Toward the Study Of the Earth Seismic Attenuation and the D" Layer with a Coupled Method of Spectral Elements and Modal
Solution.

Yann Capdeville and Barbara Romanowicz

Introduction

The new coupled method of spectral elements and modal solution for wave propagation in 3D global earth models (Capdeville et al., 2000; 2001; Chaljub et al., 2001) is well adapted to assess the study of Seismic attenuation in the crust and the D'' structure by forward modeling approach. To perform these studies, numerical developments, such as the possibility to have a layer of spectral elements between two modal solutions, are required and have been performed this year.

Research objectives

To provide a better understanding of long period seismic wave attenuation in the upper layers of the Earth by evaluating the respective contributions to observed attenuation due to intrinsic attenuation (anelasticity) and to scattering effects in the heterogeneous crust and uppermost mantle. To investigate the observed discrepancy between quality factor (Q) measured using a normal mode approach on the one hand, and a propagating surface wave approach on the other. To determine whether this discrepancy (on the order of 15% higher Q measured by modes compared to surface waves at 200 sec) may be due to the influence of scattering. To provide synthetic datasets in different synthetic Earth models to calibrate and analyze the method used to invert real data and produce 3D attenuation map of the Earth.

We also plan to study the 3D structure of the D" layer, the region of the Earth just above the Core Mantle Boundary (CMB), by performing seismic wave propagation simulations in realistic 3D D" models and comparing results with observed waveforms of seismic waves sensitive to this region of the earth (e.g. Sdiffracted). We hope to obtain new constraints on the character of 3D structure in D".

Principle of the method

The coupled method for wave propagation in global Earth models is based upon the coupling between the spectral element method and a modal solution method. The Earth is decomposed, depending on the problem addressed, in two or three spherical shells, one shell with 3D lateral heterogeneities and one or two shells with only spherically symmetric heterogeneities (see Figure 26.1).

Figure 26.1: Sandwich of spectral elements between two modal solutions. $\Gamma _1$ and $\Gamma _2$ are the two coupling interfaces.

Depending on the problem, the heterogeneous part can be mapped as the whole mantle, down to the core-mantle boundary, or restricted to the upper mantle or to the crust. In the 3D shell, the solution is sought in terms of a particular finite element method, the spectral element method, which is based on a high order variational formulation in space and a second-order explicit scheme in time. In the spherically symmetric domain, the solution is sought in terms of a modal solution in the frequency domain after expansion on the spherical harmonics basis. The spectral element method combines the geometrical flexibility of classical finite element method with the exponential convergence rate associated with spectral techniques. It avoids pole problems and allows local mesh refinement, using a non conforming discretization (see Figure 26.2),

Figure 26.2: Non conforming spectral element mesh used for the model PREM. The coupling is performed a the Core Mantle Boundary.

for the resolution of sharp variations and topographical features along interfaces. The modal solution is based on the spectral transform method and is expressed in the spherical harmonics basis allowing an accurate and isotropic representation in the spherically symmetric domain. The coupling between the spectral element method, formulated in the space and time domain, and the modal summation method, formulated in the frequency and wave-number domain, requires some original solution. Within the spectral element method, the coupling is introduced via a dynamic interface operator, a Dirichlet-to-Neumann (DtN) operator. This operator can be explicitly constructed in frequency and in generalized spherical harmonics. The back transformation in the space and time domain requires however special attention and an optimal asymptotic regularization. Such a coupling allows a significant speed-up in the simulation of the wavefield propagation, and the computation of synthetic seismograms in realistic earth models.

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

Accomplishments during 2000-2001

The accomplishments achieved during 2000-2001 include:

• The implementation of the code on the IBM SP of the NERSC.
• Technical and practical developments and optimizations of the code.
• The development of the "sandwich" coupling. Six month ago, the only configuration allowed was a spectral element outer shell coupled with a modal solution inner sphere. To study the D" at high frequency, a sandwich of spectral elements between two modal solutions is required. This implies to develop the coupling of an outer shell modal solution with a spectral element inner sphere, which has been performed with success this year.

Acknowledgements

Thanks to Dimitri Komatitsch who gave us his attenuation code for the spectral element method. The computation were made using the computational resources of the NERSC, especially the IBM SP, under repo mp342 starting project.

References

Chaljub, E. Capdeville, Y. Vilotte, J.P. and Y Maday, Solving elastodynamics in a solid heterogeneous 3D-sphere: a parallel spectral element approximation on non-conforming grids, In preparation for Journal for numerical methods in engineering, 2001

Capdeville Y., E. Chaljub, J.P. Vilotte and J.P. Montagner, Coupling Spectral Elements and Modal Solution: a New Efficient Tool for Numerical Wave Propagation in Laterally Heterogeneous Earth Models, Geophys. J. Int., submitted, 2001

Capdeville, Y., Méthode couplée éléments spectraux - solution modale pour la propagation d'ondes dans la Terre à l'échelle globale, Université Paris 7, PhD thesis, 2000.

Berkeley Seismological Laboratory, 202 McCone Hall, UC Berkeley, Berkeley, CA 94720-4760
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