6. Conclusions

 

We have studied the capabilities of two methods of nonlinear data analysis (multiresolution analysis and structure function analysis) to characterize complex, non-stationary time series, which are typically observed in solar microwave bursts. Our main findings are:

(1)

The wavelet transform permits a local decomposition of the scaling behavior of the temporal dynamics of the radio flux - in contrast to the usual global methods of data analysis based on the Fourier transform. This enables the study of non-stationary systems. Different time scales inherent in a series can be resolved, this can be done for different phases of the series separately. We have found that the occurrence of an impulsive component does not imply a significant structural change in the properties of scaling in time of the solar microwave flux. The scalegrams, obtained by averaging over the whole lifetime or a substantial fraction of a burst, do not show the dominance of a particular time scale in the considered range between 1 s and ; distinct hierarchic time structures and sequences of particular time scales are only short-living phenomena. This finding is consistent with the idea that the basic mechanism to generate the solar microwave flux is operating at a broad range of time scales as in the case of a turbulence cascade. On the other hand, the time resolution of the presently available data is limited to 0.5 s, which prevents us from studying the range of the possibly elementary time scales. It is important to note that for all observations investigated here conventional methods, such as correlation and spectral analysis, fail to yield this result.

(2)

There is no unique well-developed statistical theory for the applied analysis methods yet. To check the reliability of the estimates obtained, we have applied these techniques (power-law section of both structure function and scalegram) also to surrogates chosen to agree in some statistical properties with the data. For solar impulsive mm-wave bursts, the resemblance with the fractional Brownian motion, governed by a stochastic nonlinear evolution equation and characterized by scaling exponents H > 0.5, which reflect long-range correlations, is evident.

(3)

While not distinguishable by the method of power spectrum, the structure function and the multiresolution analysis show clearly different scaling exponents for bursts, the quiet Sun, and the sky background.

Acknowledgements

We acknowledge fruitful discussions with L. Vlahos, A. Benz, and P. Maaß. We thank Ph. Bendjoya and J.-M. Petit for providing the MRA code. The work of B.K. was supported by DARA grant 50QL9208. Finally, we thank the referee for his helpful criticism.