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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.

A new nonlinear filter for wave‐equation extrapolation‐based multiple suppression is designed in the f-k domain. The realization of the new filter in the f-k domain is an extension of the conventional f-k dip filter. However, the new demultiple filter is superior to the conventional f-k dip filter in the sense that the multiple reject zones are determined automatically (based on the information of the input original data and the multiple model traces obtained by the wave‐extrapolation method) rather than by prior information on multiple moveout (dip) range. Therefore, it can easily cope with situations such as aliasing and the mixture of energy from multiples and primaries in the same quadrant. The new filter is smooth on the boundary of the reject area. Numerical examples demonstrate that the new filter is equivalent to the conventional f-k dip filter in multiple suppression for simple situations. However, when the multiples and primaries are mixed in the same quadrant and have only slight difference in dip, the new filter offers significant advantages over the conventional technique.

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The filter for wave-equation-based water-layer multiple suppression, developed by the authors in the x-l, the lineat r-p, and the /-A domains, is extended to the parabolic r-2 domain. The multiple reject areas are determined automatically by comparing the energy on traces of the multiple model (which are generated by a wave-extrapolation method from the original data) and the original input data (multiples * primaries) in r-p space. The advantage of applying the data-adaptive 2D demultiple filter in the parabolic r-p domain is that the waves are well separated in this domain. The numerical examples demonstrate the effectiveness of such a dereverberation procedure. Filtering of multiples in the parabolic r-p domain works on both the far-offset and the near-offset traces) while the filtering of multiples in the /-A domain is effective only for the far-offset traces.Tests on a synthetic common-shot-point (CSP) gather show that the demultiple filter is relatively immune to slight errors in the water velocity and water depth which cause arrival time errors of the multiples in the multiple model traces of less than the time dimension (about one quarter of the wavelet length) of the energy summation window of the filter. The multiples in the predicted multiple model traces do not have to be exact replicas of the multiples in the input data, in both a wavelet-shape and traveltime sense. The demultiple filter also works reasonably well for input data contaminated by up to 25oh of random noise. A shallow water CSP seismic gather, acquired on the North West Shelf of Australia, demonstrates the effectiveness of the technique on real data.

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Dip moveout (DMO) processing is a partial prestack migration procedure that has been widely used in seismic data processing. The DMO process has been described in Deregowski (1986), Hale (1991) and Liner (1990). Many different DMO algorithms have been developed over the past decade. These algorithms have been designed to improve either the accuracy or the computational speed of the DMO process. Hale (1984) developed a method for performing DMO via Fourier transforms that is accurate for all reflector dips (assuming constant velocity). Hale’s method is computationally expensive because his DMO operator is temporally nonstationary, but its accuracy and simplicity have made it an industry standard. It has become a benchmark by which results from other DMO algorithms are judged. Of all the methods used to make the frequency‐domain DMO computationally efficient, the technique of logarithmic time stretching, first suggested in Bolondi et al. (1982), is widely used. After logarithmic stretching of the time axis, the DMO operator becomes temporally stationary which enables replacement of the slow temporal Fourier integration with a fast Fourier transform combined with a simple phase shift. Bale and Jakubowicz (1987) presented a log‐stretch DMO operator (hereafter referred to as Bale’s DMO) in the frequency‐wavenumber (F-K) domain without approximations, while Notfors and Godfrey (1987) suggested an approximate version of log‐stretch DMO operator (hereafter referred to as Notfors’s DMO). Surprisingly, Bale’s full log‐stretch DMO operator produces a less satisfactory impulse response than Notfors’s approximate log‐stretch DMO scheme (see Liner, 1990). Liner (1990) attributed this characteristic to the fact that Bale’s DMO derivation implicitly assumes that the Fourier transform frequency in the log‐stretch domain is not time‐dependant. He presented an exact log‐stretch DMO operator (hereafter referred to as Liner’s DMO) which was derived by transforming the time log‐stretched Hale’s (t, x) DMO impulse response into the Fourier domain. Its derivation is relatively complicated, but Liner has shown that his DMO does generate good DMO impulse responses.

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