The inversion is based on a normal mode asymptotic coupling mechanism
(NACT, *Li and Romanowicz,* 1996). The seismic data used for the
inversion are long period fundamental and overtone spheroidal modes
selected on the vertical component of the seismograms included in the
existing collection. Including higher modes provide resolution in the
transition zone. No crustal correction has been used at this point.
The direct inversion of the seismic waveforms for temperature requires
lnV/lnT (anelastic effects included) as a function
of pressure (depth) and temperature and a starting thermal model as well.
We choose an adiabat with a potential temperature of 1300C overlaid by the
geotherm for 60 m.y. old oceanic lithosphere.

To test the results of the inversion and assess the best way to address
the non-linearities, we test our temperature inversion by performing in
parallel an inversion for a physical reference model. The
chosen model is one of the best-fit adiabatic pyrolitic models (PREF)
for traveltime and fundamental mode data from *Cammarano et al.*, 2005.
The models tested span the range of elastic properties for each mineral as
inferred from mineral physics and applying different anelasticity
models that cover the range of 1-D seismic attenuation models. Note
that the thermal structure is exactly the same. An example of how a
temperature slice may be obtained by inverting with respect to a
reference model is given in Figure 13.52

The sensitivity kernels of the seismogram with respect to velocity are translated into temperature by using the partial derivatives. The kernels should be recomputed after each iteration of the inversion because of the non linearity introduced by attenuation. However, knowing how the effect changes as a function of temperature and depth, we will try to correct the model after each iteration. In figure 13.53, we show at a given depth (300 km) how the kernels change around the thermal reference temperature. Note that the kernels (the derivatives of Figure 13.53) change both towards high and low temperature.

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