## 3. The inversion methodThe 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.
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: 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 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 |