The autocorrelation function is symmetric and two-sided with a central positive maximum. Except in the case of periodicity, the Autocorrelation will damp out symmetrically in both directions from this central maximum. Moreover, it will damp out with a certain speed. If we perform a Fourier analysis on an autocorrelation, we will find the following: The autocorrelation curve can be represented as a sum of cosine curves of different frequencies and different amplitudes. At the central point of the autocorrelation, all of those cosine curves will be in phase - that is, the crest of each cosine wave will occur at the maximum (or central) value of the autocorrelation. No negative cosine curves (i.e., those that are out of phase) occur, nor do any sine curves occur. Thus, there are no phase differences; the phase is zero for every frequency. As a result, any autocorrelation function is a zero-phase waveform. A zero-phase wavelet is always two-sided.
The resolution of reflected events on a seismic trace (i.e., their distinct separation in time so that they stand out and can be identified visually as individual events) is determined by the length properties of the seismic wavelet attached to each event. A quantitative measure of the effective length of a wavelet is the second-order moment (or spread parameter) known as the variance. That moment is zero for a unit spike. The Fourier analysis of a spike yields cosine curves of all frequencies. All the cosine curves have the same amplitude, and all are in phase with their crests at time zero. A spike is the sharpest of all the zero-phase wavelets.
In seismic work, we cannot attain an infinitely large range of frequencies; instead, we must work in a narrower range governed by the frequency characteristics of the earth and our instruments. We also must take into account our ultimate objectives and the economic costs involved. With all of those limitations, we find ourselves working in a relatively narrow frequency band, such as the band from 5 Hz to 50 Hz.
Within such a band, what sort of wavelet will have the smallest spread? We want to retain the sharpness of the spike as much as possible, so we will retain the cosine waves in this range of frequencies. Because all of the retained cosine waves are still in phase with their crests at time zero, the resulting wavelet will be symmetric and zero phase. Because all of the retained cosine waves will have the same amplitude, the amplitude spectrum will be level (flat) and will cover the entire available frequency range. The resulting wavelet will have the smallest value of the variance for the given frequency band. It will have a sharp central peak. It will oscillate, however, and such secondary oscillations (side lobes) are undesirable. We can reduce these secondary oscillations (side lobes) at the expense of increasing the width of the central peak. The result will be a zero-phase wavelet whose amplitude spectrum no longer will be absolutely flat in the available frequency range but still will be smooth and broad.
We can conclude, therefore, that autocorrelation functions (which are necessarily zero phase) are the best choice for interpreter wavelets - especially autocorrelation functions with smooth and broad amplitude spectra. An additional argument for broadband zero-phase wavelets is the fact that for a given amplitude spectrum, the zero-phase wavelet has the largest central amplitude. The reason is that a zero-phase wavelet, in its Fourier analysis, is made up only of cosine waves in phase at its central point. There are no negative cosine waves and certainly no sine waves, both of which would reduce the peak amplitude. Hence, in wavelet processing, we transform the one-sided reflection wavelets (i.e., one-sided with respect to their arrival times) on the received seismic trace to their zero-phase counterparts (i.e., zero-phase wavelets with respect to their arrival times). These zero-phase interpreter wavelets maximize the detectability of the reflected-event arrival times.
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Also in this chapter
- Fourier transform
- Delay: Minimum, mixed, and maximum
- Two-length wavelets
- Illustrations of spectra
- Delay in general
- Canonical representation
- Symmetric wavelets
- Ricker wavelet
- Appendix G: Exercises