By Jan F. Eichner, Jan W. Kantelhardt, Armin Bunde, Shlomo Havlin (auth.), Jürgen Kropp, Hans-Joachim Schellnhuber (eds.)
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Additional resources for In Extremis: Disruptive Events and Trends in Climate and Hydrology
4 in a double logarithmic scale. 7) are also plotted. F. Eichner et al. 1 10 10–4 0 100 200 r 300 uncorr. −ln[Pq(r)/Pq(1)] Pq(r) RqPq(r) a) 10–3 0 2 4 6 8 0 –1 10 10 r/Rq 0 1 10 10 r/Rq 2 10 Fig. 4). (b) When plotting Pq (r ) multiplied with Rq versus r/Rq , the three curves collapse to a single stretched exponential curve (black curve). , an exponential curve (straight black line). 7 as well as for the shuffled data (from bottom to top). The symbols correspond to the quantiles q from (a) and (c).
F. Eichner et al. 8 1 γ Fig. 16 Conditional return periods for long-term correlated data with five values of γ and +/− fixed Rq = 50. (a) the single-conditional and the double-conditional return periods (Rq and ++/−− ) in units of Rq for Gaussian data. Rq+ (filled symbols, upper curve) and Rq− (filled symRq bols, lower curve) are the single-conditional return periods, where the predecessor r0 is larger or smaller than Rq . Rq++ (open symbols, upper curve) and Rq−− (open symbols, lower curve) are the double-conditional return periods where the preceding and the pre-preceding return intervals are both larger or both smaller than Rq (Rq++ = Rq (r |r0 > Rq |r−1 > Rq ), Rq−− = Rq (r |r0 < Rq |r−1 < Rq )).
8 compares Pq (r ) for simulated Gaussian-distributed data and three different correlation exponents with the five historical and reconstructed data sets introduced in Figs. 2. The solid lines, representing the theoretical curves with the measured γ values, match with Pq (r ) of the data (filled symbols). The dotted lines and the open symbols show the results for the shuffled data, when all correlations are destroyed. The shape of Pq (r ) becomes a simple exponential. 3 Power-Law Regime and Discretization Effects for Small Return Intervals The curves in Fig.