In order to highlight the characteristics of the trade-off between grain size and temperature, we show in figure 2.57 (top panel), the misfit values for isothermal structure from 100 to 400km for various constant grain sizes. We found that Q$_S$ observations are much more sensitive to T than grain size, as shown by the contour lines in figure 2.57. Note that the $\lq\lq $cold$''$ structure of the lithospheric part does not affect measurements significantly. Indeed, we test that a negligible variation of the misfit pattern is obtained when using a standard 60My old oceanic geotherm for all thermal structures in the first 80km, plus a linear gradient in the 20km below to join the isotherms. The average temperature ($<T>$) of the upper mantle is very well constrained by seismic observations for a given grain size. For example, we found that a 1 mm grain size requires $<T> \sim$1500K, while a higher temperature ($\sim$1600K) is required around 1cm and slightly increases with coarser grain sizes. Subsequently, by giving a reference temperature at 100km of 1600K, more or less consistent with the temperature expected from a 60my old oceanic geotherm at that depth, we tested linear temperature gradients down to 400km, from -1.5 $^o/km$ to 1.5 $^o/km$, for various grain sizes (middle panel of figure 2.57). Again, we found that observations are able to discriminate between different thermal gradients with depth at given grain sizes. In general, positive gradients are required at grain sizes $>1$cm, while negative ones are preferred for millimeter grain-sizes. There is an obvious trade-off, here not shown, between the Tref(100km) and the gradients below. A reference temperature of 1700K at 100km will be more compatible with positive thermal gradients. In particular, we found that an adiabatic 1300$^oC$ temperature is compatible with observations at 1cm constant grain size.

Figure 2.57: Misfit values to 0S attenuation measurements for isothermal, constant grain-size upper mantle structure (right panel) and for linear thermal gradients from 100 to 400km (left panel). For left panel, reference temperature at 100km is 1600K. Q$_S$ values are computed at period of 150s
\epsfig{file=fabio07_1_1.eps, width=5cm}\end{center}\end{figure}

In figure 2.58, we give an example of Q$_S$ depth profiles (top panel) and their predicted attenuation as a function of harmonic degree compared to 0S measurements (bottom panel) for one best-fit model (grain size=2.5cm, dT/dz(K/km)=0.2, total misfit to 0S attenuation measurements equal to 0.06) and one poorly fitting model (GS=2.5cm and dT/dz=0.8, misfit = 0.32). Note that because of the discrepancy between seismic observations, we are able to reach only minimum values of misfit equal to 0.06.

Figure 2.58: Examples of a good (solid red line) and a bad fit model (dashed).q is 1000/Q$_S$. See text for details
\epsfig{file=fabio07_1_2.eps, width=5cm}\end{center}\end{figure}

In general, Q models based on T and GS structures do not have the first-order jump that characterizes seismic models around 220 km depth. Nevertheless, the fit to attenuation measurements can resolve between our different models. This preliminary result seems to point out that the constant increase in $Q_S$ due to change in T or GS is sufficient to first order. Large uncertainties exist on the activation volume (V*). We tested the extreme values provided by Faul and Jackson, i.e. $0.6$ and $2 \times 10^{-5}$, compared to $1.2 \times 10^{-5}$ used in the model. Low values of V* require negative thermal gradients, if Tref at 100km is set to 1600K. Conversely, very positive thermal gradients with depth are consistent with a high V*. We anticipate that, in spite of the limitation imposed by V* to interpret Q seismic observations, when P,T, GS dependent models are used together with elastic data at high P and T of mantle minerals for the interpretation of seismic data, we will be able to provide more constraints on the temperature and composition structure of the upper mantle (see Cammarano and Romanowicz 2007 and 2.26..

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