GENERALIZED SINGULAR-VALUE DECOMPOSITION ANALYSIS OF HELIOSEISMIC INVERSIONS

We present the generalized singular value decomposition (GSVD) as a po werful tool for the analysis of ill-conditioned inversion problems, su ch as arise in helioseismology and other areas of astronomy. The GSVD provides rather natural bases with which to express the various proper ties of the inversion, in particular the solution, the inversion coeff icients (which relate the solution to the data) and the averaging kern els (which relate the solution to the underlying unknown function). Th ese facilitate an analysis of resolution and error properties. We appl y the theory to a helioseismic example and find inter alia that, for o ur prototypical problem, there is an effective limit to the informatio n content of the mode set in the presence of reasonable data errors wh ich is much less than might have been expected from the number of mode s. We show that various popular inversion methods can be analysed usin g the tools developed here. We find in our helioseismic application th at the methods make very similar use of the data.