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 In C(r) versus In r diagram. The first one in the range of larger In r has a gradual slope and the second one in the range of intermediate In r has a fast slope. Other two regions are settled in the range of small In r. The results lead us to claim that the series arises front the activity of at least two subsystems. The first subsystem is low-dimensional (d(f) = 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 (d(f) > 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 (T-2 approximate to 4.7 days) is longer than for transition-seasons (T-2 approximate to 4.0 days for spring, T-2 approximate to 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.