# Two-length wavelets

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 7 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 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, $\alpha$ represents the magnitude of the root and $\theta$ represents the angular frequency of the root.

Because ${|}b_{0}{|}\geq {|}b_{\rm {l}}{|}$ , the magnitude $\alpha$ of the root $Z_{1}$ is greater than or equal to one. For example, the zero of the minimum-delay wavelet (2, –1) is $Z_{\rm {l}}=2$ . Because the magnitude $\alpha$ of the root is greater than one, the root lies outside the unit circle ${|}Z{|}=1$ . 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 $H\left(Z\right)=1/B\left(Z\right)$ is the Z-transform of the inverse of the wavelet $\left(h_{0}{,\ }h_{\rm {l}}{,\ }h_{2}{,\ ..\ .}\right)$ . The zero $Z_{1}$ becomes the pole of H(Z). If $\left(b_{0}{,\ }b_{1}\right)$ is a minimum-delay wavelet, then this pole lies outside the unit circle. As a result, we can form the power series

 {\begin{aligned}H\left(z\right)=h_{0}+h_{\rm {l}}Z+h_{2}Z^{2}+\dots ,\end{aligned}} (23)

which converges at every point inside the circle of radius $\alpha$ . As a result, the coefficients $\left(h_{0}{,\ }h_{\rm {l}}{,\ }h_{2}{,\ ..\ .}\right)$ represent the inverse, which is a stable causal filter. In particular, H(Z) converges on the unit circle, so the Fourier transform $H\left(e^{-i\omega }\right)$ 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}} (24)

Thus, we see that the inverse $\left(h_{0}{,\ }h_{\rm {l}}{,\ }h_{2}{,\ ..\ .}\right)$ 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 ${\rm {let|}}b_{0}{|}\geq {|}b_{\rm {l}}{|}$ so that the reverse two-length wavelet $\left(b_{\rm {l}}^{*}{,\ }b_{0}^{*}\right)$ is a maximum-delay wavelet. Its Z-transform is $C\left(Z\right)=b_{\rm {l}}^{*}+b_{0}^{*}Z$ . The polynomial C(Z), which forms the Z-transform, is called a maximum-delay polynomial. The root (or zero) $Z_{2}$ of C(Z) is found by solving the equation $b_{\rm {l}}^{*}+b_{0}^{*}Z_{2}=0$ . The solution of this equation gives the root as $Z_{2}=-b_{\rm {l}}^{*}/b_{0}^{*}$ . Because

 {\begin{aligned}Z_{1}Z_{2}^{*}=\left(-b_{0}/b_{\rm {l}}\right){\left(-b_{1}^{*}/b_{0}^{*}\right)}^{*}=1,\end{aligned}} (25)

it follows that $Z_{2}=1/Z_{\rm {l}}^{*}$ . We recall that we wrote the expression $Z_{1}=\alpha e^{-i\theta }$ for the root of the minimum-delay wavelet. Hence, the root of the corresponding maximum-delay two-length wavelet is $Z_{2}={1/}{\left(\alpha e^{-i\theta }\right)}^{*}={\alpha }^{-1}e^{-i\theta }$ . Thus, the magnitude ${\alpha }^{-1}$ 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 $\theta$ (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 $K\left(Z\right)={\rm {l}}/C\left(Z\right)$ is the Z-transform of the inverse of the maximum-delay wavelet $\left(b_{1}^{*}{,\ }b_{0}^{*}\right)$ . The zero $Z_{2}$ becomes the pole of H(Z). Because this pole lies inside the unit circle, we can form the Laurent series

 {\begin{aligned}&K\left(z\right)=k_{-{\rm {l}}}Z^{-{\rm {l}}}+k_{-2}Z^{-2}+k_{-3}Z^{-3}+\dots ,\end{aligned}} (26)

which converges at every point outside the circle of radius ${\alpha }^{-{\rm {l}}}$ . As a result, the inverse (..., $k_{-3},k_{-2},k_{-{\rm {l}}}$ ) represents a stable anticausal filter. In particular, K(Z) converges on the unit circle, so the Fourier transform $K\left(e^{-i\omega }\right)$ exists. For example, the stable anticausal inverse of (–1, 2) is given by the coefficients in the expansion

 {\begin{aligned}{\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{aligned}} (27)

We see that the inverse (..., $\left({\ }k_{-3}{,\ }k_{-2}{,\ }k_{-{\rm {l}}}\right)$ ) 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.