INVERSE METHODS

  We have used two kinds of inverse methods for solving the 1D integral equations.

1. A Regularized Least-Squares (RLS) method with Tikhonov regularization (see e.g. [Corbard1997]). This is a global method which gives a solution at all depths which fits the data at the best in the least square sense. This is a linear method and then the value of the rotation obtained at any radius r₀is a linear combination of the data:



By replacing in Equation 4 or 6 we obtain:

 

The function in parenthesis is called 1D averaging kernel at r₀. The result obtained at r₀ will be easier to interpret as a local average of the rotation when this kernel is well-peaked and without strong oscillatory behaviour.

2. Two `local' methods which search directly the coefficients which are able to peak the averaging kernel near r₀. The two methods differ essentially in the way to localize the averaging kernel. The SOLA (Subtractive Optimally Localized Average) method [Pijpers & Thompson1992] fits the averaging kernel to a Gaussian function of given width whereas the MOLA method (Multiplicative OLA) [Backus & Gilbert1970] simply gives high weights in the minimization process to the part of the averaging kernel which are far from the target radius. In both case we use a regularizing parameter in order to establish a balance between the resolution and the error magnification reached at the target r₀.