Masters Thesis, University of Bayreuth – BITÖK (2001)

Wiederkehranalyse und Langzeitphänomene süddeutscher Abflusszeitreihen

D. Haug

This thesis examines the runoff of rivers from Southern Germany of the last decades. The aim is to detect possible changes and to identify and quantify collective phenomenons or specifics of particular gauges. This is done using the quantitative recurrence analysis (Eckmann et al. 1987, Zbilut and Webber 1992), a nonlinear, local method for examining dynamical systems. Another aim is to study the applicability of this quite new method on runoff time series. As data basis, 73 runoff time series of different rivers from the German states of Bavaria and Baden-Wuerttemberg are used. The measurements in daily resolution cover a time period of 20 to 113 years. The main advantage of the quantitative recurrence analysis is its independence from constraints like stationarity, the type of distribution, the length of the data set and the amount of noise. This is of particular interest when analysing nonstationary, nonlinear runoff time series. Basis of the recurrence analysis is the recurrence plot, the distance matrix of the embedded scalar time series. Using the program rqa5.1 (Webber 1998), five so called RQA (recurrence quantification analysis) - characteristics are calculated (recurrence W, determinism D, line entropy H, longest diagonal line L and RP (recurrence plot) trend T). Examining the temporal evolution of these characteristics, information concerning the dynamics of the underlying processes and their behaviour in the state space can be derived. This information cannot be obtained using linear standard techniques. In this thesis, the parameters for the embedding (delay τ and embedding dimension m) and the recurrence quantification analysis (radius r, window length f and minimally required diagonal line length lmin) were chosen very carefully. The applicability of the finally chosen parameters for analysing runoff time series is approved by a sensitivity analysis. Furthermore, it demonstrates that the results of the recurrence quantification analysis reflect the actual properties of the data and not those of the parameters. For the 42 Bavarian gauges, it is not possible to evaluate any changes in the runoff behaviour. The respective indicators (determinism, determinism/recurrence and line entropy) do not reveal a uniform trend for the different data sets. Two collective properties in the temporal evolution of the RQA-characteristics are detected. For most of the gauges, the ratio determinism/recurrence (D/W) and the line entropy show a dominant peak in the middle of the 1960s. Furthermore, the RP-trend of all analysed time series exhibits a significantly uniform long-term structure. This is an important result, because runoff time series do not belong to the known indicators for long-term fluctuations. Principal component analysises are performed for each Bavarian gauge using the different RQA-characteristics as variables. They indicate the statistical independence of the RQA-characteristics. RP-trend and determinism explain most of the observed temporal patterns. To detect possible connections between the uniform long-term structures in the RP-trend and different climate related long-term indices, cross-correlations are calculated between the respective time series. Here, the combination RP-trend and Southern Oscillation Index is the only one that shows a interpretable part of significant cross correlations. This thesis highlights the big potential of the quantitative recurrence analysis in studying runoff time series. This is because so far unknown properties of their nonlinear dynamics can be revealed. More studies considering known processes and different real world data sets are necessary in order to increase the understanding of the RQA-characteristics dependence on the parameters (τ, m, r, f and lmin) and on the systems themselves.

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