Whether or not we can resolve three-dimensional density structure in the mantle has long been a subject of controversy. High quality data from digital seismic instruments
emplaced this decade have renewed interest in the measurement of low frequency Earth normal modes
with the goal of extracting information on the 3D density structure (e.g. *Ishii and Tromp*, 1999; *Kuo and Romanowicz,* 1999a,b; *Resovsky and Ritzwoller*, 1999). In a study in which they inverted a dataset of mode splitting functions for structure in *V*_{s}, *V*_{p} and
up to spherical harmonic degree 6, *Ishii and Tromp* (1999) proposed that, in the lowermost mantle, high density regions were associated with low velocities, in
the two major "plume" regions in
the central Pacific and under Africa, stimulating speculations on the
mineral physics and geodynamic implications of such structure. *Ishii and Tromp* (1999) argued for the robustness of their model on the basis
of numerous experiments, including checkerboard resolution tests and exploration
of different parameterization schemes.
Using also a collection of normal mode splitting data to test inversions for three dimensional elastic structure up to degree 8, *Resovsky and Ritzwoller* [1999]
showed that the resulting density distribution depends strongly on a priori
constraints on the model parametrization and regularization.

Using a dataset of spectral waveforms, we obtained a model (*Kuo et al.,* 1999b; *Kuo,* 1999) with a different density structure than that obtained by *
Ishii and Tromp* (1999), in which, in particular low velocities are correlated with low
densities in the lowermost mantle. In order to understand these difference s, we have recently performed a series of synthetic
experiments aimed at
investigating the resolution of lateral variations in the mantle from
normal mode spectral data. Contamination effects between seismic
velocities and density are
examined in two ways: 1) by using resolution matrices computed from data kernels,
and 2) by inverting synthetic spectra computed from realistic input
Earth models.

In the experiments presented here we use a synthetic data set and
inversion scheme that mimics the actual data set and methodology which
we have used in a study of 3D mantle structure based on real mode data
(*Kuo and Romanowicz, 1999; Kuo, 1999*). This mode data set comprises the
spectra of 44 spheroidal modes observed on the vertical component,
including 7 pairs of coupled modes that provide some sensitivity to
odd
degree structure. The analysis method is a one-step inversion scheme
in which mantle structure in *V*_{s}, *V*_{p} and
is derived directly
from the observed spectra. This contrasts with the approach of some
recent studies, which use a two step procedure, involving the
computation of splitting coefficients followed by an inversion for 3D
structure (*eg. Ishii and Tromp, 1999; Resovsky and Ritzwoller, 1999*).

The resolution matrix is a useful tool to assess the leakage between model
parameters in a generalized inversion framework. The resolution matrix was computed using the data
kernels corresponding to our data set of 44 spheroidal mode spectra.
We consider two different spatial parameterizations: 1)
*s*_{max}=6, *p*_{max}=7
and 2)
*s*_{max}=6, *p*_{max}=10.

In the first series of experiments, we considered a "realistic" synthetic model, for which we chose
S velocity (*V*_{S}) to be that of model SAW12D (*Li and Romanowicz,
1996*), and P velocity (*V*_{P})to be that of P16B30 (*Masters et al,
1996*). A mock density model ()
was constructed so that the
root-mean-square amplitudes are about 25% of *V*_{S}, a common assumption in low-frequency seismology (e.g.
*Li et al., 1991*) that is in agreement with laboratory measurements at
shallow mantle pressure and temperatures (*Karato, 1993*). The mock density
perturbations are constructed by permuting and scaling (to 25%)
coefficients of *V*_{s} model
SAW12D so
that the actual patterns of
heterogeneity are not correlated with SAW12D.
We recover perturbations in *V*_{P} and *V*_{S} well for both parametrizations, but not as well for
(correlation of
60% between input and output models in the first parametrization and
50% in the second one.

In another set of experiments, we chose to fully populate
(respectively
and
)
with coefficients of
an aspherical model, keeping the other parameters zero. We could thus test how well we can resolve each parameter, independently of the others, and how much is mapped into the others. Our results show that
is well recovered, but there is some
contamination into
and
,
as expected from inspection of the resolution matrix.
The amplitude of
contamination into
is very small, and is well below the
level of signal obtained in real data inversions, so it should not be a
problem. The
amplitude of contamination into
is also not very large, but patterns at certain depths, particularly
at 2800 km, are reminiscent of patterns from recent density models inverted from normal mode splitting
coefficients (*Ishii and Tromp, 1999*), where high density features are located over Africa, and the Pacific
basin region. In the upper mantle, there is significant contamination of
patterns into
.

As seismic velocity structure in the Earth is relatively well-documented
(*Su et al., 1994; Li and Romanowicz, 1996; Masters et al, 1996; Dziewonski et al, 1997; Vasco and Johnson, 1997; van der Hilst et al., 1997 *), the study of cumulative contamination of
and
into
can be particularly informative. To investigate the latter,
we have performed a series of tests, in which we varied the depth parameterization (*p*_{max}=7 versus
*p*_{max}=10) and the input
model, keeping the input model in
to be zero.
When the input *S* model is SAW12D(*Li and Romanowicz, 1996*, the retrieved
density model is similar to density models
obtained from inversions of spectral data (*Kuo and Romanowicz,* 1999a, 1999b). On the other hand, when the input *S* model is SH12WM13 (*Su et al., 1994*), the resulting ghost density model for a
*s*_{max}=6,
*p*_{max}=10 parameterization (
)
yields patterns which
are strongly reminiscent of the published density model SPRD6 of *Ishii and Tromp* (1999)
, particularly the high density feature in the Pacific basin and over
Africa at 2800 km. From Figure
31.1b and c, the density patterns from
and SPRD6
(*Ishii and Tromp, 1999*) can be directly
compared for six depths in the mantle. The resemblance is striking,
although the amplitudes of
are much smaller
in the lower mantle, and there are slight lateral shifts in
the distribution of the patterns between
and
of SPRD6 ( *Ishii and Tromp,* 1999) . However, amplitudes retrieved from inversions
are dependent on the choice of damping parameter values. For the
*s*_{max}=6,
*p*_{max}=10 parameterization, the pattern in density(
)
at
the bottom of the mantle does not resemble that of *Ishii and Tromp,* (1999).

Since our inversion scheme is non-linear, and therefore involves several iterations, the results may not be completely represented by the resolution matrix corresponding to the first iteration only. We therefore conducted more complete tests involving the computation of synthetic seismograms and their iterative inversion, more accurately simulating the inversion process corresponding to real spectral data.

The estimated models
were obtained from inversions of
synthetic seismograms computed from input models composed of models P16B30 for
and SH12WM13 for
,
and without any density
perturbations. Aspherical structure up to harmonic degree 12 and
radial order 13 were included in the computation of synthetic
seismograms.
We started the inversions from PREM (*Dziewonski and Anderson, 1981*),
and damp the second radial derivative
to ensure radial smoothness. After 4 iterations, the
models converged to give over 99% variance reduction in the synthetic data.
Many of the features in the ghost density model retrieved resemble features
obtained in inversions with real data, both in pattern and amplitude. Some of
them could be real, of course, such as high densities beneath the Americas and
Asia at most depths in the mantle, as we expect high densities to correlate
with high velocities in subduction zones.

Current methods of retrieving three-dimensional mantle density structure from normal mode spectra
(*Kuo and Romanowicz, 1999b*) and normal mode splitting
coefficients
(*Ishii and Tromp, 1999*)
do not appear to yield reliable density models. The mantle density
models are affected by the
contamination of *V*_{P} and *V*_{S} structure into the density model
space, and depend strongly on the *a priori* starting
models in velocity, towards which the inversion is damped, at least for
certain choices of depth parameterization.

*Resovsky and Ritzwoller*, (1999) have documented the instability of
models derived from normal mode splitting coefficients when
using a sweep of *a priori* constraints. They have shown that it
is not possible to determine correlation and/or decorrelation of
with seismic velocity as a function of depth. Our work
supports their conclusions that current methods and data sets are not
sufficient to uniquely determine the density structure of the Earth.
We have shown that it is possible to retrieve models of
perturbations
purely due to contamination from *V*_{S} and *V*_{P} structure, and that
these ghost
models are consistent in pattern and amplitude with
published
models inverted from splitting coefficients (*Ishii and Tromp, 1999*), or
with
models which we determine from normal mode spectral data,
depending on the details of the test performed.

Dziewonski, A.M., X.-F. Liu, and W.-J. Su,
in *Earth's Deep Interior: the Doornbos memorial volume*,
D. J. Crossley, ed., pp. 11-50, Gordon and Breach Science, Newark, N.J., 1997.

Ishii, M. and J. Tromp,
Normal-Mode and Free-Air Gravity Constraints on Lateral Variations
in Velocity and Density of Earth's Mantle
*Science*, *285*, 1231-6, 1999.

Karato, S.,
Importance of anelasticity in the interpretation of seismic tomography,
*Geophys. Research Letters*, *20*, 1623-6, 1993.

Kuo, C., *Three-dimensional Density Structure of the Earth:
Limits for Astrophysical and Seismological Approaches*,
Ph.D. Thesis, University of California, Berkeley, California, 1999.

Kuo, C. and B. Romanowicz,
Three-dimensional Density Structure Obtained by Normal Mode
Spectral Measurements,
*Proceedings, IUGG XXII General Assembly*, A62, 1999a.

Kuo, C. and B. Romanowicz,
Density and Seismic Velocity Variations Determined from
Normal Mode Spectra,
*Eos Trans., AGU*,*80*, S14, 1999b.

Li, X.-D. and B. Romanowicz,
Global mantle shear velocity model developed using nonlinear asymptotic
coupling theory.
*J. Geophys. Res.*, *101*, 22245-72, 1996.

Li, X.-D., D. Giardini, and J.H. Woodhouse,
The relative amplitudes of mantle heterogeneity in P velocity, S velocity
and density from free-oscillation data,
*Geophys. J. Int.*, *105*, 649-57, 1991.

Masters, G., S. Johnson, G. Laske, and H. Bolton,
A shear-velocity model of the mantle.
*Philos. Trans. R. Soc. London A*, *354*, 1385-411, 1996.

Resovsky, J.S. and M.H. Ritzwoller,
Regularization uncertainty in density models estimated from
normal mode data,
*Geophys. Res. Lett.*, *26*, 2319-22, 1999.

Su, W.-j., R. L. Woodward and A. M. Dziewonski, Degree-12 model of shear velocity heterogeneity in the mantle, *J. Geophys. Res.*, *99*, 6945-6980, 1994.

Van Der Hilst, R.D., S. Widiyantoro, and E.R. Engdahl,
Evidence for deep mantle circulation from global tomography,
*Nature*, *386*, 578-84, 1997.

Vasco, D.W. and L.R. Johnson,
Whole Earth structure estimated from seismic arrival times,
*J. Geophys. Res.*, *103*, 2633-71, 1998.

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