Some experiments in extrapolating magnetic-storm intensities to extreme 100-year values

Abstract:

We examine the statistical extrapolation of historical Dst data values (limited in number, only acquired since 1957) to estimate the intensity of magnetic storms that might be expected to occur, on average, once-per-century (100-year). Such extrapolations require a statistical model; we argue, on physical grounds, that a log-normal model is plausible. We fit log-normal functions to the Dst data using a maximum-likelihood algorithm. We measure the statistical significance of the log-normal hypothesis with a Kolmogorov-Sminov test. We estimate extrapolation errors using a bootstrap procedure. In light of these results, we examine the suitability of using the standard, but more abstractly mathematical, extrapolation framework established under the Fisher-Tippett-Gnedenko theorem – this framework is often invoked in extreme-value analysis (usually, with little physical motivation). Under this theorem, the extreme-event tails of a wide range of statistical distributions can be approximated by one of three different canonical functions (Gumbel, Fréchet, or Weibull), each having very different asymptotic properties; alternatively, one can use a generalization that encompasses the asymptotic properties of all three of these functions. We fit the extreme-value functions to the Dst data, we test their significance, and estimate extrapolation errors. We then compare extreme-value results with those obtained using the log-normal model. We discuss the suitability of using extreme-value-analysis methods in situations when physics-based (or physically motivated) models are already available.