LINEAR AND PARABOLIC TAU-P TRANSFORMS REVISITED

New derivations for the conventional linear and paraboliC tau-p transf orms in the classic continuous function domain provide useful insight into the discrete tau-p transformations. For the filtering of unwanted waves such as multiples, the derivation of the tau-p transform should define the inverse transform first, and then compute the forward tran sform. The forward transform usually requires a p-direction deconvolut ion to improve the resolution in that direction. It aids the wave filt ering by improving the separation of events in the tau-p domain. The p -direction deconvolution is required for both the linear and curviline ar tau-p transformations for aperture-limited data. It essentially com pensates for the finite length of the array. For the parabolic tau-p t ransform, the deconvolution is required even if the input data have an infinite aperture. For sampled data, the derived tau-p transform form ulas are identical to the DRT equations obtained by other researchers. Numerical examples are presented to demonstrate event focusing in tau -p space after deconvolution.