Linear and parabolic τ-p transforms revisited

Abstract

New derivations for the conventional linear and parabolic τ-p transforms in the classic continuous function domain provide useful insight into the discrete τ-p transformations. For the filtering of unwanted waves such as multiples, the derivation of the τ-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 τ-p domain. The p‐direction deconvolution is required for both the linear and curvilinear τ-p transformations for aperture‐limited data. It essentially compensates for the finite length of the array. For the parabolic τ-p transform, the deconvolution is required even if the input data have an infinite aperture. For sampled data, the derived τ-p transform formulas are identical to the DRT equations obtained by other researchers. Numerical examples are presented to demonstrate event focusing in τ-p space after deconvolution.