Tests of Normal Mode Asymptotic Approximations against
Computation Using the Spectral Element Method

Yuancheng Gung, Barbara Romanowicz and Yann Capdeville

Introduction

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.

Experiments and Results

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. ${\bf u}_{\mbox{3D model}} - {\bf u}_{\mbox{reference   model}}$) 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.

Figure 30.1: Top map: the geometrical relation of the isotropic source (black dark circle), the surface receiver (triangle) and lateral variations of hypothetic 3-D model, a slow velocity anomaly of maximum amplitude -5% at 220 km depth. Top trace: CSEM synthetics for the reference model (CSEM1D) and for the 3-D model (CSEM). In the bottom panels, the differential waveforms of CSEM (dashed line) and three normal mode based asymptotic approaches (solid line) are compared. Note that in this particular case, the great circle path just grazes the anomaly.
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Figure 30.2: The geometrical relation of the dip-slip source, the surface receiver and lateral variations of the hypothetic 3D model at 150 km depth. The 3D model consists of two equal anomalies of opposite signs (fast and slow), with maximum velocity contrast of 5%. In the bottom panels, the differential waveforms of CSEM and three normal mode based asymptotic approaches are compared. Note that in this case, the great circle path passes the surface center of two anomalies.
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\epsfig{file=gung04_1_2.eps, width=8.5cm}\end{center}\end{figure}

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.

Future Work

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.

References

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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