(Texas Tech University, 1998-05) Gilliam, Xiaoning Li

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Wavelet analysis is a rapidly developing area of mathematical and application-oriented research in many disciplines of science and engineering. The wavelet transform, which localizes signals in space and scale, has become a popular tool for analyzing and understanding coherent structures in random fluid flows, especially in the atmospheric boundary layer.
In this dissertation, a statistical technique is developed to separate coherent structures in the wavelet transform from fluctuations due to incoherent noise. This technique (coherent structure detector) is based on one of the oldest methods in nonparametric statistics: the development of a randomized reference distribution. To build our randomized reference distribution, we first calculate the Discrete Fourier Transform of the signal. From this, we build a large number of exemplars by keeping the Fourier transform magnitude but randomizing the phase. Thus each exemplar has exactly the same power spectrum as the signal but is known to be incoherent. Each exemplar is then wavelet transformed to provide incoherent transform exemplars. At each scale order statistics are accumulated from the group of wavelet transform exemplars. These order statistics are used to provide the thresholds and corresponding p-values at each scale. This provides our statistical test, from which we identify the statistically significant information in the signal.