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.

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

- 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.

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.

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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