The study of geophysics requires a detailed knowledge of wavelets. A wavelet is a waveform whose value is negligibly small except in some finite region of the time scale. Accordingly, there is a time interval (or epoch) somewhere along the infinite time scale over which we can say that the wavelet exists.
Two important types of symmetric wavelets occur that, in a sense, represent opposite extremes. One type is the rectangular wavelet, and the other is the sine cardinal (or sinc) wavelet. (The sine cardinal function is usually called the sinc function.) The rectangular wavelet has a sinc spectrum, and the sinc wavelet has a rectangular spectrum.
The rectangular wavelet with unit-time duration (i.e., with unit width) has unit height; it is centered at t = 0. Its spectrum is given by the sinc function. This sinc spectrum is equal to unity at f = 0; it is equal to zero for f = … –3, –2, –1, 1, 2, 3, … , and it falls off asymptotically in the same manner that falls off. The left side of Figure 19 shows two examples of the rectangular wavelet, one with width 1 and the other with width 2. The right side of Figure 19 shows those wavelets’ respective frequency spectra. We see that the frequency spectrum of a rectangular wavelet is real valued (that is, it has no imaginary component). We also see that the frequency spectrum of a rectangular wavelet is symmetric about the origin. So far, all is well. However, the frequency spectrum has regions of negative values. It is well-known that a negative value is 180° out of phase with the corresponding positive value.
Thus, in the regions where the frequency spectrum is negative, the phase spectrum is equal to 180°. Of course, in the regions where the frequency spectrum is positive, the phase spectrum is equal to 0°. Therefore, the rectangular wavelet is not a zero-phase wavelet. It is one to be avoided in geophysics.
The sinc wavelet for width 2 is equal to unity at t = 0, is equal to zero for t = …, –3, –2, –1, 1, 2, 3, …, and falls off asymptotically as . Its spectrum is rectangular with unit bandwidth and unit height, and it is centered at f = 0. The left side of Figure 20 shows two examples of the sinc wavelet. The right side of Figure 20 shows the wavelets’ respective frequency spectra. We see that the frequency spectrum of a sinc wavelet is real valued, symmetric about the origin, and zero phase.
The rectangular wavelet is useful for theoretical purposes; however, it is unsuitable for many practical problems because of the strong side lobes in its frequency spectrum. The rectangular wavelet, which is symmetric, is nonzero phase. The side lobes make the effective bandwidth of the rectangular wavelet very wide. To be suitable for most practical purposes, a wavelet should have the narrowest possible frequency spectrum commensurate with the need for the wavelet to have adequate time resolution.
The sine cardinal (sinc) wavelet, on the other hand, does have a frequency spectrum restricted to a finite band. The sinc wavelet, which is also symmetric, is zero phase. However, the sinc wavelet has a strong precursor and a strong tail that will overlap into the regions of adjacent wavelets. This lack of time resolution will deteriorate any detection method.
Let us now take a rectangular wavelet of width 0.01 and modulate it (i.e., multiply it) by a sinusoid of frequency Hz. Figure 21 shows the result. This modulated rectangular wavelet is a nonzero phase wavelet.
No wavelet exists that can be restricted to a finite time interval with a spectrum simultaneously restricted to a finite frequency band. However, various tapered wavelets are restricted to a finite time interval and do have spectra with much lower side lobes than is the case for the spectrum of the rectangular wavelet. A tapered wavelet must rise in amplitude less sharply than the rectangular wavelet does so its spectrum will contain less high-frequency energy. In fact, the occurrence of sharp discontinuities in a wavelet’s shape or derivatives should be avoided whenever possible, to achieve reduced spectral energy in spectral side-lobe regions.
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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
- Zero-phase wavelets
- Ricker wavelet
- Appendix G: Exercises