Introduction

The frequency content of seismic data decreases with time as a result of absorption as the wavefield travels through the earth. Therefore, for seismic interpretation it is desirable to bandpass-filter the seismic traces with different filters at different times. Usual practice consists of applying a few filters in predefined time windows which are made to overlap to provide a smooth transition between them. There are at least two problems with this approach: on the one hand, the overlap zones are rather arbitrary and phase distortion can be expected in them. On the other hand, with this approach we are limited to a few filters corresponding to the few chosen time windows.

A well-known alternative is the theory of non-stationary convolution and combination Margrave (1998); Rickett (1999). This theory allows the design of arbitrary filters that can be made to change in a sample-by-sample manner. The design of the filters themselves is done in the frequency domain and its application to the data can be done in either the time domain, the frequency domain or a mixed time-frequency domain. In the time domain the process is similar to stationary filtering, with the columns of the convolutional matrix representing the delayed impulse responses of the filters applied to each sample, rather than the more familiar Toeplitz matrix of the stationary case. In the frequency domain, the convolutional matrix is nearly diagonal with the departure from diagonal being a direct indication of the degree of non-stationarity of the filters. In the mixed domain, the non-stationary filtering is performed via a slow generalized Fourier transform.

Aside from the flexibility in choosing the domain of computation, we can also choose between non-stationary convolution and combination. The former is more appropriate when the spectra of the filters vary slowly and the latter when the change is sudden. Both non-stationary convolution and combination, however, reduce to stationary convolution in the limit of stationarity. This of course means that stationary filtering is a particular case of non-stationary filtering when the filters are kept constant for all samples.

In this paper I show the implementation of this algorithm for seismic trace filtering and for forward and inverse NMO correction. For the first application I used a set of randomly-generated seismic traces as well as a few traces of an actual seismic line. It will be shown by a time-frequency analysis of the data before and after the filter that it is indeed possible to change the spectrum of the seismic trace in a sample-by-sample basis without noticeable frequency distortions. For the NMO application I used a few CMP gathers consisting of five hyperbolic reflections and background Gaussian noise. It will be shown that we can pose the NMO-correction problem as a time-variant filtering problem and that we can control the accuracy of the underlying fractional sample interpolation as an input parameter.