Astron. Astrophys. 324, 298-310 (1997)

3. The inversion method

The problem of inverting Eq. (2) is intrinsically an ill-posed problem because of its global (or integral) nature (e.g. Craig & Brown 1986). Furthermore this is strengthened in the helioseismic case because of the lack of modes able to sound the deepest and shallowest layers of the Sun: only a small percentage of the observed p-modes have their corresponding rotational kernels Eq. (3) with significant amplitude below or with their lower turning point between and the surface (see Fig. 1). Then the solution is not well constrained at these depths and the global nature of the problem implies that this leads to difficulties in the whole domain and that there is no unique solution for the problem.

Fig. 1. diagram showing the modes included in the LOWL 2 year dataset. Solid lines indicate the values of that correspond to modes with turning points =0.4, 0.85, 0.95 from the left to the right.

In order to discretize Eq. (2), we project the unknown rotation rate on a tensorial product of B-splines (see Appendix A):

Then, we apply a regularized least-squares method on values of both observed splittings and observed surface rotation in order to find the set of coefficients . The aim of regularization is to stabilize the inversion process by ruling out rapidly oscillating solutions which are physically unacceptable.

In our inversion we adopt a Tikhonov regularization method (Tikhonov & Arsenin 1977) by solving:

where is the least-squares term (see Appendix B for details) and is of the form:

with:

This can be regarded as a measure of smoothness of the rotation . The functions and are used to assign different weights to the smoothing terms for different positions r and µ. It should be noticed, however, that well chosen functions together with first derivatives () can lead to the definition of flatness given by Sekii (1991): . The choice of the so-called trade-off parameters and depends on the data from which we perform the inversion and is discussed in the next section.

Finally, let us define the which characterizes how the observed splittings are approached by the solution :

This value corresponds to the first term in the sum that defines Eq. (B2).

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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