Fractal Analyses of Century-Long Daily Variations in Air Temperatures in Kiev

1. Fractal Analyses of Century-Long Daily Variations in the Mean, Maximal and Minimal Air Temperatures in Kiev Vladimir Lutkovsky Sergey Shumov UKRAINIAN HYDROMETEOROLOGICAL RESEARCH INSTITUTE

2. Introduction Now it is obviously and reliably established, that the climate of the Earth never was stationary, it continuous varied , these changes were rather considerable and sometimes happened very promptly. The surface of the Earth was covered by glaciers, and then these glaciers disappeared and in considerable territory of the Earth the tropical climate was erected. In the past such ice periods there were some tens, their recurring was irregular, and the gaps between them made from 40 thousand up to several hundreds thousand years. Last ice period begun to recede all 20 thousand years back. It is necessary to note, that in epoch of major icing the level of World ocean was lowered more than on 100 ì below modern. It is confirmed by the independent geological data. Now level of world ocean varies very slowly. For a period with 1890 on 1950 yy it has increased all on 10 ñì, and since 1950 remains practically to stationary values (with oscillations ± 3 cm) [1]. Some scientists consider, that apparent warming is bound not to ejection of a great many « greenhouse gases », and with is unusual by a high level of intensity radiated by the Sun within almost all last century of energy. This warming will follow natural cooling, that is confirmed by that fact, that the upper stratums of world ocean already have begun to cool down and warming epoch will be soon finished]. So, still there is unclosed a problem: « can be whether warming of a climate is a result of antropogeneous action? Or it is prime a beginning of a new natural warming cycle? ». Maybe, the antropogeneous action could provoke climatic shift and to call a new warming cycle? The unique answer is not present. The considerable pinch of a scientific level of examinations of all problems, bound with global warming, especially of problems of change of solar activity and increasing of greenhouse gases is necessary. The problem of change of a climate today is not only scientific, but also economic and political problem.

3. Materials and methods At the analysis of a climate input data are the time series containing values of those or other climatic exponents (temperature, precipitates, damp etc.) for a certain period. Obviously, that the a series is longer, the information from it more it is possible to extract. Generally there are two basic purposes of the analysis of time series: (1) Definition of a nature of a series; (2) Prediction (prediction of the future values of time series for the present and last values). As well as majority of other views of the analysis, the analysis of time series guesses, that the data contain systematic component (usually including a little component) and casual noise (error), which impedes detection a regular component. The majority of research techniques of time series includes various expedients of a filtration of a noise permitting to see regular component more clearly. The majority of regular component time series belongs to two classes: they are either trend, or seasonal component. The trend represents to a blanket systematic linear or nonlinear component, which can vary in time. Seasonal component is periodically varying component. Both these of a view a regular component frequently are present at a series simultaneously. Traditionally for the analysis of the data on a course of temperatures in a climatology the trends are used. Thus, as a rule, the second problem is solved only: a prediction of the future values of a series. At the same time trend nothing speaks about, as far as a series is inconvertible. Besides at build-up of a trend the major value has a choice of a method of build-up: usual whether usual , moving average etc. The results obtained by different methods, can considerably differ. Let's illustrate abovementioned on an example of the trend analysis of a course of temperatures (data from [3]). As it is shown (fig. 1), results of trend analysis Figure 1. moving 10-year's average values of annual temperature of air in Kiev under the archival data are strongly depend on the moment of a beginning of observations. So if to select period 1910- 1934 yy., it is possible to conclude about warming, during 1945-1970 yy.- stability, with 1990 on 2000 yy.- about sharp warming.

4. Input data Sliding 10-year's medial values of annual temperature of air (ToC) in Kiev under the archival data

5. The Big Question Real Observations stochastic irregular behavior with self similarity Nonlinear Law We all know that the self similarity (scale invariance) is synonymous with fractal type behavior Fractal analysis is a convenient tool for study of self-similar properties of natural object. Fractal dimension characterize scaling properties of the object. Concerning application to time series, fractal analysis provide information about the degree of chaos and order in the time series being analyzed. Question The question we are all asking ourselves is, how fractals can be used to predict most adequately the future values

6. Rescaled Range (R/S) Analysis Method Rescaled range analysis studies the distribution of events by grouping observed data into clusters of different sizes and studying the scaling behavior of the statistical parameters with the cluster sizes.

7. In the Beginning In 1951, Hurst defined a method to study natural phenomena such as the flow of the Nile River. Process was not random, but patterned. He defined a constant, K, which measures the bias of the fractional Brownian motion. In 1968 Mandelbrot defined this pattern as fractal. He renamed the constant K to H in honor of Hurst. The Hurst exponent gives a measure of the smoothness of a fractal object where H varies between 0 and 1.

8. Random and not Random It is useful to distinguish between random and non-random data points. If H equals 0.5, then the data is determined to be random. If the H value is less than 0.5, it represents anti-persistence. If the H value varies between 0.5 and 1, this represents persistence. (what we get)

9. Calculation of R/S Start with the whole observed data set that covers a total duration and calculate its mean over the whole of the available data

10. Calculation of R/S Sum the differences from the mean to get the cumulative total at each increment point, V(N,k), from the beginning of the period up to any point, the result is a series which is normalized and has a mean of zero

11. Calculation of R/S Calculate the range

12. Calculation of R/S Calculate the standard deviation

13. Calculation of R/S Plot log-log plot that is fit Linear Regression Y on X where Y=log R/S and X=log n where the exponent H is the slope of the regression line. Calculations of Hurst parameters and fractal dimensions were carried out by using of the free software Fractan 4.4., worked out by V. Sychov in 2003 (Institute of Mathematical Problems of Biology RAS).

14. We explored a series with average annual temperatures for a period 1924- 2010 yy for Kiev. If toanalyze the data of a trend (fig. 2), it is obtained, that for 100 years the increase of temperature should be equal to 1.72 degrees, that misses the really available data. At the same time Hurst index, obtained for this purposeof a series, is 0.58, that speaks, that a series is feeblly persistent.The explanation to so high value of a slope angle of a trend can be found if to conduct the analysis not onannual values of temperature, and to analyze) changes of monthly average temperature (fig. of 3) anddaily temperatures (fig.4) for all surveyed period. A trend of mean annual temperatures & value of an exponent H for a period 1924-2010 yy

15. A trend of mean monthly temperatures & value of an exponent H for a period 1924-2010 yy

16. Trends of mean daily temperatures & value of an exponent H for a period 1924-2010 yy

17. Conclusion Using the data obtained with the help of both approaches (classical and fractal), it is possible to make a conclusion, that for today a situation in Kiev region on a climate following: the steady grouth of winter temperatures and steady diminution of summer is observed. I.e. in the long term it is possible to expect leveling of an difference between winter and summer periods.

18. References (1)L.A. Solntsev, D.I. Iudin, M.S. Snegireva, D.B. Gelashvili Nizhni Novgorod University memoirs, 2007,№ 4, pp.88-91. (2) Terez E.I. Steady development and problems of change of a global climate change on the Earth //Memoirs of the Tavria national university ). - 2004. – V. 17 (56), # 1. - pp 181-205. (3) < http: // www.rian.ru/science >. (4) Secular course of medial temperature of air and totals of precipitates on Kiev (5) Delignieres D. Et al. // Journal of Mathematical Psychology - 2006. - V. 50. - P. 525-544. (6) Hurst H.E., Black R.P., Simaika Y.M. Long-term storage: An experimental study. - L.: Constable, 1965.

19. All my colleagues and friends from Ukrainian Hydromet. Institute & its management for financial support Acknowledgements Thank you for your attention!