Linear and parabolic tau -p transforms revisited

New derivations for the conventional linear and parabolic tau -p transforms 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 transform. The forward transform usually requires a p-direction deconvolution to improve the resolution in that direction. It aids the wave filtering by improving the separation of events in the tau -p domain. The p-direction deconvolution is required for both the linear and curvilinear tau -p transformations for aperture-limited data. It essentially compensates for the finite length of the array. For the parabolic tau -p transform, the deconvolution is required even if the input data have an infinite aperture. For sampled data, the derived tau -p transform formulas are identical to the DRT equations obtained by other researchers. Numerical examples are presented to demonstrate event focusing in tau -p space after deconvolution.