Imperfections in the crystalline structure of any mineral govern deformation and viscoelastic relaxation (anelasticity) at seismic frequencies. The study of different creep mechanisms and their mutual relevance at different P-T conditions (e.g., Frost and Ashby 1982) is useful in understanding the possible physical mechanisms that may also be responsible for seismic attenuation (i.e. at much higher frequencies). Recently, accurate experimental data of shear attenuation for mantle minerals at seismic frequencies (e.g., Jackson et al. 2002) are starting to provide a better understanding of such phenomena. A grain-boundary sliding mechanism seems compatible with laboratory experiments. Temperature and grain-size dependence for olivine polycrystalline samples have been accurately measured and modeled (Faul and Jackson, 2005). Pressure dependence, represented by activation volume, remains mostly unknown, however.

Within the Earth, viscoelastic relaxation causes dissipation and dispersion of seismic waves, or what is commonly referred to as intrinsic attenuation. Seismic attenuation mostly affects the amplitude of the waveforms. However, other effects related to the 3-D elastic structure of the Earth (focusing, scattering) and noise in the data make it difficult to retrieve information on the intrinsic attenuation structure of the Earth (see for a review, Romanowicz, 1998). Nevertheless, observations of attenuation of free oscillation and surface waves constrain the radial (1-D) attenuation profile of the Earth's upper mantle well enough , in spite of the well known discrepancy between the two datasets.

Here we use the modified Burgers model defined by Faul and Jackson (2005) to predict the quality factor (Q$_S$) for a range of simple thermal and grain-size structures for the shallow upper mantle (down to 400 km). We assume the QL6 (Durek and Ekström, 1996) attenuation profile below that depth. We computed the Q$_S$ values as a function of harmonic degree for fundamental and overtone spheroidal and toroidal modes, assuming a background reference velocity model. We found no distinguishable effects when testing two alternative velocity models (PREM Dziewonski and Anderson, 1980 and inverted PREF Cammarano and Romanowicz, 2007), indicating a weak sensitivity to the background velocity model used. We compared predicted values with seismic observations. Here, we show only comparisons with fundamental spheroidal modes (0S). We used five different compilations (see figure 2.58) based on attenuation of free oscillations and surface waves. Note that we do not consider here the discrepancy between the two types of observations. We defined a misfit function as

\end{displaymath} (29.1)

and we computed the total misfit for each given structure. Consistent with the frequency of surface waves, we used a reference period of 150s in the computations of Q$_S$ profiles with the Faul and Jackson model. The frequency dependence determined by their work is 0.27. Choosing a different period, within the band of surface waves, has a secondary effect on fitting observations compared to the unconstrained pressure dependence, as we shall discuss later.

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