Long-range correlations and rare events in boundary layer wind fields

doi:10.1016/j.physa.2004.07.012

Physica A: Statistical Mechanics and its Applications

Volume 345, Issues 3–4, 15 January 2005, Pages 713-721

https://doi.org/10.1016/j.physa.2004.07.012 Get rights and content

Abstract

We analyse the statistics of extreme events in the boundary layer wind fields. We focus on the wind speeds that are long-term correlated and show that its distribution of the return intervals of rare events is a stretched exponential and the stretching exponent, to within the numerical errors, is identical to the long-term correlation exponent $γ$ . We also analyse correlations among the return intervals which are themselves long-term correlated with an exponent close to $γ$ . This work complements the results obtained by Bunde et al. (Physica A 330 (2003) 1).

Introduction

Extreme value statistics is the study of rare events in a statistical setting [1]. For instance, events like gusts in wind fields, extreme precipitation sometimes leading to floods or historical highs and lows attained in stock market indices are amenable for study within the ambit of extreme value statistics. Thus, in general, if we have N recorded values of events at discrete time intervals, say, $y_{i}, i = 1, 2, \dots, N$ , then the rare events would be defined as those that exceed a given threshold q, i.e., $y_{i} > q$ . In this sense, the rare events, for a suitably chosen threshold q, are associated with the tail of the distribution $f_{q} (y - 〈 y 〉)$ , where $〈 y 〉$ is the sample mean. In principle, ultimately we would be interested in obtaining predictions for the probability of the occurrence of rare events. An important pre-requisite step is the necessity to characterise these rare events, particularly addressing questions like how the time intervals between the occurrence of rare events are distributed and the correlations among them.

In recent years, it has been shown that many physical phenomena generating weather records, heart rhythms, DNA sequences or economic data are all long-term correlated. This is done using principally the method of detrended fluctuation analysis (DFA) [2]. This property of long-term correlatedness leads to slower than exponential fall off in the (auto) correlation function for the given data, typically a power law. If $y_{i}$ are long-term correlated with zero mean, then its autocorrelation is given by $C_{y} (s) = \frac{1}{N - s} \sum_{i = 1}^{N - s} y_{i} y_{i + s} .$ That goes as $C_{y} (s) \sim s^{- γ}$ , where $γ$ is the long-term exponent and $0 < γ < 1$ . The DFA technique identifies an exponent $α$ that represents the decay, $s^{- α}$ , of the detrended fluctuations with the change in size s of the integrated data window which is subjected to detrending. The long-term correlations are related to DFA through the relation between their exponents, $α = 1 - γ / 2$ [3]. Does the presence of long-term correlations affect extreme value statistics? This question has been answered in the affirmative by the work of Bunde et al. [4]. Let r denote the time interval between two successive occurrences of extreme values defined with respect to a threshold q. If the data is uncorrelated then the return intervals are also uncorrelated and are Poisson distributed as $P_{q} (r) = (\frac{1}{〈 r 〉}) \exp (\frac{- r}{〈 r 〉}),$ where $〈 r 〉$ is the mean return interval. This implies that the successive rare events are more likely to be clustered together rather than take place in an isolated fashion at random intervals. The principal result in Ref. [4] is that the distribution of the return intervals of long-term correlated records is a stretched exponential with a stretching exponent that is identical to the long-term correlation exponent $γ$ , i.e., $P_{q} (r) \sim \exp [- {(\frac{r}{〈 r 〉})}^{γ}] .$ It is also shown that the return intervals are long-term correlated. This empirical result has been obtained based on simulations using long-term correlated data. In this work, we perform extreme value analysis with experimentally obtained wind data and show that the boundary layer wind speeds are characterised by the value of long-term exponent $γ \sim 0.15$ –0.2, which is independent of the threshold q. The distribution of return intervals is a stretched exponential and follows the prediction in Ref. [4].

Section snippets

Return interval distribution

In this work, we analyse the atmospheric boundary layer wind speed measured with a cup anemometer at a height of about 10 m from the ground with a sampling rate of 8 Hz. The experiment was performed in the island of Zeeland in Denmark during June 1987. The experimental terrain was featureless and data were recorded for several days. These data have been previously studied and reported in several contexts including modelling based on stochastic approaches [5] and its long-term correlation

Long-term exponent

In this section we present empirical results to show that the long-term correlation exponent for boundary layer wind is $\approx$ 0.2. Following Ref. [4] we expect the return interval distribution for the wind data to be described by a stretched exponential given by $P_{q} (r) \sim N_{0} \exp [- {(\frac{r}{〈 r 〉})}^{γ}],$ where $N_{0}$ is the normalisation constant. It is important to note that $γ$ is identical to the one appearing in the correlation function in Eq. (1). Often, it is difficult to determine $γ$ from Eq. (1) and for our present

Correlations in return intervals

In this section we study the correlations among the return intervals using detrended fluctuation analysis since direct estimate of correlations is unlikely to be useful due to non-stationarities in the data. The broad outline of DFA goes as follows : the integrated time-series profile is subdivided into N subsequences each of length s. Data in each subsequence is regressed against a polynomial of order k to give DFA $(k)$ . The rms deviation of the profile from the best-fit polynomial gives the

Discussion and conclusion

We have analysed the boundary layer wind fields for the statistics of its extreme values. For the wind speed data, the distribution of extreme values is a stretched exponential with an exponent $γ \sim 0.15$ –0.2 and this is identical to the long-term correlation exponent. The return intervals themselves are also long-term correlated whose exponent is numerically close to $γ$ identified above. Based on the empirical evidence from several cases of boundary layer wind fields studied in this work, we

Acknowledgements

One of us (MSS) thanks Dr. R.B. Govindan for useful discussions on DFA methods.

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Long-range correlations and rare events in boundary layer wind fields

Abstract

Introduction

Section snippets

Return interval distribution

Long-term exponent

Correlations in return intervals

Discussion and conclusion

Acknowledgements

Boundary-Layer Meteorol.

Phys. Rev. Lett.

Physica A

Physica A

Phys. Rev. E

Phys. Rev. Lett.

Phys. Rev. Lett.

Physica A

Physica A