NG41B-1545: "Bunched Black Swans" in Complex Geosystems: Cross-Disciplinary Approaches to the Additive and Multiplicative Modelling of Correlated Extreme Bursts

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Authors: Nicholas W Watkins1, 2, Martin Rypdal3, Ola Lovsletten3

Author Institutions: 1. British Antarctic Survey (NERC), Cambridge, United Kingdom; 2. Centre for Fusion Space and Astrophysics, University of Warwick, Coventry, United Kingdom; 3. University of Tromso, Tromso, Norway

For all natural hazards, the question of when the next “extreme event”ù (c.f. Taleb’s “black swans”ù) is expected is of obvious importance. In the environmental sciences users often frame such questions in terms of average “return periods”ù, e.g. “is an X meter rise in the Thames water level a 1-in-Y year event ?”ù. Frequently, however, we also care about the emergence of correlation, and whether the probability of several big events occurring in close succession is truly independent, i.e. are the black swans “bunched”ù. A “big event”ù, or a “burst”ù, defined by its integrated signal above a threshold, might be a single, very large, event, or, instead, could in fact be a correlated series of “smaller”ù (i.e. less wildly fluctuating) events. Several available stochastic approaches provide quantitative information about such bursts, including Extreme Value Theory (EVT); the theory of records; level sets; sojourn times; and models of space-time “avalanches”ù of activity in non-equilibrium systems. Some focus more on the probability of single large events. Others are more concerned with extended dwell times above a given spatiotemporal threshold: However, the state of the art is not yet fully integrated, and the above-mentioned approaches differ in fundamental aspects. EVT is perhaps the best known in the geosciences. It is concerned with the distribution obeyed by the extremes of datasets, e.g. the 100 values obtained by considering the largest daily temperature recorded in each of the years of a century. However, the pioneering work from the 1920s on which EVT originally built was based on independent identically distributed samples, and took no account of memory and correlation that characterise many natural hazard time series. Ignoring this would fundamentally limit our ability to forecast; so much subsequent activity has been devoted to extending EVT to encompass dependence. A second group of approaches, by contrast, has notions of time and thus possible non-stationarity explicitly built in. In record breaking statistics, a record is defined in the sense used in everyday language, to be the largest value yet recorded in a time series, for example, the 2004 Sumatran Boxing Day earthquake was at the time the largest to be digitally recorded. The third group of approaches (e.g. avalanches) are explicitly spatiotemporal and so also include spatial structure. This presentation will discuss two examples of our recent work on the burst problem. We will show numerical results extending the preliminary results presented in [Watkins et al, PRE, 2009] using a standard additive model, linear fractional stable motion (LFSM). LFSM explicitly includes both heavy tails and long range dependence, allowing us to study how these 2 effects compete in determining the burst duration and size exponent probability distributions. We will contrast these simulations with new analytical studies of bursts in a multiplicative process, the multifractal random walk (MRW). We will present an analytical derivation for the scaling of the burst durations and make a preliminary comparison with data from the AE index from solar-terrestrial physics. We believe our result is more generally applicable than the MRW model, and that it applies to a broad class of multifractal processes.

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