Figure 4: SOLA inversion of MDI a1-coefficients and GOLF l=1,2
sectoral splittings. The corresponding 1D averaging kernels are shown below
each point of the solution. Vertical and horizontal error bars represent
respectively the error and the resolution defined as the FWHM of the averaging
kernel.
Figure 5: MOLA inversion of the same data as in Figure 4. Five
regularizing parameters have been used at each target location. The full
line shows an RLS inversion of the corrected `truncated sectoral splittings'
(see text) and the dotted line gives 1sigma errors on this solution.
In this preliminary work, we have chosen, as a first step, to use only
l=1,2 GOLF sectoral splittings in order to reduce the difficulties
discussed in the previous sections. Figures 4
and 5 show the results obtained by inverting
the a1-coefficients together with l=2 sectoral splittings by using
SOLA and MOLA inverse methods. In SOLA method the trade-off parameter is
rescaled at each target location to obtain more localized averaging kernels.
For MOLA method five trade-off parameters have been used at each target
location. In the case of low regularization MOLA averaging kernels have
some small oscillatory parts. The two methods fall in good agreement down
to 
where
the rotation rate increases up to 
nHz.
Below 
both
methods fail to peak kernels. On Figure 5
we have shown a RLS solution obtained by inverting sectoral splittings.
As expected the two solutions (
and 
) differ
in the convection zone where the rotation rate vary with latitude. In the
core, it is difficult to use RLS method because it is a global method and
if one try to obtain a well localized averaging kernel down to 
then we have to decrease the regularization and the solution becomes very
oscillating everywhere with big error bars. With an optimal L-curve choice
of the regularizing parameter the solution is constant (445nHz see
Figure 5) below 0.4R but averaging
kernels (not shown on the plot) computed below this point are still localized
near 0.4R so that, with this method, there is no conclusion on the
core rotation in terms of weighted average of the true rotation. Nevertheless,
we can use this solution and look at the residuals for each mode. The global
normalized 
of the inversion is 1.2. Now, if we look only at low-degree GOLF modes,
the `partial normalized 
'
is around 0.5 showing that within error bars GOLF data are in good agreement
with a constant rotation below 0.4R and that GOLF errors are probably
not underestimated. Furthermore, looking at the residuals for each individual
splittings can help in the signal analysis by pointing out some modes with
high residuals that may be reanalyzed in order to become more confident
on the result.
Figure 6: MOLA and RLS inversion of the corrected `truncated
sectoral splittings'. Low regularization has been used in MOLA inversion.
Following the discussion of the previous sections we have also inverted
the sectoral (or `truncated sectoral') splittings corrected by using the
latitudinal dependence of the rotation found by a 2D RLS inversion of MDI
data. The result is shown on Figure 6
in the case of low regularization. As expected the solution corresponds
to the equatorial rotation profile as found by the RLS method in the convection
zone and the error bars increase compared to the use of a1-coefficients
alone. The solution in the core is a little bit higher than found by a1-inversion
but remains compatible within error bars showing also an increasing rotation
rate below 
.
We have also tried to include GOLF l=3 sectoral splittings in
our inversions. This leads to a more important increase of the rotation
rate below 0.2R (around 
nHz). But in this case more work is needed in order to become more confident
in our result. In particular, in that case, we have to test the effects
of the various corrections suggested in Section 4
and to look at their influence on the estimation of the uncertainties on
the core rotation.
Figure 7: The same as Figure 5 but using GONG a1-coefficients
for l=1,2,3 instead of GOLF sectoral splittings.
Finally, we have done the same analysis with GONG low-degree data [Rabello-Soares
& Appourchaux1998]. In the case of GONG data we have a1-coefficients
so that, as quoted before, it may be more suited to carry an inversion
of these coefficients using Equation 6
rather than using the `truncated sectoral' splittings. In order to compare
the result with the previous ones, the GONG a1-coefficients for low-degrees
have been used together with the MDI a1-coefficients of higher degree modes.
Figure 7 shows that these data tend to
produce a slightly decreasing rotation rate below 
.
Therefore there is still a relatively important difference between the
solutions obtained with the different low-degrees data. These differences
are significant only if the error bars obtained on the solutions are not
underestimated
and may be related to the important dispersion of individual splittings measurements (cf. Figure 1). Furthermore, we must notice that whereas MDI and GOLF splittings are for the same year of observations(5/96-5/97), the GONG data are for the year before (5/95-5/96) and therefore the results may not be directly compared. Therefore this result needs to be confirmed in future works and we have also to inverse GONG a1-coefficients for all the modes which should be a more self-consistent dataset.