# Two-length wavelets

Series | Geophysical References Series |
---|---|

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

Chapter | 7 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | 9781560801481 |

Store | SEG Online Store |

Two-length wavelets are either minimum-delay or maximum-delay wavelets. We can pair every two-length wavelet with another two-length wavelet - namely, its reverse . Such a pair is called a dipole. One of the members of the dipole is a minimum-delay wavelet and the other is a maximum-delay wavelet. The minimum-delay wavelet is the one that has the larger coefficient (in magnitude) at the front, whereas the maximum-delay wavelet is the one that has the smaller coefficient (in magnitude) at the front.

An example of a dipole is (2, 1) and (1, 2). For this dipole, the minimum-delay wavelet is (2, 1) and the maximum-delay wavelet is (1, 2). Another example is the dipole (*i*, 0.5) and (0.5, – *i*). For that dipole, the minimum-delay wavelet is (*i*, 0.5) and the maximum-delay wavelet is (0.5, – *i*).

What is the significance of the root of the *Z*-transform of a two-length wavelet? Let so that a minimum-delay two-length wavelet. Its *Z*-transform is . The polynomial *B*(*Z*), which forms the *Z*-transform, is called a *minimum-delay polynomial*. The root (or zero) of *B*(*Z*) is found by solving the equation . The solution of this equation gives the root as , which we designate by . Here, represents the magnitude of the root and represents the angular frequency of the root.

Because , the magnitude of the root is greater than or equal to one. For example, the zero of the minimum-delay wavelet (2, –1) is . Because the magnitude of the root is greater than one, the root lies outside the unit circle . In general, the zero (or root) of the *Z*-transform of a two-length minimum-delay wavelet lies outside (or on) the unit circle.

The reciprocal is the *Z*-transform of the inverse of the wavelet . The zero becomes the pole of *H*(*Z*). If is a minimum-delay wavelet, then this pole lies outside the unit circle. As a result, we can form the power series

**(**)

which converges at every point inside the circle of radius . As a result, the coefficients represent the inverse, which is a stable causal filter. In particular, *H*(*Z*) converges on the unit circle, so the Fourier transform exists. For example, the stable causal inverse of (2, –1) is given by the coefficients in the expansion

**(**)

Thus, we see that the inverse is the damped (in the direction of positive time) geometric series (0.5, 0.25, 0.125, 0.0625, …) (Figure 2).

Again, let so that the reverse two-length wavelet is a maximum-delay wavelet. Its *Z*-transform is . The polynomial *C*(*Z*), which forms the *Z*-transform, is called a *maximum-delay polynomial*. The root (or zero) of *C*(*Z*) is found by solving the equation . The solution of this equation gives the root as . Because

**(**)

it follows that . We recall that we wrote the expression for the root of the minimum-delay wavelet. Hence, the root of the corresponding maximum-delay two-length wavelet is . Thus, the magnitude of the root of the maximum-delay wavelet is the reciprocal of the magnitude of the root of the corresponding minimum-delay wavelet. Because the root of the minimum-delay wavelet lies outside (or on) the unit circle, it follows that the root of the maximum-delay wavelet lies inside (or on) the unit circle. It is important to observe that both the roots have the same angular frequency (Figure 3).

The reciprocal is the *Z*-transform of the inverse of the maximum-delay wavelet . The zero becomes the pole of *H*(*Z*). Because this pole lies inside the unit circle, we can form the Laurent series

**(**)

which converges at every point outside the circle of radius . As a result, the inverse (..., ) represents a stable anticausal filter. In particular, *K*(*Z*) converges on the unit circle, so the Fourier transform exists. For example, the stable anticausal inverse of (–1, 2) is given by the coefficients in the expansion

**(**)

We see that the inverse (..., ) is the damped (in the direction of negative time) geometric series (0.0625, 0.125, 0.25, 0.5) (Figure 4).

What are equal-delay wavelets? Any two-length wavelet whose root is on the unit circle is both a minimum-delay and a maximum-delay wavelet. Such a two-length wavelet is called an *equal-delay wavelet. *In other words, an equal-delay wavelet is one that is both a minimum-delay and a delay wavelet. For such a wavelet, the roots of its *Z*-transform lie on the unit circle. It might be said that an equal-delay wavelet represents two wavelets in one.

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Delay: Minimum, mixed, and maximum | Illustrations of spectra |

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Frequency | Synthetics |

## Also in this chapter

- Wavelets
- Fourier transform
- Z-transform
- Delay: Minimum, mixed, and maximum
- Illustrations of spectra
- Delay in general
- Energy
- Autocorrelation
- Canonical representation
- Zero-phase wavelets
- Symmetric wavelets
- Ricker wavelet
- Appendix G: Exercises