Inversions in Astronomy and the Sola Method

Abstract

A brief overview of applications of inversions within astronomy is presented here and also an inventory of the techniques commonly in use. Most of this paper is concerned with a presentation of a recent modification of the well-known Backus & Gilbert method (1967, 1968, 1970).

In general inversions in astronomy arise when observational (experimental) data are a convolution of some quantity of astrophysical interest and a known or measured effect. The latter can be a known property of the instrument used for the observation, an effect of projection on the sky or, as in helioseismology, a convolution along the ray path of a seismic wave in the Sun. Since the measured data is sampled discretely and suffers from measurement errors of various kinds, it is rare that an exact analytical inversion can be carried out. Furthermore what distinguishes astronomy from most other experimental physical sciences is that both the sampling and the data errors are difficult or impossible to control. A number of numerical inversion techniques are currently in use that try to deal with these difficulties in various ways. A particularly useful reference that describes the basics of many inversion and deconvolution techniques are two chapters in the book by Press et al. (1992). Almost all the techniques described there have been applied in astronomy in one form or another. Another useful although somewhat older reference is the book by Craig and Brown (1986) which discusses many techniques and applications of inversions in astronomy. A much more recent set of papers giving a good overview of the current use of techniques of inverse problems in astronomy can be found in an issue of the journal ‘Inverse Problems’ (cf. Brown, 1995).

The second part of this paper is focussed on a small selection of astronomical inversion problems, where the method of Subtractive Optimally Localized Averages (SOLA) has been used. The SOLA method is an adaptation of the Backus & Gilbert method (1967, 1968, 1970). It was originally developed for application in helioseismology (Pijpers & Thompson, 1992, 1993b, 1994) where the Backus & Gilbert method is computationally too slow. Apart from achieving a considerable speedup in the SOLA formulation, the strength of the method lies in that it provides a good a priori estimate of the error due to data error propagation and similarly a good a priori estimate of the achievable resolution. The latter property in particular turns out to be of importance in the problem of reverberation mapping of active galactic nuclei (Pijpers & Wanders, 1994). The freedom to choose a desired resolution within OLA is also particularly useful if the ‘known’ function under the integral sign is a measured quantity with associated measurement noise, as in reverberation mapping, because it allows a better control of the propagation of this measurement noise as well as of the usual measurement noise outside of the integral sign.