Wavelets-based estimation of multivariate evolutionary spectra and its application to nonstationary downburst winds

This paper presents a wavelets-based approach for estimating evolutionary power spectral density (EPSD) functions of vector-valued nonstationary processes. This approach is based on the presumption that the time-varying rate of the modulation function of the process is slow as compared to the localized wavelet functions. This assumption leads to the relationships between wavelet coefficients and EPSDs, which are used for estimating EPSDs through wavelet coefficients calculated from time history samples. A parametric study is implemented to examine the efficacy of the approach for a host of evolutionary processes. Special emphases are placed on the influence of the wavelet functions, the number of time history samples and the wavelet scales on the accuracy of the estimations. The results of this study illustrate that this approach offers accurate EPSD estimation provided that sufficient numbers of time history samples are available and that the changes in the modulation functions with time are relatively slow as compared to the wavelets. Compared with the time-frequency spectra directly defined in terms of wavelet coefficients reported in literature, the EPSDs calculated from the wavelet coefficients using the proposed approach shed more physical insights to the processes. Finally, the proposed approach is used to analyze downburst measurement data, leading to an improved understanding of the time-varying frequency characteristics of nonstationary winds.