlation
coefficients in these regions suggest that the models
may be significantly anti-correlated. In the bulk lower
M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124
119
Fig. 5. Map views of laterally varying ratios of density-to-shear-velocity (left; R-to-S), density-to-compressional-velocity (center; R-to-P),
and Poisson’s ratio (right) at six discrete depths. Blue indicates regions where the relative perturbation is higher than average and red
indicates values that are lower than average. Perturbations from the reference ratio as in PREM (Dziewoński and Anderson, 1981) are
shown at 0.05, 0.02, and 0.05% levels for the three ratios, respectively.
mantle, the correlations between density and seismic
velocities initially decrease monotonically but flatten or increase at the core-mantle boundary. Simple
radially-dependent scaling relations between density
and velocity have been calculated both theoretically
and experimentally (e.g., Anderson et al., 1968;
Anderson, 1987; Karato, 1993) and have been used
extensively in geodynamic modeling (e.g., Hager and
Clayton, 1989; Forte et al., 1994), but their meaning
is ambiguous when the models are not highly correlated (Ishii and Tromp, 2001). Therefore, we calculate
laterally varying ratios of density-to-shear-velocity,
density-to-compressional-velocity, and Poisson’s ratio
(Fig. 5). Note that Poisson’s ratio is another representation of the ratio between shear and compressional
velocities. The ratios are usually dominated by the
pattern of models with stronger lateral variations,
although effects due to density variations can be discerned in the density-to-compressional-velocity ratio.
Anomalies associated with superplumes clearly stand
out in these plots.
5. Discussion
The resolution of these mantle models can be addressed most conveniently by looking at the resolution
matrix of the inversion (see Ishii and Tromp, 2001, for
the definition of the resolution matrix and for various
resolution tests). If the model parameters are perfectly
resolved, i.e., if no damping is used in the inversion,
the resolution matrix becomes the identity matrix,
therefore deviations of the resolution matrix from
the identity matrix reflect the effects of damping on
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Fig. 6. Resolution matrix of the SPR inversion when Chebyshev polynomials are used as the radial basis functions. The elements are
arranged by model, i.e., shear (S) velocity, compressional (P) velocity, or density, and in increasing Chebyshev polynomial order (left to
right). Parts of the matrix corresponding to model parameters at degree 2 order 1 (top), degree 4 order 1 (middle), and degree 6 order 1
(bottom).
M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124
model parameters. In Fig. 6 components of the resolution matrix are shown. Because neither the sensitivity
kernels nor the applied damping depend upon angular
order t, the resolution for different values of t within
the same angular degree is nearly identical. The effect of stronger damping for higher-order Chebyshev
polynomials is evident, because higher-order model
coefficients are poorly resolved (diagonal elements
with high values of k are close to zero). In addition,
stronger damping on higher spherical harmonic degrees and the compressional-velocity models manifest
themselves as poorer resolution of the related parameters. Fig. 6 also shows that there does not appear
to be substantial leakage from model to model (indicated by smaller off-diagonal elements in comparison
to diagonal components).
To address the effects of off-diagonal terms associated with density model parameters and to understand
the full effects of damping and the starting model, we
perform a resolution test. In this test, splitting-function
coefficients of all modes used in the inversion are calculated using shear- and compressional-wave models
SKS12WM13 and P16B30, respectively, a zero model
for density, and an inner-core model described in Ishii
et al. (2002). These synthetic data (without noise) are
inverted with the same damping, weighting, and starting models as in inversions based upon the real data.
This test focuses on the leakage of power from velocity heterogeneity to density, because any density variations observed in the retrieved model are due to the
inversion process. The RMS amplitudes and correlation between the two density models obtained from
the real and synthetic data are compared in Fig. 7.
The influence of the velocity models on the density is
small, and the density distribution obtained from this
resolution test is considerably different from that presented in Fig. 2. Both the RMS amplitude and correlation indicate that the depths affected most strongly
by velocity structure are between 2000 and 2500 km.
The weak trade-off between the velocity models and
density is in contrast with a similar analysis by Kuo
and Romanowicz (2002), who used less than 20% of
the modes considered in our study.
We assume in our analysis that the mantle exhibits
only isotropic variations. In the bulk mantle, this assumption is reasonable given that most modes have
kernels with broad sensitivity in radius. However, this
is not a valid assumption in the upper mantle, where
121
Fig. 7. Resolution test for density model. Comparison of density
models obtained from the inversion of data and from a resolution
test where shear and compressional velocity models are used as
the input models. The RMS amplitudes of the density model (solid
curve) and that obtained from the resolution test (dashed curve)
are shown in the left panel, and the correlation between the two
models is shown in the right panel.
there is evidence of significant anisotropic variations
(e.g., Ekström and Dziewoński, 1998). To investigate
the effects of anisotropy near the Earth’s surface, we
perform an inversion without the splitting-function coefficients of surface-wave equivalent modes. Two alterations to the models are observed when we use this
subset of the data. The first is a change in pattern.
When correlation coefficients are calculated between
models with and without the Masters et al. (2000b)
data, they are smaller near the surface, as expected.
The minimum correlation coefficient is observed for
the density models and occurs around 150 km depth.
However, it is still large at 0.83, so the effect of pattern modification due to the omission of the surface
waves is not very significant, especially in the lower
mantle. The second effect is related to the amplitude
of the models. Because we have excluded a significant number of data, the constraints on amplitude are
weaker and hence the resulting mantle models have
amplitudes that are closer to those of Ishii and Tromp
(2001).
The choice of Chebyshev polynomials as the radial
basis function is somewhat arbitrary, although their
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Fig. 8. Comparison of density models based upon various radial basis functions. Density models using layers (left), Chebyshev polynomials
(center), and cubic B-splines (right) at six discrete depths. Blue indicates regions where the relative perturbation is higher than average
and red indicates values that are lower than average. The scale is fixed at a saturation level of ±0.5% for all maps.
gradual variation is consistent with the smoothly varying sensitivity kernels. The disadvantage of global basis functions, such as Chebyshev polynomials, is that
the termination of polynomials at some order (in our
case, kmax = 13) can lead to structure due to “ringing”
near the end points (i.e., near the surface and the
core-mantle boundary). In what follows, we present
results for density models using two local basis functions: layers and cubic B-splines (de Boor, 1978). In
these inversions, the number of unknowns in the radial
direction is kept constant (i.e., kmax = 13) and both
layers and B-spline knots are spaced evenly throughout the mantle. It should be remembered that damping
in the radial direction has an entirely different meaning
for local and global basis functions. Therefore, damping parameters in the radial direction are chosen so that
the traces of the resolution matrices (i.e., the number
of resolved model parameters) are similar, while keeping damping in the lateral directions the same. The
fits to data with local basis functions are then similar
to that obtained with Chebyshev polynomials. In general, the observed patterns in the density distribution
are compatible between models with different radial
bases, including features near the core-mantle boundary (Fig. 8). The models are least correlated near the
surface and the core-mantle boundary, as expected,
but the correlation coefficients remain well above the
95% significance level. Furthermore, the amplitudes
of these models are relatively consistent with one another (Fig. 9). The two peaks in RMS amplitude in the
transition zone and around 2300 km depth are present
regardless of the choice of basis function or damping
scheme, implying that they are robust features and not
the results of “ringing”.
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123
resolution matrix indicates that the model parameters are well-resolved, with limited leakage between
models.
Unlike our previous inversions, where the amplitude of the density model needed to be constrained
by gravity data, the amplitude of the new density
model is stable using only the modal data set. The
density RMS amplitude is generally smaller than that
of the velocity models, and exhibits two peaks around
600 and 2300 km depth. These depths coincide with
poor or negative correlations between density and
velocities. Experiments with local radial basis functions demonstrate that the two maxima in the density
RMS amplitude are robust, and that the pattern of the
density distribution does not depend on the choice
of the basis or damping scheme. The presence of
heavier material at the locations of superplumes suggests that these large-scale anomalies differ from the
ambient mantle not only in temperature but also in
chemistry.
Acknowledgements
Fig. 9. Comparison of the RMS amplitude of density based upon
various radial basis functions. Plot of the RMS amplitudes of
density models based upon Chebyshev polynomials (grey curve),
cubic B-splines (solid black curve), and layers (dashed black
curve).
M.I. was supported by a Julie Payette Research
Scholarship from the Natural Sciences and Engineering Research Council of Canada. Contribution
Number 8887, Caltech Division of Geological and
Planetary Sciences.
6. Conclusions
We present three-dimensional models of shear velocity, compressional velocity, and density within
the mantle based upon a mode data set that includes inner-core sensitive free oscillations. The
low-frequency inner-core modes possess substantial
sensitivity to the mantle, and we simultaneously invert
for mantle heterogeneity and inner-core anisotropy.
The mantle models obtained from this inversion do
not trade-off significantly with inner-core anisotropy,
and the improved database supports inversions for
shear-velocity, compressional-velocity, and density structure with a maximum radial expansion of
order 13. The models obtained from our inversions
exhibit a similar distribution of positive and negative
anomalies compared to our previous study, including
the regional anti-correlation of velocities and density near the core-mantle boundary. Analysis of the
References
Anderson, D.L., 1987. A seismic equation of state II. Shear
properties and thermodynamics of the lower mantle. Phys. Earth
Planet. Inter. 45, 307–323.
Anderson, O.L., Schreiber, E., Liebermann, R.C., Soga, N., 1968.
Some elastic constant data on minerals relevant to geophysics.
Rev. Geophys. 6, 491–524.
Bolton, H., 1996. Long Period Travel Times and the Structure of
the Mantle. Ph.D. Thesis, U.C. San Diego.
Christensen, U., 1984. Instability of a hot boundary layer and
initiation of thermo-chemical plumes. Ann. Geophys. 2, 311–
320.
Davies, G.F., Gurnis, M., 1986. Interaction of mantle dregs with
convection: lateral heterogeneity at the core-mantle boundary.
Geophys. Res. Lett. 13, 1517–1520.
de Boor, C., 1978. A Practical Guide to Splines. Springer Verlag,
New York.
Durek, J.J., Romanowicz, B., 1999. Inner core anisotropy inferred
by direct inversion of normal mode spectra. Geophys. J. Int.
139, 599–622.
124
M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124
Dziewoński, A.M., 1984. Mapping the lower mantle: determination
of lateral heterogeneity in P velocity up to degree and order 6.
J. Geophys. Res. 89, 5929–5952.
Dziewoński, A.M., Anderson, D.L., 1981. Preliminary reference
Earth model. Phys. Earth Planet. Inter. 25, 297–356.
Dziewoński, A.M., Liu, X.-F., Su, W.-J., 1997. Lateral
heterogeneity in the lowermost mantle. In: Crossley D.J.
(Ed.), Earth’s Deep Interior. Gordon and Breach, pp. 11–
50.
Edmonds, A.R., 1960. Angular Momentum in Quantum
Mechanics. Princeton University Press, Princeton, NJ.
Ekström, G., Dziewoński, A.M., 1998. The unique anisotropy of
the Pacific upper mantle. Nature 394, 168–172.
Forte, A.M., Mitrovica, J.X., 2001. Deep-mantle high-viscosity
flow and thermochemical structure inferred from seismic and
geodynamic data. Nature 410, 1049–1056.
Forte, A.M., Woodward, R.L., Dziewoński, A.M., 1994. Joint
inversions of seismic and geodynamic data for models of
three-dimensional mantle heterogeneity. J. Geophys. Res. 99,
21857–21877.
Giardini, D., Li, X.-D., Woodhouse, J.H., 1988. Splitting functions
of long-period normal modes of the Earth. J. Geophys. Res.
93, 13716–13742.
Hager, B.H., Clayton, R.W., 1989. Constraints on the structure
of mantle convection using seismic observations, flow models,
and the geoid. In: Peltier, W.R. (Ed.), Mantle Convection: Plate
Tectonics and Global Dynamics. Gordon and Breach, Newark,
NJ, pp. 657–763.
Hansen, U., Yuen, D.A., 1988. Numerical simulations of
thermal-chemical instabilities at the core-mantle boundary.
Nature 334, 237–240.
He, X., Tromp, J., 1996. Normal-mode constraints on the structure
of the Earth. J. Geophys. Res. 101, 20053–20082.
Inoue, H., Fukao, Y., Tanabe, K., Ogata, Y., 1990. Whole mantle
P-wave travel time tomography. Phys. Earth Planet. Inter. 59,
294–328.
Ishii, M., Tromp, J., 1999. Normal-mode and free-air gravity
constraints on lateral variations in velocity and density of the
Earth’s mantle. Science 285, 1231–1236.
Ishii, M., Tromp, J., 2001. Even-degree lateral variations in the
mantle constrained by free oscillations and the free-air gravity
anomaly. Geophys. J. Int. 145, 77–96.
Ishii, M., Tromp, J., Dziewoński, A.M., Ekström, G., 2002. Joint
inversion of normal mode and body wave data for inner core
anisotropy: 1. Laterally homogeneous anisotropy, J. Geophys.
Res. 107, 10.1029/2001JB000712.
Karato, S., 1993. Importance of anelasticity in the interpretation
of seismic tomography. Geophys. Res. Lett. 20, 1623–
1626.
Kuo, C., Romanowicz, B., 2002. On the resolution of density
anomalies in the Earth’s mantle using spectral fitting of normal
mode data. Geophys. J. Int. 150, 162–179.
Li, X.-D., Romanowicz, B., 1996. Global mantle shear-velocity
model developed using nonlinear asymptotic coupling theory.
J. Geophys. Res. 101, 22245–22272.
Li, X.-D., Giardini, D., Woodhouse, J.H., 1991. Large-scale threedimensional even-degree structure of the Earth from splitting
of long-period normal modes. J. Geophys. Res. 96, 551–577.
Masters, G., Laske, G., Gilbert, F., 2000a. Autoregressive
estimation of the splitting matrix of free-oscillation multiplets.
Geophys. J. Int. 141, 25–42.
Masters, G., Laske, G., Gilbert, F., 2000b. Matrix autoregressive
analysis of free-oscillation coupling and splitting. Geophys. J.
Int. 143, 478–489.
Masters, G., Laske, G., Bolton, H., Dziewoński, A.M., 2000c.
The relative behavior of shear velocity, bulk sound speed,
and compressional velocity in the mantle: Implications for
chemical and thermal structure. In: Karato, S., Forte, A.M.,
Libermann, R.C., Masters, G., Stixrude, L. (Eds.), Earth’s Deep
Interior: Mineral Physics and Tomography From the Atomic
to the Global Scale, vol. 117. American Geophysical Union,
Washington DC, pp. 63–87.
Mooney, W.D., Laske, G., Masters, T.G., 1998. Crust 5.1: a global
crustal model at 5◦ × 5◦ . J. Geophys. Res. 103, 727–747.
Resovsky, J.S., Ritzwoller, M.H., 1995. Constraining odd-degree
Earth structure with coupled free-oscillations. Geophys. Res.
Lett. 22, 2301–2304.
Resovsky, J.S., Ritzwoller, M.H., 1998. New constraints on deep
Earth structure from generalized spectral fitting: application to
free oscillations below 3 mHz. J. Geophys. Res. 103, 783–810.
Resovsky, J.S., Ritzwoller, M.H., 1999a. A degree 8 mantle shear
velocity model from normal mode observations below 3 mHz.
J. Geophys. Res. 104, 993–1014.
Resovsky, J.S., Ritzwoller, M.H., 1999b. Regularization uncertainty
in density models estimated from normal mode data. Geophys.
Res. Lett. 26, 2319–2322.
Ritzwoller, M.H., Resovsky, J.S., 1995. The feasibility of normal
mode constraints on higher degree structures. Geophys. Res.
Lett. 22, 2305–2308.
Romanowicz, B., 2001. Can we resolve 3D density heterogeneity
in the lower mantle? Geophys. Res. Lett. 28, 1107–1110.
Su, W.-J., 1992. The Three-Dimensional Shear-Wave Velocity
Structure of the Earth’s Mantle. Ph.D. Thesis, Harvard
University
Tackley, P.J., 1998. Three-dimensional simulations of mantle
convection with a thermo-chemical basal boundary layer: D”?
In: Gurnis, M. et al. (Eds.), The Core-Mantle Boundary Region,
Geodynamics Series, vol. 28. American Geophysical Union,
Washington DC, pp. 231–253.
Tromp, J., 1995. Normal-mode splitting due to inner-core
anisotropy. Geophys. J. Int. 121, 963–968.
Tromp, J., Zanzerkia, E., 1995. Toroidal splitting observations from
the great 1994 Bolivia and Kuril Islands earthquakes. Geophys.
Res. Lett. 22, 2297–2300.
van der Hilst, R.D., Widiyantoro, S., Engdahl, E.R., 1997. Evidence
for deep mantle circulation from global tomography. Nature
386, 578–584.
Woodhouse, J.H., Dahlen, F.A., 1978. The effect of a general
aspherical perturbation on the free oscillations of the Earth.
Geophys. J. R. Astron. Soc. 53, 335–354.