ABSTRACT

This chapter discusses the interesting properties of wavelets such as multiscaling, compact support, vanishing moments and orthogonality. The wavelets are independently developed in the fields of mathematics, quantum physics, engineering, and seismic geology to utilize their specific properties that make them suitable for various applications such as signal processing, image compression, turbulence, earthquake prediction, and solution of partial differential equations. The continuous wavelet transforms was introduced by Morlet as a kind of modified windowed transform to analyze geophysical signals. The high frequency signal often contains noise which can be easily filtered, and small detail coefficients are often eliminated to save computer memory. A lifting scheme provides a method to generate some customized wavelets and scaling functions suitable for such problems. The concept of a lifting scheme is also used to customize the bi-orthogonal wavelets. Similar to splitting of the signal, the odd grid points correspond to the detail of the signal and so the associated basis functions are wavelets.