Various types of wavelets are used in practice (Hubral and Tygel, 1989). Accurate estimation of wavelets is essential in seismic processing (Ulrych et al., 1995). A wavelet commonly used in generating synthetic seismograms is the Ricker wavelet, named after Norman Ricker (1896–1980). The Ricker wavelet is noncausal and symmetric about its origin time. Furthermore, the Ricker wavelet has the important quality that it is a zero-phase signal. (A zero-phase signal is also an autocorrelation; see above.) In continuous time, the Ricker wavelet is given by the second derivative of the Gaussian function (Clay, 1990, p. 285):
The parameter b is called the wavelet breadth and measures separation (in time) of the wavelet’s two valleys, which occur on each side of the central lobe. Let us sample the infinitely long continuous Ricker wavelet of breadth 10, at a time spacing of one unit, and make it of finite length, say, 23. The resulting discrete 23-length Ricker wavelet (Figure 22) is
What is the Z-transform of the discrete Ricker wavelet? The Z-transform of a noncausal wavelet involves both positive and negative powers of Z. For example, the Z-transform of the discrete version of the continuous Ricker wavelet introduced above is
If we multiply R(Z) by , we obtain a polynomial. This polynomial is the Z-transform of the Ricker wavelet, shifted just enough to make the resulting wavelet causal. The resulting wavelet is called the shifted Ricker wavelet.
Going back to high-school algebra days, we recall that as soon as a teacher presented a polynomial to a class, the students were asked to factor it. The same holds for Z-transforms - we factor them. Factoring yields 22 roots. Because the Ricker wavelet coefficients are real, all complex roots occur in complex-conjugate pairs. The absolute values of these roots are
We see that 12 of the roots have an absolute value of unity. Any root with absolute-value unity corresponds to a two-length equal-delay wavelet in the composition of the discrete Ricker wavelet being considered. We do not like equal-delay wavelets for reasons put forth earlier in this chapter. We want all the two-length constituent wavelets to be either strictly minimum-delay wavelets or strictly maximum-delay wavelets (Robinson et al., 1978). We therefore modify the Ricker wavelet by adding a little bit of white noise to the center value of the autocorrelation function - say, 1% of this value - which is an operation called prewhitening. In other words, we multiply the center value of this autocorrelation function by 1.01. The modified Ricker wavelet now becomes
We note that only the center value has been altered: it is now 1.01 instead of 1.00. When the modified Ricker wavelet is plotted, it looks so much like the true Ricker wavelet of Figure 22 that it is not worthwhile to show it.
Next let us compute the roots of this new Z-transform of the modified Ricker wavelet. The roots’ absolute values are now
No root has an absolute value of unity, and hence none of the two-length constituent wavelets is an equal-delay wavelet.
What is the component signal for the modified Ricker wavelet? Consider the above absolute values of the roots of the modified Ricker wavelet: one-half of them are greater than one, whereas the other half are less than one. Take the 11 roots with absolute values greater than one. With these 11, form a polynomial. The coefficients of this polynomial make up what we shall call a component wavelet (Robinson and Treitel, 1985). This component wavelet is necessarily a minimum-delay wavelet because (by construction) each root of its Z-transform polynomial has magnitude greater than one (Figure 23). In Figure 23, we also show the time-reverse of this component wavelet, which is necessarily a maximum-delay wavelet.
- Hubral, P., and M. Tygel, 1989, Analysis of the Rayleigh pulse: Geophysics, 54, 654–658.
- Ulrych, T. J., R. D. Velis, and M. D. Sacchi, 1995, Wavelet estimation revisited: The Leading Edge, 14, 1139–1143.
- Clay, C. S., 1990, Elementary exploration seismology: Prentice Hall.
- Robinson, E. A., D. Loewenthal, and S. Treitel, 1978, Numerical testing of minimum-delay, positive- real, and positive-definite digital filters: Journal of Computational Physics, 29, 421–430.
- Robinson, E. A., and S. Treitel, 1985, The right-half autocorrelation theorem, in O. D. Anderson, J. K. Ord, and E. A. Robinson, eds., Time series analysis, theory and practice 6: North Holland Publishing Co., 105–132.
|Previous section||Next section|
|Symmetric wavelets||Appendix G: Exercises|
|Previous chapter||Next chapter|
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
- Symmetric wavelets
- Appendix G: Exercises