Using the Coupled Method of Spectral Elements and Modal Solution to study the D'' layer.

Yann Capdeville, Barbara Romanowicz, Akiko To

Introduction and Research objectives

The D" region, which encompasses the last 300 km or so of the deep mantle, is thought to be both a thermal and a chemical boundary and the site of vigorous dynamic processes. Its structure is believed to hold the key to many largely unanswered questions in deep Earth geodynamics, such as the ultimate fate of subducting slabs, the origin of hotspots, the amount of bottom heating driving global mantle circulation, electro-magnetic coupling between the core and the mantle, and the nature of chemical heterogeneity in the deep mantle. Strong lateral variations in D'' as well as the presence of topography on the fluid-solid core mantle boundary, has to be addressed by forward modeling approaches. However, present global waveform modeling can not handle wave propagation in this region of the Earth. Another difficulty is that the relevant frequency cutoff is high (0.1 Hz or higher) which leads to very high costs of numerical simulations. The coupled method of Spectral Elements and Modal Solution is very well adapted to such a problem by allowing to restrict the 3D model in the D" layer and to use a spherically symmetric model is enough in the rest of the Earth. Until now, this method is the only one able to address 3D wave propagation in this region at realistic frequencies. By producing synthetic seismograms using the coupled method in different D" layer models and confront them to real data, we hope to provide significant advances to our understanding of this region, especially using both amplitude and time shift informations of data.

Method

This work is based upon an extension to the coupling scheme of the Spectral Element Method (SEM) with a normal mode solution in spherical geometry. This extension allows us to consider a thin spherical shell of spectral elements between two modal solutions above and below. The SEM is based on a high order variational formulation in space and a second-order explicit scheme in time. It combines the geometrical flexibility of the classical finite element method with the exponential convergence rate associated with spectral techniques. In the inner sphere and outer shell, the solution is sought in terms of a modal solution in the frequency domain after expansion on the spherical harmonics basis. The SEM has been shown to obtain an excellent accuracy in solving the wave equation in complex media but is still numerically expensive for the whole Earth for high frequency simulations. On the other hand, modal solutions are well known and numerically cheap in spherically symmetric models. By combining these two methods we take advantage of both, allowing high frequency simulations in global Earth models with 3D structure in a limited depth range. Within the spectral element method, the coupling is introduced via a dynamic interface operator, a Dirichlet-to-Neumann (DtN) operator which can be explicitly constructed in the frequency and generalized spherical harmonics domain using modal solutions in the inner sphere and outer shell. The presence of the source and receivers in the top modal solution shell requires some special treatment. The accuracy of the method is checked against the mode summation method in simple spherically symmetric models and shows very good agreement for all type of waves, including diffracted waves traveling on the coupling boundary.

Simulations in a 3D D'' layer model based on the tomographic model SAW24B16 has been performed up to a corner frequency of 1/12 s The mesh used is presented on Fig. 37.1.

Figure 37.1: Sandwich of spectral elements between two modal solutions for D'' application with SAW24B16 model.

The comparison with data using deep events shows surprisingly good results for the 3D model even when the observed waveform amplitudes differ significantly from the ones predicted in the spherically symmetric reference model (PREM ). Some exemples of synthetics are given on Fig. 37.2.

Figure 37.2: Comparison between data (thin line), synthetics in PREM (dotted line) and synthetics in SAW24B16 + PREM (bold line) for an event in the Fidji Island recorded at different stations around the world.

A description of the method with illustrations can be found on http://seismo.berkeley.edu/~yann

Accomplishments during 2001-2002

The accomplishments achieved during 2000-2001 include:

• Technical and practical developments and optimizations of the code on the IBM SP of the NERSC.
• Comparison of synthetics seismograms in the tomographic model SAW24B16 and data for several deep events. These comparisons will lead to an improved model of the D'' layer.

Acknowledgements

The computation were made using the computational resources of the NERSC, especially the IBM SP, under repo mp342.

We thank the Miller Insitute for their support.

References

Y. Capdeville, B. Romanowicz and A. To, Coupling Spectral Elements and Modes in a spherical earth: an extension to the ``sandwich'' case. Submitted to Geosphys. J. Int., 2002.

Capdeville Y., C. Larmat, J.P. Vilotte and J.P. Montagner, Numerical simulation of the scattering induced by a localized plume-like anomaly using a coupled spectral element and modal solution method. In press in Geoph. Res. Lett., 2001.

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. accepted in Geophys. J. Int., 2001.

Chaljub, E. Capdeville, Y. and Vilotte, J.P., Solving elastodynamics in a solid heterogeneous 3D-sphere: a parallel spectral element approximation on non-conforming grids. Submitted to J. Comp Phy., 2001.

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