Applying the Spectral Element Method to Model 3D Attenuation in the Upper Mantle

Vedran Lekic, Barbara Romanowicz and Yann Capdeville (IPGP)

Introduction

Observations of surface waves and overtones provide a unique opportunity for probing the physical state and dynamics of the upper mantle, due to their global coverage and sensitivity to structure at depths approaching 1000 km. Earth structure affects seismic waveforms through elastic effects of (de)focusing and scattering, as well as anelastic effects that result in amplitude decay and velocity dispersion. Because anelastic processes appear to be thermally activated, temperature has a stronger effect on seismic attenuation than does chemical heterogeneity. This is not the case with seismic velocity, which can strongly depend on both chemical composition and temperature. These differences - combined with a three-dimensional model of both seismic velocity and attenuation - translate into a unique opportunity to separate the effects of temperature variation from those of compositional heterogeneity within the mantle.

Past Work and Challenges

Normal mode summation methods coupled with approximate first-order perturbation techniques have been successfully applied to modeling of surface waves. Recent high-resolution global tomographic models based on these techniques can resolve regions of fast and slow seismic velocities as small as a few hundred kilometers across (e.g. Mégnin and Romanowicz, 2000). Yet, modeling of the 3D distribution of seismic attenuation has lagged behind due to difficulties in separating the effects on seismic waveforms of focusing and scattering of seismic energy in the heterogeneous earth from those due to attenuation intrinsic to the medium. Even when the elastic structure is known perfectly, first-order perturbation techniques are inadequate in modeling wave propagation near the source or receiver, near nodes of the radiation pattern, in locations of strong heterogeneity, and at times long after the source time. Spectral element methods, on the other hand, allow accurate modeling of seismic wave propagation, including the effects of (de)focusing of energy and multiple scattering.

Extracting structural information from seismic waveforms requires calculating the sensitivity kernels of the waveform to parameters like elastic wave velocity and attenuation at each point in the earth. The conventional way of computing sensitivity kernels, called the path average approximation (PAVA), assumes that the wave is only sensitive to structure along the great circle path joining the source with the receiver, and that this sensitivity is only a function of depth. Higher-order asymptotic approaches, such as non-linear asymptotic coupling theory (NACT: Li and Romanowicz, 1995), are capable of more accurately modeling the wave's actual sensitivity within the plane defined by the great circle path. Yet, long period surface waves observed at teleseismic distances are sensitive to structure both along and off the great circle path. While the (de)focusing effects that arise from velocity gradients transverse to the great circle path can be incorporated into the NACT formalism (NACT+F: Gung and Romanowicz, 2004), it is also now within reach to use more exact formalisms - such as the full Born approximation - in order to more accurately calculate sensitivity kernels.

Method

We propose a hybrid approach to tomography, in which we calculate the propagation of seismic waves through an arbitrary 3D medium exactly, using the coupled Spectral Element Method (cSEM: Capdeville et al., 2003), and compute the sensitivity kernels approximately, using perturbation theory. This approach allows us to iteratively converge on the correct model as long as the sign of the sensitivity kernels is correct. We apply this approach to 3-component fundamental and overtone waveforms, low-passed at 60 sec, and recorded at more than 100 stations of the IRIS/GSN, GEOSCOPE, GEOFON, and various regional broadband networks. Because of the increased computational costs associated with using cSEM, we at first restrict our focus to $\sim $70 events. We ensure excellent global coverage by including major arc surface waves, which provide complimentary sensitivity to minor arc paths. Furthermore, we select deep focus events in all regions where deep seismicity is present. Figure 28.1 shows the events used in our study, color-coded by centroid depth. Resolution tests indicate that our data coverage is sufficient for resolving structures $\sim $900 km at the surface, and that vertical smearing is limited.

Figure 28.1: Events used in our study color coded by centroid depth.
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Preliminary Results and Future Work

We have modified cSEM to include the effects of ellipticity, topography/bathymetry, radial anisotropy, 3D attenuation, and lateral variations in crustal velocities and depth to the Moho discontinuity. Because their sensitivities are concentrated near the surface, fundamental mode waves are especially strongly affected by 3D Moho topography; in contrast, crustal velocities play a secondary role. For long continental paths, waveforms corrected for crustal structure display phase delays of 100 sec. Interestingly, first order perturbation theory underpredicts by as much as 50 percent the cSEM phase delays over continents. Along oceanic paths, the differences between cSEM and first order perturbation theory are negligible. Figure 28.2 shows synthetic waveforms calculated by cSEM and NACT for typical long continental and oceanic paths.

We adopt an iterative waveform inversion approach, in which we solve for elastic and anelastic structure in successive steps. As a starting model, we adopt a high-resolution 3D elastic model of Panning and Romanowicz (2006), and a one-dimensional attenuation model constructed from separate upper mantle (QL6: Durek and Ekstrom, 1996) and lower mantle (PREM: Dziewonski and Anderson, 1981) models. Following our first iteration for elastic structure, which is performed in order to render compatible the starting model with our new dataset, we shall proceed to invert for source parameters. Because rotations of the radiation pattern can result in large amplitude changes - especially near the radiation nodes - inverting for the source mechanism is crucial to developing high resolution models of mantle attenuation.

Conclusion

We have compiled a waveform dataset of fundamental mode surface waves and overtones with good global lateral and depth coverage. We have modified cSEM to make it capable of accurately modeling wave propagation in an elliptic earth with surface topography / bathymetry, oceans, and 3D crustal structure. We find that first order perturbation theory inadequately predicts the effects of crustal structure on waveforms for long continental paths. Because of the magnitude of this deficiency, we expect our use of cSEM to allow us to model more accurately mantle structure below continents. Furthermore, the use of cSEM will allow us to better model amplitudes of seismic waveforms, and therefore to separate the often elusive signal of attenuation from that due to elastic structure.

Figure 28.2: Synthetic seismograms from mode summation (blue) and cSEM (red) for a long continental (top) and oceanic (bottom) path.
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Acknowledgements

Support for V. Lekic was provided by the Berkeley Fellowship.

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.

Durek, J.J. and G. Ekström, A radial model of anelasticity consistent with long-period surface-wave attenuation. BSSA, 86, 144-158, 1996.

Dziewonski, A.M. and D.L. Anderson, Preliminary reference Earth model. Phys. Earth and Planetary Int., 25, 297-356, 1981.

Gung, Y. and B. Romanowicz., Q tomography of the upper mantle using three-component long-period waveforms. Geophys. J. Int., 157, 813-830, 2004.

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.

Megnin, C. and B. Romanowicz, The three-dimensional shear velocity structure of the mantle from the inversion of body, surface and higher-mode waveforms. Geophys. J. Int., 143, 709-728, 2000.

Panning, M. and B. Romanowicz, A three-dimensional radially anistropic model of shear velocity in the mantle. Geophys. J. Int., in press, 2006.

Figure 28.3: Horizontal slices through the isotropic (a), radial (b) and azimuthal (c) anisotropic portion of the model. Anomalies are with respect to average in (a,b). Red arrows in (c) indicate the absolute plate motion direction in a hotspot reference frame.
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