Data-Adapted Wavelets for Analysis of Observational Data

提出了一种逼近任意给定目标函数的双正交小波的构造方法。当给定的数据是目标函数的叠加时，这种方法被认为是有用的。然后，双正交小波提供了将给定数据分解为事件元素的有效方法。双正交小波是通过拉格朗日乘子法最小化目标函数与主小波之差的L2范数得到的。作为一个例子，将该方法应用于一些目标函数，产生接近这些函数的双正交小波。用这些自适应双正交小波对一些地震信号进行小波展开，发现它们比Meyer和Daubechies小波展开具有更小的熵。本课题还研究了小波在人工地震信号构建、振荡分析、分数阶导数粘弹性模型等方面的工程应用。

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.