Computation Using the Spectral Element Method

We evaluate normal mode asymptotic methods by comparing the corresponding
3D synthetics with those computed
using the coupled spectral element/normal mode method (CSEM)(*Capdeville et al,* 2003). Three normal mode based asymptotic
approaches are compared: (1)path average approximation (PAVA) (*Woodhouse and Dziewonski*, 1984), in which only along-branch
mode coupling effects are considered; (2)Non-linear asymptotic coupling theory (NACT)(*Li and Romanowicz,*
1995), which includes the across-branch mode coupling effects; and (3) NACT+F, an extension of NACT with focusing
terms computed using higher order asymptotic theory (*Romanowicz,* 1987; *Romanowicz et al.*, 2004).
Systematic waveform comparisons are implemented. We find that NACT and NACT+F
provide much better fit, and
the off-great-circle effects, which result in focusing/defocusing and not seen by PAVA or NACT, are
well explained by NACT+F.

Two 3-D synthetic Earth models are used to test the validity of three normal mode based analytical approaches, PAVA, NACT and NACT+F. The CSEM is used to provide the accurate reference synthetics in the 3-D test models.

The synthetic models are parameterized laterally using spherical harmonics up to degree 16, and radially using cubic splines. To examine more closely the small perturbation of the seismograms caused by the 3-D heterogeneities, the differential waveforms (i.e. ) for CSEM and normal mode techniques are compared. Two representative results are shown here. Figure 30.1 shows the results for an isotropic source in a 3-D model with an ellipsoidal anomaly centered at the 220 km depth, and Figure 30.2 shows the results for a dip-slip source in a 3-D model with two opposite ellipsoidal anomalies centered at 150 km depth.

From the above results, we find that (1) when the path just grazes the anomaly, both PAVA and NACT fail to match CSEM synthetics, since they are insensitive to off-path structure, while NACT+F predicts the expected focusing effects fairly well; and (2) when the path passes through two anomalies with opposite signs, the effects of heterogeneities are cancelled out in the PAVA formalism (there is nearly no perturbation in PAVA differential waveform, as seen in Figure 30.2), and they are well explained in NACT and NACT+F, particularly for the overtone phases.

In summary, we have confirmed, through a series of synthetic experiments, that NACT and NACT+F are much better than PAVA in explaining waveform perturbations in a 3D heterogeneous Earth model. We also verified that the focusing effects are predicted well by the higher order asymptotic approximation, NACT+F.

NACT+F is potentially very important in Q tomography, in which the major technical difficulty encountered is how to
discriminate anelastic signals from elastic effects on the amplitude of seismic data. It has been shown that while the elastic
focusing/defocusing effects are not significant at low degrees ( 8) )*Selby and Woodhouse*, 2002), they need to be
included to achieve a higher resolution Q model.

With NACT+F, we propose a two-step iterative waveform inversion procedure for a next generation of Q model. In the first step, 3D elastic models are inverted with the radial anisotropy and focusing effects taken into account. In the second step, anelastic 3D models are inverted, and 3D elastic models from step one are used to correct the phase shift and focusing effects due to elastic anomalies prior to anelastic tomography.

Capdeville Y., E. Chaljub, J.P. Vilotte and J.P. Montagner, Coupling the spectral element
method with a modal solution for elastic wave propagation in global Earth models, *Geophys. J. Int., 152*, 34-66, 2003.

Li, X.-D. and B. Romanowicz, Comparison of global waveform inversions with and without considering cross-branch
modal coupling, *Geophys. J. Int., 121,* 695-709, 1995.

Romanowicz, B., Multiplet-multiplet coupling due to lateral heterogeneity: asymptotic effects on the amplitude and frequency of
Earth's normal modes, *Geophys. J. R. Astr. Soc., 90*, 75-100, 1987.

Romanowicz, B., Y. Gung and Y. Capdeville, Long period seismograms in a 3D earth: tests of normal mode asymptotic approximations
against computations using the Spectral Element Method, *in preparation*, 2004.

Selby, N.D. and J.H. Woodhouse, The Q structure of the upper mantle:
constraints from Rayleigh wave amplitudes, *J. Geophys. Res., 107*, 933-940, 2097, doi:10.1029/2001JB000257, 2002.

Woodhouse, J.H. and A.M. Dziewonski, Mapping the upper mantle: Three dimensional modeling of Earth's structure by inversion of
seismic waveforms, *J. Geophys. Res., 89*, 5,953-5,986, 1984.

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