- Sensitivity to even-degree structure
- Resolution tests
- Additional sensitivity using coupled modes
- Sensitivity to P- and S-wave velocities
- References

We have developed an algorithm in which the observed spectra for several modes are simultaneously and directly inverted for intrinsic structure, explicitly imposing the requirement that the spectra are modeled only by Earth stucture.

Our dataset consists of spectra measured from 28 mantle-sensitive spheroidal modes well
isolated in the frequency band between 0.4 to 4 mHz,
extracted from two 1994 great earthquakes (Bolivia 6/9/94, *M*_{w}=8.2
and Kurile Islands 10/4/94, *M*_{w}=8.3).
The final models of S-wave velocity perturbations (Kuo et al., 1997) from the 28 modes
are stable in that they explain 75% of the data spectra, converge
after 2 iterations, and are independent
of the starting model, whether it be PREM (Dziewonski and Anderson, 1981) or
SAW12D of U.C. Berkeley (Li and Romanowicz, 1996).

In order to assess the validity of the inversions, tests on the algorithm's ability to resolve the structure of an input S-velocity perturbation model were performed. Synthetic normal mode spectra were constructed by forward modelling in 3D models, and then subsequently inverted for structure, using PREM as a starting model.

The models of harmonic degree 4 generated from synthetically derived normal mode spectra stably converge to 99.9% variance reduction after 2 iterations using our data set of 28 modes . Lateral degree 8 in SAW12D is well resolved by the twenty-five degree 8 sensitive modes, producing resolution models of 99.9% variance reduction (Figure 20.1). The ability to resolve degree 10 in SAW12D degrades as the data set reduces to 21 modes, in which the output model can explain 88% of the synthetic spectra.

Optimal resolution is evaluated by the variance reduction amplitude and correlation of lateral heterogeneity at depth between input and output models. The shape of correlation profiles is dependent on damping parameters, lateral and radial parameterization, and the sensitivity of the mode kernels. Radially, the models were parameterized by Legendre polynomials in two schemes: one in which the Legendre polynomials spanned the whole radius of the mantle, and one in which sets of Legendre polynomials separately spanned the upper mantle and lower mantle. The resolution tests indicate that the modes are not sensitive to a sharp 670 km boundary as indicated in model SAW12D, leading whole mantle radial parameterization to be the optimal configuration.

Normal modes coupled through Earth structure and the Coriolis forces in the frequency range of interest could expand our data set. The inclusion of coupled modes in the direct inversion scheme allows for additional sensitivity to even and odd degree structures. Presently, 6 mode pairs increase the even degree sensitivity from 28 to 40 modes, and of these particular mode pairs, 4 are sensitive to odd degree structure. Models derived from 28 isolated modes and 12 coupled modes fit 74% of the data. The slight decrease in fit compared to the isolated case may be due to the scant sensitivity to odd degrees. At certain depths, the correlation to SAW12D is improved by the addition of the coupled modes. Near the core-mantle boundary, the high-velocity feature surrounding a low-velocity Pacific seen in SAW12D is manifested more prominently in the model when the data includes coupled modes (Figure 20.2). This can be attributed to either the increased sensitivity to even and odd degree structure, or perhaps the removal of contamination in the isolated mode inversion from even degrees to the odd degrees. The root-mean-square amplitudes of the odd degree heterogeneity are an order of magnitude less than those of the even degrees using this modal data set.

The S-wave velocity perturbation models were derived by imposing a constant scaling relationship of the P-wave velocity and density sensitivity kernels to the S-wave velocity kernel based on the proportionality of P- to S-wave velocity perturbations inferred from normal mode ``splitting functions'' (Li et al., 1991a; Li et al., 1991b). The heterogeneity of P- and S-wave velocities has been modelled in a number of arrival time studies (Robertson and Woodhouse, 1995; Vasco and Johnson, 1998). We have investigated the P- and S-wave velocity heterogeneity ( and , respectively) by independently inverting for these two velocity parameters by releasing the scaling relationship between sensitivity kernels. Determination of the scaling relationship between and from the P- and S-wave models is of interest, particularly its behavior in the lower mantle,

Results of resolution tests on an input model in which / increases linearly in the lower mantle indicate that a local radial parameterization scheme be adopted, as opposed to the global parameterization established by the use of Legendre polynomials. A spline parameterization will be used for the radial basis functions. The model parameterized by whole mantle radial parameterization most successfully retrieved the input model in which the to ratio is constantly scaled.

Dziewonski, A.M. and D.L. Anderson, Preliminary reference Earth model,
*Phys. Earth Planet. Inter.**25*, 297-356, 1981.

Kuo, C., J. Durek, and B. Romanowicz, Direct Inversion of Normal
Mode Spectra for Mantle Heterogeneity (abstract), *Eos Trans.
AGU,* *78* (46), Fall Meet. Suppl., F460, 1997.

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*, 649-657, 1991a.

Li, X-D., D. Giardini, and J.H. Woodhouse, Large-scale three-dimensional
even-degree structure of the Earth from splitting of long-period normal modes,
*J. Geophys. Res.*, *96*, 551-577, 1991b.

Li, X.-D. and B. Romanowicz, Global mantle shear velocity model
developed using nonlinear asymptotic coupling theory,*J. Geophys. Res.*,
*101*, 22245-22272, 1996.

Robertson, G.S. and J.H. Woodhouse, Evidence for proportionality of *P*
and *S* heterogeneity in the lower mantle, *Geophys. J. Int.*, *123*,
85-116, 1995.

Vasco, D.W. and L.R. Johnson, Whole Earth structure estimated from
seismic arrival times, *J. Geophys. Res.*,*103*, 2633-2671, 1998.

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