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.

- The fast wavelet transform
- 1-D seismic signal decomposition and reconstruction
- 2-D bandpass
- 2-D and 3-D segmentation

1/13/1998