Eva Koscielny-Bunde

Multifractal detrended fluctuation analysis of nonstationary time series

We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent.

Detecting long-range correlations with detrended fluctuation analysis

We examine the detrended fluctuation analysis (DFA), which is a well-established method for the detection of long-range correlations in time series. We show that deviations from scaling which appear at small time scales become stronger in higher orders of DFA, and suggest a modified DFA method to remove them. The improvement is necessary especially for short records that are affected by non-stationarities. Furthermore, we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of DFA.

Power-law persistence and trends in the atmosphere: A detailed study of long temperature records

We use several variants of the detrended fluctuation analysis to study the appearance of long-term persistence in temperature records, obtained at 95 stations all over the globe. Our results basically confirm earlier studies. We find that the persistence, characterized by the correlation C(s) of temperature variations separated by s days, decays for large s as a power law, C(s) approximately s(-gamma). For continental stations, including stations along the coastlines, we find that gamma is always close to 0.7. For stations on islands, we find that gamma ranges between 0.3 and 0.7, with a maximum at gamma=0.4. This is consistent with earlier studies of the persistence in sea surface temperature records where gamma is close to 0.4. In all cases, the exponent gamma does not depend on the distance of the stations to the continental coastlines. By varying the degree of detrending in the fluctuation analysis we obtain also information about trends in the temperature records.

Multifractality of river runoff and precipitation: comparison of fluctuation analysis and wavelet methods

We study the multifractal temporal scaling properties of river discharge and precipitation records. We compare the results for the multifractal detrended fluctuation analysis method with the results for the wavelet-transform modulus maxima technique and obtain agreement within the error margins. In contrast to previous studies, we find non-universal behaviour: on long time scales, above a crossover time scale of several weeks, the runoff records are described by fluctuation exponents varying from river to river in a wide range. Similar variations are observed for the precipitation records which exhibit weaker, but still significant multifractality. For all runoff records the type of multifractality is consistent with a modified version of the binomial multifractal model, while several precipitation records seem to require different models.

Analysis of daily temperature fluctuations

We study daily temperature fluctuations over more than 50 yr in two places on the globe that are separated by more than 3000 km. We analyze the temperature fluctuations ΔTi with respect to the mean noon temperature 〈Ti〉 averaged, for each day of the year, over the whole year, ΔTi = Ti − 〈Ti〉. We find that the ΔTi are correlated and can be characterized for up to at least 103 days by a power law correlation with an exponent α ≅ 0.65.

Long term persistence in the atmosphere: global laws and tests of climate models

The persistence of short term weather states is a well known phenomenon: A warm day is more likely to be followed by a warm day than by a cold one and vice versa. Using advanced methods from statistical physics that are able to distinguish between trends and persistence we have shown recently that this rule may well extend to months, years and decades, and on these scales the decay of the persistence seems to follow a universal power law. Here we review these studies and discuss, how the law can be used as an (uncomfortable) test bed for the state-of-the-art climate models. It turns out that the models considered display wide performance differences and actually fail to reproduce the universal power law behavior of the persistence. It seems that the models tend to underestimate persistence while overestimating trends, and this fact may imply that the models exaggerate the expected global warming of the atmosphere.

Comment on "Scaling of atmosphere and ocean temperature correlations in observations and climate models".

In a recent letter [K. Fraedrich and R. Blender, Phys. Rev. Lett. 90, 108501 (2003)], Fraedrich and Blender studied the scaling of atmosphere and ocean temperature. They analyzed the fluctuation functions F(s) ~ s^alpha of monthly temperature records (mostly from grid data) by using the detrended fluctuation analysis (DFA2) and claim that the scaling exponent alpha over the inner continents is equal to 0.5, being characteristic of uncorrelated random sequences. Here we show that this statement is (i) not supported by their own analysis and (ii) disagrees with the analysis of the daily observational data from which the grid monthly data have been derived. We conclude that also for the inner continents, the exponent is between 0.6 and 0.7, similar as for the coastline-stations.

Atmospheric Persistence Analysis: Novel Approaches and Applications

The persistence of short-term weather states is a well known phenomenon: a warm day is more likely to be followed by a warm day than by a cold one and vice versa. In a recent series of studies we have shown that this rule may well extend to months, years, and decades, and on these scales the decay of the persistence seems to follow a universal power-law. We review these studies and discuss how the law can be used to test the state-of-the-art chmate models.

Detrended Fluctuation Studies of Long-Term Persistence and Multifractality of Precipitation and River Runoff Records

We studied and compared the autocorrelation behaviour and the temporal multifractal properties of long daily river discharge and precipitation records from 42 hydrological stations and 99 meteorological stations around the globe. To determine the scaling behaviour in the presence of trends , we applied detrended fluctuation analysis (DFA) and multifractal DFA . We found that the daily runoffs are characterised by a power-law decay of the autocorrelation function above some crossover time that usually is several weeks.