Application of Wavelets to Observational Data Analysis

研究了小波分析在观测数据中的应用。以地震加速度数据为例，提出了一种由双正交小波展开和拉格朗日乘子法组成的数据校正方法。该方法基于小波展开，可在时频域对数据进行局部校正。此外，我们设计了一种生成双正交小波的算法，该算法将一类变线性算子对角/半对角化为尺度变换，以减少数据校正中的数值任务，例如积分。将该算法应用于Riesz势、导数Hilbert变换和Abel变换。数值检验表明，该表示矩阵的元素在非对角线区域衰减迅速。这意味着矩阵实际上可以被视为带对角线矩阵，并允许我们快速计算。我们还研究了小波在摩擦和振荡吸收等问题上的工程应用。

Application of wavelet analysis to observational data is studied. Taking an acceleration data of earthquake as an example, we propose a data correction method consisting of biorthogonal wavelet expansion and Lagrange multiplier method. This method is based on wavelet expansion and enables us to correct the data locally in time-frequency domain. Moreover we devised an algorithm to generate biorthogonal wavelets which diagonalize/semi-diagonalize a class of linear operators in-variant to scale transformation, in order to reduce numerical task in the data correction including, integration, for example. We applied this algorithm to Riesz potential, derivative Hilbert transformation and Abel transformation. Numerical inspection shows that elements of the representation matrices decay rapidly in the off-diagonal region. This means that the matrices can accually be treated as band-diagonal ones, and permits us fast calculation. We also studied engineering application of wavelets to problems including friction and oscillation absorption.