A typical one-dimensional (radial) inversion problem, is given by the
following integral relation
where di are the data available with error . The function f(r), where r is the independent spatial variable, is the one we want to obtain by inverting relation (1 (click here)). The basis functions Ki(r), called kernels, are known functions that give the contribution of each radial point to the value of the data di. In helioseismic inversion problems, di could be the frequency differences between a solar model and the actual Sun, then f(r) is related to the difference in some structural parameter between the Sun and the solar model. On the other hand, if di are rotational splittings, or any combination of them, then f(r) gives information about the solar rotational rate.
Inversion of relation (1 (click here)) is an ill-posed problem (Thompson 1995), hence the solution is unstable and shows undesired high-frequency oscillations that must be avoided. This problem has been solved by several methods that can be classified in two different techniques: Regularized Least Squares (RLS, Craig & Brown 1986) and Optimal Localized Averages (OLA, Backus & Gilbert 1968, 1970). Both give the solution as a linear combination of the data, but in a different way.
RLS requires the discretization of the integral relation to be inverted.
In our case, Eq. (1 (click here)) is transformed into a matrix relation
where D is the data vector, with elements di and dimension M, f is the solution vector to be determined at N tabular points, A is the matrix with the kernels, of dimension , and is the vector containing the errors in D.
The RLS solution is the one that minimizes the quadratic difference
, with a constraint given by a smoothing matrix, H,
introduced in order to avoid the instabilities in the solution.
The general relation to be minimized is
where is a escalar introduced to give a suitable weight to the constraint matrix H in the solution. Hence, the function f is approximated by
while the error is calculated as
In general, the weighting factor , as well as the form of the smoothing condition H must be choosen before the inversion is done, without an a priori knowledge of the behaviour of the solution. Something similar happens with the number and distribution of points in the inversion mesh. These must be fixed in a previous step in the inversion procedure, without any information of how optimal such a mesh is. Therefore, there is some degree of arbitrariness in the procedure that can lead to significantly different solutions.
The OLA technique is more sophisticated and tries to obtain,
at any point r0 of the mesh, a combination of the kernels
(with coefficients qi, ), called the averaged kernel
, that mimics a -function around r0,
but with a moderate error propagation. With such an averaged kernel,
it is possible to calculate the solution as
It is necessary to include a parameter (analogous to the weight of the smoothing function in RLS) that is a trade-off between the "Dirac delta behaviour'' of and its error propagation. The main problem with this method is in choosing the trade-off parameter. When dealing with two functions f1(r) and f2(r) (for instance sound speed and density), and hence two trade-off parameters, this method is particularly problematical.