FREQUENCY WAVENUMBER APPROACH OF THE τ‐p TRANSFORM: SOME APPLICATIONS IN SEISMIC DATA PROCESSING*

Abstract

AbstractThe τ‐p transform is an invertible transformation of seismic shot records expressed as a function of time and offset into the τ (intercept time) and p (ray parameter) domain. The τ‐p transform is derived from the solution of the wave equation for a point source in a three‐dimensional, vertically non‐homogeneous medium and therefore is a true amplitude process for the assumed model. The main advantage of this transformation is to present a point source shot record as a series of plane wave experiments.The asymptotic expansion of this transformation is found to be useful in reflection seismic data processing. The τ‐p and frequency‐wavenumber (or f‐k) processes are closely related. Indeed, the τ‐p process embodies the frequency‐wavenumber transformation, so the use of this technique suffers the same limitations as the f‐k technique. In particular, the wavefield must be sampled with sufficient spatial density to avoid wavenumber aliasing.The computation of this transform and its inverse transform consists of a two‐dimensional Fast Fourier Transform followed by an interpolation, then by an inverse‐time Fast Fourier Transform. This technique is extended from a vertically inhomogeneous three‐dimensional medium to a vertically and laterally inhomogeneous three‐dimensional medium.The τ‐p transform may create artifacts (truncation and aliasing effects) which can be reduced by a finer spatial density of geophone groups by a balancing of the seismic data and by a tapering of the extremities of the seismic data.The τ‐p domain is used as a temporary domain where the attack of coherent noise is well addressed; this technique can be viewed as ‘time‐variant f‐k filtering’. In addition, the process of deconvolution and multiple suppression in the τ‐p domain is at least as well addressed as in the time‐offset domain.