HOW TO USE l=2, 3 SECTORAL SPLITTINGS?

As already quoted, GOLF data for l=2,3 are not a₁-coefficients but sectoral splittings. Therefore one may want to use the Equation 4 in order to infer the equatorial rotation rate. There are two difficulties with this approach:

1. As already mentioned in Section 2.1, the relation Equation 4 is not valid for low-degree l and therefore may not be suited for the determination of the core rotation. This may be corrected by assuming that the latitudinal dependence of the rotation rate (i.e. ) is known (taken from some previous 2D inversions for example). With this assumption, we can correct the observed sectoral splitting prior to inversion by adding for each mode the difference between the sectoral splittings computed from Equation 1 and from Equation 4. This difference is plotted on Figure 2 as a function of the degree l.
2. MDI data do not provide the sectoral splittings for high l but only upon 18 odd indexed a-coefficients. This number of coefficients is however high enough so that taking their sum as sectoral splittings is a good approximation. Nevertheless, the error on these sums (called `truncated sectoral splittings' in the following) is always higher than the error on a₁ alone.

Another possibility is to consider the sectoral splittings l=2,3 as a₁-coefficients at first approximation. In this approach we do not need to correct the data because Equation 6 is valid even for low degrees. Nevertheless, we can also use other dataset (MDI for example) or our knowledge of the latitudinal dependence of the rotation (taken from a previous 2D inversion for example) in order to estimate a₃ for modes l=2 and a₃, a₅ for modes l=3.

With the same approximation as in Equation 2 we can write for the a-coefficients:

so that, for both approaches, the solution obtained at r₀ can ever be seen as an average of the rotation of the form:

The 2D averaging kernels (defined by the term in parentheses in Equation 10) can therefore be estimated from the 1D averaging kernels Equation 8 obtained at a target location r₀ by adding the angular part of the 2D rotational kernel that corresponds to the data really inverted.

Figure 3 shows on the left the 2D averaging kernels (defined by the term in parentheses in Equation 10) obtained at by inverting GOLF sectoral splittings together with MDI `truncated sectoral' splittings i.e.:

whereas, on the right, it shows the 2D averaging kernel obtained at the same target location by inverting MDI a₁- coefficients together with GOLF sectoral splittings for l=1,2,3 i.e.:

Figure 3: 2D averaging kernels for 1D inversions computed at (see Section 4). Left panel: inversion of GOLF sectoral splittings for with MDI `truncated sectoral'(see text) splittings for l>3. Right panel: inversion of GOLF sectoral splittings for with MDI a₁-coefficients. The peaks near the surface are truncated on the plot for clarity.

The two corresponding 1D averaging kernels (cf. Figures. 6 and 5) are the integral over latitude of these 2D kernels and are very similar: well localized near and without contributions near the surface. The angular part of the averaging kernel for sectoral splittings strongly depend on the degree l and this leads to the oscillatory behaviour of the 2D kernel with very high peaks near the surface. From this plot it is clear that the interpretation of the result obtained by inverting sectoral splittings is possible only if we have already a good knowledge of the latitudinal dependence of the rotation. Furthermore, as already pointed out, this knowledge is needed in order to correct the low-degree sectoral splittings for which the 1D approximation is not valid. At the opposite, for coefficients is independent of l (cf. Equation 12). Therefore the surface oscillatory behaviour on the right panel of Figure 3 comes only from the use of sectoral splittings for l=2,3. In this case, the knowledge of the rotation profile is needed only near the surface in order to interpret the result.

We can summarize the results of this study in few points:

1. It is clear that it is better to use a₁-coefficients than sectoral splittings when we have them.
2. In the case of GOLF data, we have access only to sectoral splittings for l=2,3. If we want to use them, it seems more reasonable to try to correct them by doing some assumptions on and a₅ for these modes and to use the exact 1D integral rather than correcting all the modes in order to use the approximated 1D integral Equation 4.
3. In both approaches we can in principle obtain a result easy to interpret if the latitudinal variation of the rotation is assumed to be known exactly. But by inverting a₁ for l>3 together with sectoral splittings for l<3, we just have to make assumptions on the surface rotation.
4. Several ways can be followed for correcting the results obtained by using sectoral splittings in a₁ inversions. We can either
• take a guess rotation and integrating its surface part with the 2D kernel in order to correct the solution after the inversion or
• take a₃ and a₅ from other datasets or
• calculate these coefficients from the guess rotation and subtract them from sectoral splittings before the inversion.

The best is probably to compare the effects of all these corrections and to compare with the inversion of the corrected `truncated sectoral' splittings. In any case assumptions are needed on the latitudinal dependence of the rotation. As this dependence can not be known exactly it should be interesting to study in future works how these assumptions increase the uncertainties on the solution.

The next Section shows some preliminary results obtained with these different approaches for the use of the combined MDI and GOLF data.