Jan F. Eichner

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.

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.

The Statistics of Return Intervals, Maxima, and Centennial Events Under the Influence of Long-Term Correlations

We review our studies of the statistics of return intervals and extreme events (block maxima) in long-term correlated data sets, characterized by a power-law decaying autocorrelation function with correlation exponent γ between 0 and 1, for different distributions (Gaussian, exponential, power-law, and log-normal). For the return intervals, the long-term memory leads (i) to a stretched exponential distribution (Weibull distribution), with an exponent equal to γ, (ii) to long-term correlations among the return intervals themselves, yielding clustering of both small and large return intervals, and (iii) to an anomalous behavior of the mean residual time to the next event that depends on the history and increases with the elapsed time in a counterintuitive way. We present an analytical scaling approach and demonstrate that all these features can be seen in long climate records. For the extreme events we studied how the long-term correlations in data sets with Gaussian and exponential distribution densities affect the extreme value statistics, i.e., the statistics of maxima values within time segments of fixed duration R. We found numerically that (i) the integrated distribution function of the maxima converges to a Gumbel distribution for large R similar to uncorrelated signals, (ii) the deviations for finite R depend on both the initial distribution of the records and on their correlation properties, (iii) the maxima series exhibit long-term correlations similar to those of the original data, and most notably (iv) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the previous maximum. Finally we evaluate the effect of long-term correlations on the estimation of centennial events, which is an important task in hydrological risk estimation. We show that most of the effects revealed in artificial data can also be found in real hydro- and climatological data series.

Statistics of Return Intervals and Extreme Events in Long-term Correlated Time Series

We review our studies of the statistics of return intervals and extreme events (maxima) in long-term power-law correlated data sets characterized by correlation exponents γ between 0 and 1 and different (Gaussian, exponential, power-law, and log-normal) distributions. We found that the long-term memory leads (i) to a stretched exponential distribution of the return intervals (Weibull distribution with an exponent equal to γ ), (ii) to clustering of both small and large return intervals, and (iii) to an anomalous behavior of the mean residual time to the next extreme event that increases with the elapsed time in a counterintuitive way. For maxima within time segments of fixed duration R we found that (i) the integrated distribution function converges to a Gumbel distribution for large R similar to uncorrelated signals, (ii) the speed of the convergence depends on both, the long-term correlations and the initial distribution of the values, (iii) the maxima series exhibit long-term correlations similar to those of the original data, and most notably (iv) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the previous maximum. Most of the effects revealed in artificial data can also be found in real hydro- and climatological data series.