Data-Adapted Wavelets for Analysis of Observational Data

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A construction method is proposed for a biorthogonal wavelet which approximates an arbitrary given target function. This method is expected to be useful in the cases where the given data is a superposition of the target functions dilated, and translated. The biorthogonal wavelet then provides an efficient decomposition of the given data into the elements of events. The biorthogonal wavelet is obtained by Lagrange's multiplier method minimizing the L2 norm of the difference between the target function and the primary wavelet. As an example, this method is applied to some target functions to produce biorthogonal wavelets close to these functions. Some seismic signals are decomposed by wavelet expansion with some of these adapted-biorthogonal wavelets, which are found to have less entropy than the expansions with Meyer and Daubechies wavelets. In this research project, engineearing applications of' wavelets were also studied in several fields, including construction of artificial seismic signals, oscillation analysis, fractional derivative viscoelasticity models.