A Short Bibliography of Structure Functions

“Structure functions” (SFs) are often used for the analysis of variability in astronomy. Today I spent some time tracking down some of the original references and I thought I’d record what I found for posterity.

It’s surprisingly difficult to find pedagogical references for SFs as used in the astronomical context. Lindsey and Chie below refer to them as “Kolmogorov structure functions,” and there’s a Wikipedia article on that topic, but it’s far more abstract than what astronomers would want: in fact, I’m pretty sure that it’s totally unrelated, but it’s so abstract that I can’t actually tell. The basic references define SFs in terms of “increments” on random processes but I can’t find a lot of explanatory material in that direction either. For pedagogical background, Lindsey and Chie below reference generic textbooks on random processes (e.g., “Theory of Stationary Random Functions,” A. M. Yaglom, Prentice-Hall, 1962).

One of the original references, if not the original reference, seems to be Kolmogorov (1941), in “Doklady Akademiia Nauk SSSR” 30 301, which is translated and available online in 1991 RSPSA 434 9. (This is the Kolmogorov turbulence paper.) Unsurprisingly it isn’t very helpful pedagogically.

The next step along one particular intellectual path is the apparent introduction of SFs into the engineering literature by Lindsey and Chie (1976) [DOI], with an overview in their Appendix I. A few years later Rutman (1978) [DOI] also presented SFs in a very similar context (in that paper’s Section VIII). I haven’t digested the papers but Rutman is much more highly-cited than Lindsey and Chie. Prokhorov et al (1975) [DOI] appear to have introduced SFs in the IEEE Proceedings first but don’t explain things as clearly.

It looks like SFs were then brought to astronomy by Rickett, Coles, & Bourgois (1984), though a close second are Simonetti, Cordes, & Heeschen (1985). The latter reference Lindsey & Chie and Rutman and have an appendix that gives a nice rundown of some of the basic useful properties of SFs in the same style as those. Rickett et al have a less helpful explanation.

After writing the first version of this bibliography, I found Emmanoulopoulos et al (2010), which gives a nice and thorough astronomical SF bibliography and also discusses caveats of SFs in astronomical usage. Another couple recent papers going into SF properties in various specialized astronomical circumstances are Hughes, Aller, & Aller (1992) and de Vries et al (2005). There are many more relevant recent papers out there than just these, though.