Arthur E. Barnes

Instantaneous frequency and amplitude at the envelope peak of a constant‐phase wavelet

Robertson and Nogami (1984) have shown that the instantaneous frequency at the peak of a zero‐phase Ricker wavelet is exactly equal to that wavelet’s average Fourier spectral frequency weighted by its amplitude spectrum. Bodine (1986) gave an example which shows this is also true for constant‐phase bandpass wavelets. Here I prove that this holds for any constant‐phase wavelet. I then develop an equation expressing this quantity as a function of propagation time through an attenuating medium. A corresponding equation is derived for the amplitude of the envelope peak. Taken together, these may aid in the analysis of seismic data as suggested by Robertson and Nogami (1984), Bodine (1986), and Robertson and Fisher (1988).

Another look at NMO stretch

The normal moveout (NMO) correction is applied to seismic reflection data to transform traces recorded at non-zero offset into traces that appear to have been recorded at zero offset; this introduces undesirable distortions called NMO stretch (Buchholtz, 1972). NMO stretch must be understood because it lengthens waveforms and thereby reduces resolution. Buchholtz (1972) gives a qualitative assessment of NMO stretch, Dunkin and Levin (1973) derive its effect on the amplitude spectra of narrow waveforms, while Yilmaz (1987, p. 160) considers its effect on dominant frequencies. These works are approximate and do not show how spectral distortions vary through time.

Weighted average seismic attributes

Local weighted averaging of instantaneous seismic attributes improves their interpretability by removing spikes and reducing rapid and confusing variations. Averaging in a window weighted by the instantaneous power produces a local measure that equals a Fourier spectral average, facilitating quantitative analysis. Local 1-D frequency and bandwidth are scalars, but local 2-D and 3-D seismic attributes derive from average vector wavenumbers, which may require velocity information. The direction of the average vector wavenumber provides average dip and azimuth, and its magnitude provides a measure of average wavelength or frequency. A related measure of bandwidth is a scalar in all dimensions; it includes contributions from both instantaneous bandwidth and the variance of instantaneous frequency or wavenumber. It quantifies data similarity.

Shaded relief seismic attribute

Shaded relief is employed throughout geophysics to display digital data as illuminated apparent topography. Such displays facilitate geologic understanding because apparent topography often suggests true geology. First developed for elevation data (Batson et al., 1975), shaded relief has long been used with gravity and magnetic data (Paterson and Reeves, 1985; Kowalik and Glenn, 1987), it is a natural product of synthetic aperture radar and side-scan sonar, and it is routinely applied to interpreted seismic horizons (Hoetz and Watters, 1992).

A filter to improve seismic discontinuity data for fault interpretation

Seismic-discontinuity attributes quantify the degree to which neighboring seismic traces vary from each other (Bahorich and Farmer, 1995). They detect abrupt lateral changes in seismic data character caused by faults, diapirs, stratigraphic changes, and noise. An important application is to highlight fault discontinuities for seismic data interpretation. For this purpose, other discontinuities are considered noise and should be removed (Ashbridge et al., 2000; Pedersen et al., 2002; AlBinHassan and Marfurt, 2003).

The complex seismic trace made simple

Many geophysicists find the complex seismic trace to be complex. With its Hilbert transforms and imaginary traces, the complex seismic trace may well seem unnatural, a kind of mathematical trick. Yet we geophysicists find Fourier transforms, with their complex exponentials and negative frequencies, natural and reasonable. This is because we intuitively understand what Fourier transforms are all about, but we often lack comparable understanding of complex traces. And without intuitive understanding the complex trace seems more difficult than it really is.

On: “3-D instantaneous frequency used as a coherency/continuity parameter to interpret reservoir compartment boundaries across an area of complex turbidite deposition,” (B. A. Hardage et al., GEOPHYSICS, Vol. 63, 1520–1531)

Hardage et al. are to be commended for a thoughtful and valuable study on instantaneous frequency applied to reservoir characterization. Of particular note is the observation that anomalous instantaneous frequencies often correspond to lower seismic coherency and form interpretable patterns. This observation has an interesting background.

Pulse distortion in depth migration; discussion and reply

Tygel et al. have written an excellent and rigorous discussion of pulse distortion in seismic reflection data caused by prestack depth migration. Such distortion is easily understood by recognizing that it is more or less the same effect as normal moveout (NMO) stretch combined with frequency shifting due to poststack time migration.

Teaching yes, practice no

The method introduced by B.J. Evans for accomplishing deterministic deconvolution by long division is the method of polynomial division in a different guise. Evans’ notation is nonetheless intriguing and may have considerable merit as a teaching device. But for deconvolving geophysical data, long division is impractical because it can fail disastrously. This is shown empirically in the following.

Response by Arthur E. Barnes to the reply by the authors

I appreciate the thoughtful and thorough response given by Tygel et al. They point out that even for a single dipping reflector imaged by a single non‐zero offset raypath, pulse distortion caused by “standard processing” (NM0 correction‐CMP sort‐stack‐time migration) and pulse distortion caused by prestack depth migration are not really the same, because the reflecting point is mispositioned in standard processing. Within a CMP gather, this mispositioning increases with offset, giving rise to “CMP smear.” CMP smear degrades the stack, introducing additional pulse distortion. Where i‐t is significant, and where lateral velocity variations or reflection curvature are large, such as for complex geology, the pulse distortion of standard processing can differ greatly from that of prestack depth migration.