Generalized singular value decomposition analysis of helioseismic inversions

We present the generalized singular value decomposition (GSVD) as a powerful tool for the analysis of ill-conditioned inversion problems, such as arise in helioseismology and other areas of astronomy. The GSVD provides rather natural bases with which to express the various properties of the inversion, in particular the solution, the inversion coefficients (which relate the solution to the data) and the averaging kernels (which relate the solution to the underlying unknown function). These facilitate an analysis of resolution and error properties. We apply the theory to a helioseismic example and find inter alia that, for our prototypical problem, there is an effective limit to the information content of the mode set in the presence of reasonable data errors which is much less than might have been expected from the number of modes. We show that various popular inversion methods can be analysed using the tools developed here. We find in our helioseismic application that the methods make very similar use of the data.