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 Vs, Vp 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  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 Vs, Vp 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) smax=6, pmax=7 and 2) smax=6, pmax=10.
In the first series of experiments, we considered a "realistic" synthetic model, for which we chose S velocity (VS) to be that of model SAW12D (Li and Romanowicz, 1996), and P velocity (VP)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 VS, 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 Vs model SAW12D so that the actual patterns of heterogeneity are not correlated with SAW12D. We recover perturbations in VP and VS 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 (pmax=7 versus pmax=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 smax=6, pmax=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 smax=6, pmax=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 VP and VS 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 VS and VP 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.
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