Estimation of the Primary Seismic Pulse *

Abstract

A seismic trace after application of suitable amplitude recovery may be treated as a stationary time‐series. Such a trace, or a portion of it, is modelled by the expression where j represents trace number on the record, t is time, αj is a time delay, α (t) is the seismic wavelet, s(t) is the reflection impulse response of the ground and nj is uncorrelated noise. With the common assumption that s(t) is white, random, and stationary, estimates of the energy spectrum (or auto‐correlation function) of the pulse α(t) are obtained by statistical analysis of the multitrace record. The time‐domain pulse itself is then reconstituted under the assumption of minimum‐phase. Three techniques for obtaining the phase spectrum have been evaluated: (A) use of the Hilbert transform, (B) Use of the z‐transform, (C) a fast method based on inverting the least‐squares inverse of the wavelets, i.e. inverting the normal time‐domain deconvolution operator. Problems associated with these three methods are most acute when the z‐transform of α(t) has zeroes on or near the unit circle. Such zeroes result from oversampling or from highly resonant wavelets. The behaviour of the three methods when the energy spectra are perturbed by measurement errors is studied. It is concluded that method (A) is the best of the three. Examples of reconstituted pulses are given which illustrate the variability from trace‐to‐trace, from shot‐to‐shot, and from one shot‐point medium to another. There is reasonable agreement between the minimum‐phase pulses obtained by this statistical analysis of operational records and those estimated from measurements close to the source. However, this comparison incorporates a “fudge‐factor” since an allowance for absorption has to be made in order to attenuate the high frequencies present in the pulse measured close to the shot.

Attenuation Estimation From Correlated Sequences

“…Processing reduces multiple reflections, enhancing primary events. Absorption of seismic waves in the earth results in preferential attenuation of high frequencies with depth, and as stated earlier has a linear-with-frequency characteristic [17]. Processing of seismic data for attenuation estimation avoids any time-variant steps, meaning that no change in deconvolution operators or filters are allowed over the total time interval involved; the data discussed here was specially processed for attenuation estimation.…”

“…In the case when 2 is nonzero, its percentage effect on variance reduction can be calculated via 100 1 varfP j 2 = 0g 0 varfP j 2 g varfP j 2 = 0g : (17) This is illustrated in Fig. 2.…”

“…The acoustic attenuation coefficient of soft biological tissue has been observed to have a linear-with-frequency characteristic [7]. Likewise a linear form has also been justified in seismology [4], [17], and the ubiquitous linear assumption for attenuation is made in this correspondence. Let (f) = f say, where is a constant, so that log S 22 (f) S 11 (f) = c 0 21T f (1) where c = log c0 and the coefficient is called the logarithmic decrement and is measured in nepers (better known as the natural log of a voltage ratio).…”

An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method

“…There are three methods to obtain the phase spectrum: (1) Hilbert transform (or Kolmogoroff factorization) method, (2) z transform method, and (3) Wiener-Levinson inverse method. An overview of these methods can be found in WHITE and O'BRIEN (1974), LINES and ULRYCH (1977) and CLAERBOUT (1985).…”

“…WHITE and O'BRIEN (1974) suggest that, even for noisy environments, Hilbert transform method produces the best results for minimum phase wavelets. Therefore, Hilbert transform method, which operates in frequency domain, is used to obtain the minimum and zero phase wavelets in this study.…”