Daubechies wavelets are becoming a popular image processing and data compression tool because they are base functions of finite length and represent sharp edges by a small number of coefficients. Therefore, data which show a dense Fourier representation may yield a sparse matrix when transformed by wavelets. Additionally, wavelet transformations can be implemented extremely efficiently; the cost of algorithms performing wavelet transformations increases only linearly with the length of the transformed vector. The localization of its base function and its efficiency seem to make wavelet transformation a promising candidate for seismic processing. However, wavelets have not been widely used in the field of geophysics.
Daubechies 1992 gives a thorough theoretical introduction to various wavelets. Besides Daubechies wavelets, which are compact and orthogonal minimum phase filters, she discusses nonorthogonal wavelet bases, called frames, and near-zero phase filters.
Strang 1989 and Press 1991 give short introductions to Daubechies wavelet transformation. Strang talks mostly about continuous Daubechies wavelets and discrete Haar transform. Press complements Strang's more theoretical paper by presenting simple algorithms for computing the wavelet transform.
Beyond its success in the compression of ``edgy'' data, the wavelet transform is still a tool in search of applications. Recent research in replacing standard Fourier operations by wavelet manipulations Beylkin (1992) may provide efficient and more physical operators than Fourier methods. Examples relevant to geophysics are interpolation, shift, and differentiation techniques.
In the geophysical literature little has been written about wavelet transformation. The case of two coefficients is known as Haar transform and was formerly discussed by Ottolini 1990. Abma and Muir 1992 discuss the Haar and sliding Fourier transform. They also visualize the positioning of the roots corresponding to different wavelet filters.
Complementing Abma's paper and generalizing Ottolini's Haar transformation this paper formulates the wavelet transform explicitly as a matrix recursion scheme. Requiring the base filter coefficients of this recursion to be compact, orthogonal, and approximative, Daubechies wavelets of any length are easily derived. A pyramidal algorithm based on this matrix recursion scheme performs the transformation based on Daubechies wavelets of lengths from 4 to 20 samples. In the electronic version of this report, interactive figures using xtpanel Cole and Nichols (1992) allow readers to explore the transformation by submitting their own input and choosing the wavelet's length. Finally, I apply this algorithm to compress and interpolate seismic data.