Two-length wavelets

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

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 $ \left(b_{0}{,\ }b_{1}\right) $ with another two-length wavelet - namely, its reverse $ \left(b_{l}^{*}{\ ,\ }b_{0}^{*}\right) $. 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 $ {|}b_{0}{|}\geq {|}b_{1}{|} $ so that $ \left(b_{0}{,\ }b_{1}\right) $ a minimum-delay two-length wavelet. Its Z-transform is $ B\left(z\right)= $ $ b_{0}+b_{1}Z $. The polynomial B(Z), which forms the Z-transform, is called a minimum-delay polynomial. The root (or zero) $ Z_{1} $ of B(Z) is found by solving the equation $ b_{0}+ $ $ b_{1}Z=0 $. The solution of this equation gives the root as $ Z_{\rm {l}}=-b_{0}/b_{\rm {1}} $, which we designate by $ Z_{\rm {l}}=\alpha e^{-i\theta } $. Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha represents the magnitude of the root and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta represents the angular frequency of the root.

Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {|}b_0{|}\ge {|}b_{{\rm l}}{|} , the magnitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha of the root Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{1} is greater than or equal to one. For example, the zero of the minimum-delay wavelet (2, –1) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{{\rm l}}=2 . Because the magnitude $ \alpha $ of the root is greater than one, the root lies outside the unit circle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {|}Z{|}=1 . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H\left(Z\right)=1/B\left(Z\right) is the Z-transform of the inverse of the wavelet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(h_0{,\ }h_{{\rm l}}{,\ }h_{2}{,\ ..\ .}\right) . The zero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{1} becomes the pole of H(Z). If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(b_0{,\ }b_{1}\right) is a minimum-delay wavelet, then this pole lies outside the unit circle. As a result, we can form the power series

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} H\left(z\right)=h_0+h_{{\rm l}} Z+h_{2}Z^{2}+ \dots, \end{align}

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which converges at every point inside the circle of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha . As a result, the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(h_0{,\ }h_{{\rm l}}{,\ }h_{2}{,\ ..\ .}\right) represent the inverse, which is a stable causal filter. In particular, H(Z) converges on the unit circle, so the Fourier transform Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H\left(e^{-i\omega }\right) exists. For example, the stable causal inverse of (2, –1) is given by the coefficients in the expansion

$ {\begin{aligned}{\frac {\rm {l}}{2-Z}}={\frac {\rm {l}}{2}}+{\frac {\rm {l}}{4}}Z+{\frac {\rm {l}}{8}}Z^{2}+{\frac {\rm {l}}{16}}Z^{3}+\dots .\end{aligned}} $

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Thus, we see that the inverse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(h_0{,\ }h_{{\rm l}}{,\ }h_{2}{,\ ..\ .}\right) 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm let|}b_0{|}\ge {|}b_{{\rm l}}{|} so that the reverse two-length wavelet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(b^{*}_{{\rm l}}{,\ }b^{*}_0\right) is a maximum-delay wavelet. Its Z-transform is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C\left(Z\right)=b^{*}_{{\rm l}}+b^{*}_0Z . The polynomial C(Z), which forms the Z-transform, is called a maximum-delay polynomial. The root (or zero) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{2} of C(Z) is found by solving the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b^{*}_{{\rm l}}+b^{*}_0Z_{2}=0 . The solution of this equation gives the root as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{2}=-b^{*}_{{\rm l}}/b^{*}_0 . Because

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} Z_{1}Z^{*}_{2}=\left(-b_0/b_{{\rm l}} \right){\left(-b^{*}_{1}/b^{*}_0\right)}^{*}=1, \end{align}

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it follows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{2}=1/Z^{*}_{{\rm l}} . We recall that we wrote the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{1}=\alpha e^{-i\theta } for the root of the minimum-delay wavelet. Hence, the root of the corresponding maximum-delay two-length wavelet is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{2}={1/}{\left(\alpha e^{-i\theta }\right)}^{*}= {\alpha }^{-1}e^{-i\theta } . Thus, the magnitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\alpha }^{-1} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K\left(Z\right)={\rm l}/C\left(Z\right) is the Z-transform of the inverse of the maximum-delay wavelet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(b^{*}_{1}{,\ }b^{*}_0\right) . The zero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z_{2} becomes the pole of H(Z). Because this pole lies inside the unit circle, we can form the Laurent series

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} &K\left(z\right)=k_{-{\rm l}} Z^{-{\rm l}}+k_{-2}Z^{-2}+k_{-3}Z^{-3}+ \dots, \end{align}

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which converges at every point outside the circle of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\alpha }^{-{\rm l}} . As a result, the inverse (..., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k_{-3},k_{-2},k_{-{\rm l}} ) represents a stable anticausal filter. In particular, K(Z) converges on the unit circle, so the Fourier transform Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K\left(e^{-i\omega }\right) exists. For example, the stable anticausal inverse of (–1, 2) is given by the coefficients in the expansion

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{{\rm l}} {2Z-1}=\frac{{\rm l}}{2}Z^{-1}+\frac{{\rm l}}{{4}}Z^{-2}+\frac{{\rm l}}{{8}}Z^{-3}+\frac{{1}}{{16}}Z^{-{4}}+ .\dots \end{align}

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We see that the inverse (..., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\ }k_{-3}{,\ }k_{-2}{,\ }k_{-{\rm l}}\right) ) 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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Table of Contents (book)

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

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