Filtering

Filters play an important role in seismic processing, be it in modelling the propagation medium,
or in limiting the observation of the section to certain frequencies, wavenumbers or to the dominant signal,
or in suppressing certain noises.

Digital filters can operate in the space domain by using the convolution product. A sampled signal can be
represented by a polynomial of variable z and/or by a set of coefficients.

The convolution product of two signals is the product of the polynomials in z domain of these signals.

All linear filters can be presented by the relationship between two polynomials in z.
A filter for whom the polynomial is a fraction, whose denominator is equal to 1, is a moving average filter or MA.
A filter for whom the polynomial is a fraction, whose numerator is equal to 1, is an autoregressive filter or AR.
An ARMA filter is the product of an AR filter by an MA filter.
The order of the filter is fixed by the number of polynomial terms in z from which it is made up.
We study the behaviour of the most commonly used filters, this behaviour being analysed in terms of impulsional
response and transfer function (modulus and phase of their Fourier transform).

We study in particular:

Boxcar filters,

a first order MA filter,

a first order AR filter,

a second order MA filter,

a second order AR filter,

a Butterworth ARMA filter.

In seismic processing, the desired information, obtained by an instrument which introduces distortions,
is disturbed by interfering events (for example multiples, ghost effects...) which can be represented by
convolutive models. Distortions, inherent to measure, can be softened by adapting the measure process,
but they cannot be totally eliminated. The aim of deconvolution is to propose processing methods which allow
softening, ideally suppressing, distortions introduced by measure and interfering events.

Deconvolution is an inverse filter which, by its nature, has a troublesome tendency of diminishing the signal
to noise ratio. A method classically used and relatively robust to noise is the deconvolution with
the Wiener filter.

We will successively show:

Inverse filter,

AR2 inverse filter,

Wiener filter: spiking deconvolution,

Wiener filter: Study of the variability of a seismic section,

Wiener filter: predictive deconvolution.