Resolution performance of Wiener filters

Abstract

To improve the resolution of seismic events, one often designs a Wiener inverse filter that optimally (in the least‐squares sense) transforms a measured source signature into a spike. When this filter is applied to seismic data, the bandwidth of any noise which is present increases along with the bandwidth of the signal. Thus the signal‐to‐noise ratio is degraded. To reduce signal ambiguity it is common practice to prewhiten the Wiener filter. Prewhitening the filter improves the output signal‐to‐ambient noise ratio, but at the same time it reduces resolution. The ability to resolve the temporal separation between events is determined by the resolution time constant which we define as the ratio of signal energy to peak signal power from the filter. For unfiltered wavelets the resolution time constant becomes the reciprocal of resolving power recently described by Widess (1982). For matched filter signals the resolution time constant can be regarded as the inverse of the frequency span of the signal. Although it is satisfying that the resolution time constant definition agrees with other measures of resolution, this more general definition has two major advantages. First, it incorporates the effect of filtering; second, it is easily generalized to incorporate the effects of noise by assuming that the filter is a Wiener filter. For a given amount of noise the Wiener filter is a generalization of the matched filter. Marine seismic wavelets demonstrate how reducing the noise level improves the resolution of a Wiener filter relative to a matched filter. For these wavelets a point of diminishing return is reached, such that, to realize a further small increase in resolution, a large increase in input signal‐to‐noise ratio is required to maintain interpretable information at the output.

Use of a Priori Information for the Deconvolution of Ultrasonic Signals

“…The spectrum of the estimated signal is well reconstructed but only for the frequencies inside the band-pass of the acoustic pulse. In general, this is not sufficient to increase significantly the resolution except if the SNR is very high [5]. …”

Spectral sparse Bayesian learning reflectivity inversion

“…The most popular method to retrieve reflectivity is inverse filtering within a least‐squares framework (e.g., Berkhout 1977; Robinson and Treitel 1980; Bickel and Martinez 1983). This method is based on the assumption that the earth's reflectivity has a spectrum of white noise, so that the autocorrelation of the wavelet can be replaced by that of seismic data.…”

Swell Noise Suppression by Wiener Prediction Filter

“…Wiener filter has successfully been used in deconvolution of the seismic data for many years (Berkhout, 1977;Bickel and Martinez, 1983;Gülünay, 1986;Peacock and Treitel, 1969;Robinson, 1967;Treitel and Robinson, 1969). In recent years, it has also been used to attenuate different high amplitude noises in the seismic data.…”