INTRODUCTION TO WAVELETS

Mathematical wavelets are a basis function for decomposing signals. The simplest example is the half-cycle square-wave family illustrated in Figure 1. Wavelets resemble Fourier sinusoidal basis functions in that they vary in wavelength, are orthogonal to each other, fully decompose and recompose the signal. The are superior to Fourier functions in that they are finite in length (compact support) and faster to compute. They may be inferior to Fourier functions in that they don't obey the derivative identity necessary for solving the wave equation. (At least one that obeys the derivative identity hasn't been brought to my attention yet.)

Mathematical wavelets have their origin in the local frequency transforms of Goupillaud et al. (1984). Since then they have been given a rigorous mathematical foundation (see references in Strang 1989). The tutorial article by Strang provides a concrete introduction for applied mathematicians.

Wavelets are based on recursive scaling functions (dilatations), that is, functions that can be generated by stretching and shifting one or more instances of itself. I study the simplest wavelet family--half-cycle square-wave (Figure 1a)-in this article. These wavelets scale by powers of two. Strang gives algorithms for constructing more sophisticated wavelets with better secondary mathematical properties such as smoothness and interpolatibility.

Wavelet analysis of seismic signals is generally applicable whereever the geometric properties of Fourier analysis have been used: frequency decomposition, dip decomposition, Gaussian decomposition, filtering, segmentation, etc.