Wavelet analysis: basic theory and some applications

The basic theory of the wavelet transform, an effective investigation tool for inhomogeneous processes involving widely different scales of interacting perturbations, is presented. In contrast to the Fourier transform, with the analysing function extending over the entire axis of time, the two-parametric analysing function of the one-dimensional wavelet transform is well localised in both time and frequency. The potential of the method is illustrated by analysing familiar model series (such as harmonic, fractal, and those with various types of singularities) and the long-term variation of some meteorological characteristics (Southern Oscillation index and global and hemispheric temperatures). The analysis of a number of El Nino events and of the temporal behaviour of the Southern Oscillation index reveals periodic components, local periodicity features and time scales on which self-similarity structures are seen. On the whole, both stochastic and regular components seem to be present. The global and hemispheric temperatures are qualitatively similar in structure, the main difference — presumably due to the greater amount of land and stronger anthropogenic factor — being that the warming trend in the Northern Hemisphere is slightly stronger and goes first in time.