A typical one-dimensional (radial) inversion problem, is given by the
following integral relation

where *d*_{i} 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 *K*_{i}(*r*), called kernels, are known functions that
give the contribution of
each radial point to the value of the data *d*_{i}.
In helioseismic inversion problems, *d*_{i} 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 *d*_{i} 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 *d*_{i} 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 *r*_{0} of the mesh, a combination of the kernels
(with coefficients *q*_{i}, ), called the averaged kernel
, that mimics a -function around *r*_{0},
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 *f*_{1}(*r*) and *f*_{2}(*r*)
(for instance sound speed and density), and hence two
trade-off parameters, this method is particularly problematical.

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