Ondículas de doble longitude

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
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


(23)

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


(24)

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).

Figure 2.  The causal inverse of the minimum-delay wavelet (2, –1).

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


(25)

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).

Figure 3.  The zero of the minimum-delay two-length wavelet and the zero of the corresponding zero of the maximum-delay two-length wavelet.

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


(26)

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


(27)

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).

Figure 4.  The anticausal inverse of the maximum-delay wavelet (–1, 2).

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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