Inversions in Astronomy and the Sola Method

  • Frank P. Pijpers
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 90)

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.

Keywords

Entropy Agated Convolution Tral Deconvolution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Backus, G.E., Gilbert, J.F., Geophysical Journal of the Royal Astronomical Society, 13, (1967) 247.CrossRefGoogle Scholar
  2. [2]
    Backus, G.E., Gilbert, J.F., Geophysical Journal of the Royal Astronomical Society, 16, (1968) 169.MATHCrossRefGoogle Scholar
  3. [3]
    Backus, G.E., Gilbert, J.F., Philosophical Transactions of the Royal Society of London A, 266, (1970) 123.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Binney, J.,de Vaucouleurs, G., Monthly Notices of the Royal Astronomical Society, 194, (1981) 679.Google Scholar
  5. [5]
    Blandford, R.D., McKee, CF., Astrophysical Journal, 255, (1982) 419.CrossRefGoogle Scholar
  6. [6]
    Brinks, E., Shane, W.W., Astronomy & Astrophysics Suppl. Str., 55, (1984) 179.Google Scholar
  7. [7]
    Brouw, W.N., in Methods in Computational Physics. Vol. 14—Radio astronomy, 131 (Academic Press, New York) (1975).Google Scholar
  8. [8]
    Brown, J.G., Inverse Problems, 11, 635 (see also the 10 subsequent papers in the same journal issue) (1995).Google Scholar
  9. [9]
    Christensen-Dalsgaard, J., Schou, J., Thompson, M.J., Monthly Notices of the Royal Astronomical Society, 242, (1990) 353.Google Scholar
  10. [10]
    Christiansen, W.N., Högbom, J.A., Radiotlescopes 2nd Ed., (Cambridge Univ. Press, Cambridge) (1985).Google Scholar
  11. [11]
    Cox, J.P., Theory of stellar pulsation, (Princeton Univ. Press, Princeton) (1980).Google Scholar
  12. [12]
    Craig, I.J.D., Brown, J.C., Inverse problems in Astronomy: a guide to inversion strategies for remotely sensed data, (Adam Hilger, Bristol & Boston) (1986). [13] Gough, D.O., Solar Physics, 100, 65Google Scholar
  13. Hörne, K., 1985, Monthly Notices of the Royal Astronomical Society, 213, (1985) 129.Google Scholar
  14. [14]
    Home, K., in Proc. Reverberation mapping of the Broad-Line Region of Active Galactic Nuclei, Astronomical Society of the Pacific Conference series, 69, (1994) 23.Google Scholar
  15. [15]
    Högbom, J.A., Astronomy & Astrophysics Supplements, 15, (1974) 417.Google Scholar
  16. [16]
    Lucy, L.B., Astronomical Journal, 79, (1974) 745.CrossRefGoogle Scholar
  17. [17]
    Lucy, L.B., Astronomical Journal, 104, (1992) 1260.CrossRefGoogle Scholar
  18. [18]
    Lucy, L.B., Astronomy & Astrophysics, 289, (1994) 983.Google Scholar
  19. [19]
    Marsh, T.R., Hörne, K., Monthly Notices of the Royal Astronomical Society, 235, (1988) 269.Google Scholar
  20. [20]
    Narayan, R., Nityananda, R., Annual reviews of Astronomy & Astrophysics, 24, (1986) 127.CrossRefGoogle Scholar
  21. [21]
    Peterson, B.M., Publications of the Astronomical Society of the Pacific, 105, (1993) 247.CrossRefGoogle Scholar
  22. [22]
    Pijpers, F.P., in Proc. Reverberation mapping of the Broad-Line Region of Active Galactic Nuclei, Astronomical Society of the Pacific Conference series, 69, (1994) 69.Google Scholar
  23. [23]
    Pijpers, F.P., Thompson, M.J., Astronomy & Astrophysics, 262, (1992) L33.Google Scholar
  24. [24]
    Pijpers, F.P., Thompson, M. J., in Proc. GONG’ 92 Seismic investigation of the Sun and stars, Astronomical Society of the Pacific Conference series, 42, (1993a) 241.Google Scholar
  25. [25]
    Pijpers, F.P., Thompson, M.J., in Proc. GONG’ 92 Seismic investigation of the Sun and stars, Astronomical Society of the Pacific Conference series, 42, (1993b) 245.Google Scholar
  26. [26]
    Pijpers, F.P., Thompson, M.J., Astronomy & Astrophysics, 281, (1994) 231.Google Scholar
  27. [27]
    Pijpers, F.P., Wanders, L, Monthly Notices of the Royal Astronomical Society, 271, (1994) 183.Google Scholar
  28. [28]
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., Numerical Recipes: the art of scientific computing 2nd Ed., (1992) 530–602 779–817 (Cambridge Univ. Press, Cambridge).Google Scholar
  29. [29]
    Richardson, W.H., Journal of the Optical Society of America, 62, (1972) 55.CrossRefGoogle Scholar
  30. [30]
    Richichi, A., Salinari, P., Lisi, F., Astrophysical Journal, 326, (1988) 791.CrossRefGoogle Scholar
  31. [31]
    Roberts, D.H., Léhar, J., Dreher, J.W., Astronomical Journal, 93, (1987) 968.CrossRefGoogle Scholar
  32. [32]
    Rybicki, G.B., Press, W.H., Astrophysical Journal, 398, (1992) 169.CrossRefGoogle Scholar
  33. [33]
    Schou, J., Christensen-Dalsgaard, J., Thompson, M.J., Astrophysical Journal, 433, (1994) 389.CrossRefGoogle Scholar
  34. [34]
    Schwarz, U.J., Astronomy & Astrophysics, 65, (1978) 345.Google Scholar
  35. [35]
    Skilling, J., Bryan, R.K., Astrophysical Journal, 211, (1984) 111.MATHGoogle Scholar
  36. [36]
    Skilling, J., Gull, S.F., in Maximum-Entropy and Bayesian Methods in Inverse Problems eds. C.R. Smith, W.T. Grandy, Jr., (Reidel, Dordrecht) (1985).Google Scholar
  37. [37]
    Starck, J.-L., Bijaoui, A., Lopez, B., Perrier, C, Astronomy & Astrophysics, 283, (1994) 349.Google Scholar
  38. [38]
    Thompson, A.R., Moran, J.M., Swenson, G.W. Jr., Interferometry and Synthesis in Radio Astronomy (Wiley, New York) (1986).Google Scholar
  39. [39]
    Tsumuraya, F., Miura, N., Baba, N., Astronomy & Astrophysics, 282, (1994) 699.Google Scholar
  40. [40]
    Wakker, B.P., Schwarz, U.J., Astronomy & Astrophysics, 200, (1988) 312.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Frank P. Pijpers
    • 1
    • 2
  1. 1.Theoretical Astrophysics CenterInstitute of Physics and Astronomy, Aarhus UniversityNy MunkegadeDenmark
  2. 2.Uppsala Astronomical ObservatoryUppsalaSweden

Personalised recommendations