Fundamentals of the Discrete Wavelet Transform for Seismic Data Processing

The discrete wavelet transform has been an exciting topic of mathematical research for about 10 years now. Some of the early wavelet research had seismic applications explicitly in mind, but elds as diverse as quantum physics and voice coding have also provided insights leading to the development of the modern theory. Our purpose is to explain and illustrate the eeect of the discrete wavelet transform on seismic data, thus providing the information necessary for researchers to assess its possible use in their areas of data processing. An analysis of the discrete wavelet transform of dipping segments with a signal of given frequency band leads to a quantitative explanation of the known division of the two-dimensional wavelet transform into horizontal, vertical and diagonal emphasis panels. The results must be understood in a \fuzzy" sense: since wavelet mirror lters overlap, the results stated can be slightly violated with violation tending to increase with shortness of the wavelet chosen. The speciic angles that delimit the three wavelet panels can be stated in terms of the Nyquist frequency F t in the rst dimension and the Nyquist frequency F x in the second dimension. The angle HV = arctan(F t =F x) approximately separates the dips < HV that appear in the horizontal panel from those greater dips that appear in the vertical panel. Similarly, the angles arctan(F t =2F x), arctan(2F t =F x) approximately bound the dips appearing in the diagonal panel. These results are simple and probably have been observed by other researchers, but we haven't found a prior reference for them. As an example of seismic processing using the wavelet transform, some simple examples of de-aliasing spatially aliased data are discussed.