Predictability of daily temperature series determined by maximal Lyapunov exponent Jan Skořepa, Jiří Mikšovský, Aleš Raidl Department of Atmospheric Physics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic, Introduction Positive maximal Lyapunov exponent is a mark of chaotic behavior. It shows how fast initially close trajectories diverge from each other in the phase space. Generally speaking, predictability is time for which we can predict future state of the atmosphere. In this poster we adopt concept of predictability as time when separation of trajectories reaches its limit value [1]. Meteorological series are usually too short to show exact exponential growth. Concept of nonlinear Lyapunov exponent (NLLE) by Li & Ding allows for quantification of non- exponential divergence of trajectories in the phase space [1]. Here, we compare this approach to the classical Lyapunov exponent method described by Rosenstein [2], using tools from the TISEAN program package [3] as well our own implementations. Methods and Data We investigated global ERA-40 daily temperature data in 500 hPa level from years 1960 to Three methods were used to estimate the Lyapunov exponents and evaluate predictability: 1) The Rosenstein method - single variable series, embedding dimension 6, and time delay is 1 day. Procedure lyap_r from the TISEAN package [3] was applied. 2) Our implementation of the NLLE method, considering five geographically closest points to form vector in the phase space. Time delay was not used. 3) Our method (denoted div-t) for multidimensional data – time delay was not used. The respective algorithm involved: a)subtraction of annual cycle of temperature b)13 x 9 geographically closed series forming an analyzed set c)PCA on the analyzed set: n highest eigenvalues explaining 95 % of total variance of the set d)phase space reconstruction with embedding dimension n without time delay We illustrate divergence of trajectories S(t) in Figs. 1 and 2. For all the three methods we determined maximal Lyapunov exponent as (t) = [S(t)-S(0)]/t. We set time of predictability as time when S(t) is higher than value of average of S(t) minus 1.5 times standard deviation of S(t) (see dotted line in Fig. 1 and Fig. 2). Discussion and results Roughly speaking, all methods show the same general behavior of S(t) function: rapid growth in the first few days and then constant value or slow oscillation related to the annual cycle. (The NLLE method chooses neighbors in phase space in such a way that the annual cycle is suppressed.) The geographical structure of Lyapunov exponents is quite nontrivial, and the results depend on the method used. We distinguish two main patterns: one expressed more for the Rosenstein method has up to seven zonal belts, meanwhile the other pattern expressed for the NLLE method and div-t has only three belts, showing contrast between lower and higher latitudes. References [1] Li, J., & Ding, R. (2011). Temporal–Spatial Distribution of Atmospheric Predictability Limit by Local Dynamical Analogs. Mon. Wea. Rev., 139(10), doi: /mwr-d [2] Rosenstein, M., Collins, J., & De Luca, C. (1993). A practical method for calculating largest Lyapunov exponents from small data sets. Physica D: Nonlinear Phenomena, 65(1-2), doi: / (93)90009-p [3] Hegger, R., Kantz, H., & Schreiber, T. (1999). Practical implementation of nonlinear time series methods: The TISEAN package. Chaos, 9(2), 413. doi: / Acknowledgements The study was supported by the Charles University in Prague, project GA UK No , and Czech Science Foundation, project P209/11/0956. Fig. 1. Divergence of nearby trajectories S(t) for three methods in a particular location. Fig. 2. Same as Fig. 1 for longer time. We can see annual cycle for the Rosenstein method. (1) (2) (10) Time of predictability Value of time of predictability is highly sensitive to the position of saturation-approximating threshold (see Fig. 2), therefore method of estimation plays a key role. We designed our method of predictability estimation to generate a similar range of values for all three approaches (see maps in the fourth row). Results show a rather indistinct pattern of five belts, which is expressed best for the NLLE method and does not appear at all for the div-t method. RosensteinNLLEdiv-t