Ales Raidl

Estimating the fractal dimension,K 2 -entropy, and the predictability of the atmosphere

The series of mean daily temperature of air recorded over a period of 215 years is used for analysing the dimensionality and the predictability of the atmospheric system. The total number of data points of the series is 78527. Other 37 versions of the original series are generated, including “seasonally adjusted” data, a smoothed series, series without annual course, etc.Modified methods of Grassberger & Procaccia are applied. A procedure for selection of the “meaningful” scaling region is proposed. Several scaling regions are revealed in the lnC(r) versus Inr diagram. The first one in the range of larger lnr has a gradual slope and the second one in the range of intermediate lnr has a fast slope. Other two regions are settled in the range of small lnr. The results lead us to claim that the series arises from the activity of at least two subsystems. The first subsystem is low-dimensional (df=1.6) and it possesses the potential predictability of several weeks. We suggest that this subsystem is connected with seasonal variability of weather. The second subsystem is high-dimensional (df>17) and its error-doubling time is about 4–7 days.It is found that the predictability differs in dependence on season. The predictability time for summer, winter and the entire year (T2≈4.7 days) is longer than for transitionseasons (T2≈4.0 days for spring,T2≈3.6 days for autumn).The role of random noise and the number of data points are discussed. It is shown that a 15-year-long daily temperature series is not sufficient for reliable estimations based on Grassberger & Procaccia algorithms.

Evolutionary Reconstruction of Chaotic Systems

This chapter discusses the possibility of using evolutionary algorithms for the reconstruction of chaotic systems. The main aim is to show that evolutionary algorithms are capable of the reconstruction of chaotic systems without any partial knowledge of internal structure, i.e. based only on measured data. Five different evolutionary algorithms are presented and tested in a total of 13 and 12 versions in two different versions of experiments. System selected for numerical experiments here is the well-known logistic equation. For each algorithm and its version, 100 repeated simulations were conducted. According to obtained results it can be stated that evolutionary reconstruction is an alternative and a promising way as to how to identify chaotic systems.

Initial Errors Growth in Chaotic Low-Dimensional Weather Prediction Model

The growth of small errors in weather prediction is exponential. As an error becomes larger, the growth rate decreases and then stops with the magnitude of the error about at a value equal to the size of the average distance between two states chosen randomly.

Evolutionary Synchronization of Chaotic Systems

This chapter introduces a simple investigation on deterministic chaos synchronization by means of selected evolutionary techniques. Five evolutionary algorithms has been used for chaos synchronization here: differential evolution, self-organizing migrating algorithm, genetic algorithm, simulated annealing and evolutionary strategies in a total of 15 versions. Experiments in this chapter has been done with two different coupled systems (master — slave) — Rossler-Lorenz and Lorenz-Lorenz. The main aim of this chapter was to show that evolutionary algorithms, under certain conditions, are capable of synchronization of, at least, simple chaotic systems, when the cost function is properly defined as well as the parameters of selected evolutionary algorithm. This chapter consists of two different case studies. For all algorithms each simulation was 100 times repeated to show and check the robustness of proposed methods and experiment configurations. All data were processed to obtain summarized results and graphs.

Global Patterns of Nonlinearity in Real and GCM-Simulated Atmospheric Data

We employed selected methods of time series analysis to investigate the spatial and seasonal variations of nonlinearity in the NCEP/NCAR reanalysis data and in the outputs of the global climate model HadCM3 of the Hadley Center. The applied nonlinearity detection techniques were based on a direct comparison of the results of prediction by multiple linear regression and by the method of local linear models, complemented by tests using surrogate data. Series of daily values of relative topography and geopotential height were analyzed. Although some differences of the detected patterns of nonlinearity were found, their basic features seem to be identical for both the reanalysis and the model outputs. Most prominently, the distinct contrast between weak nonlinearity in the equatorial area and stronger nonlinearity in higher latitudes was well reproduced by the HadCM3 model. Nonlinearity tends to be slightly stronger in the model outputs than in the reanalysis data. Nonlinear behavior was generally stronger in the colder part of the year in the mid-latitudes of both hemispheres, for both analyzed datasets.

Time Evolution of Initial Errors in Lorenz’s 05 Chaotic Model

Initial errors in weather prediction grow in time and, as they become larger, their growth slows down and then stops at an asymptotic value. Time of reaching this saturation point represents the limit of predictability. This paper studies the asymptotic values and time limits in a chaotic atmospheric model for five initial errors, using ensemble prediction method (models data) as well as error approximation by quadratic and logarithmic hypothesis and their modifications. We show that modified hypotheses approximate the models time limits better, but not without serious disadvantages. We demonstrate how hypotheses can be further improved to achieve better match of time limits with the model. We also show that quadratic hypothesis approximates the models asymptotic value best and that, after improvement, it also approximates the models time limits better for almost all initial errors and time lengths.

Estimations of Initial Errors Growth in Weather Prediction by Low-dimensional Atmospheric Model

Initial errors in weather prediction grow in time. As errors become larger, their growth slows down and then stops at an asymptotic value. Time of reaching this value represents the limit of predictability. Other time limits that measure the error growth are doubling time τ d, and times when the forecast error reaches 95%, 71%, 50%, and 25% of the limit of predictability. This paper studies asymptotic value and time limits in a low-dimensional atmospheric model for five initial errors, using ensemble prediction method as well as error approximation by quadratic and logarithmic hypothesis. We show that quadratic hypothesis approximates the model data better for almost all initial errors and time lengths. We also demonstrate that both hypotheses can be further improved to achieve even better match of the asymptotic value and time limits with the model.