# Two-length wavelets

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
Other languages:
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 ${\displaystyle \left(b_{0}{,\ }b_{1}\right)}$ with another two-length wavelet - namely, its reverse ${\displaystyle \left(b_{l}^{*}{\ ,\ }b_{0}^{*}\right)}$. 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 ${\displaystyle {|}b_{0}{|}\geq {|}b_{1}{|}}$ so that ${\displaystyle \left(b_{0}{,\ }b_{1}\right)}$ a minimum-delay two-length wavelet. Its Z-transform is ${\displaystyle B\left(z\right)=}$ ${\displaystyle b_{0}+b_{1}Z}$. The polynomial B(Z), which forms the Z-transform, is called a minimum-delay polynomial. The root (or zero) ${\displaystyle Z_{1}}$ of B(Z) is found by solving the equation ${\displaystyle b_{0}+}$ ${\displaystyle b_{1}Z=0}$. The solution of this equation gives the root as ${\displaystyle Z_{\rm {l}}=-b_{0}/b_{\rm {1}}}$, which we designate by ${\displaystyle Z_{\rm {l}}=\alpha e^{-i\theta }}$. Here, ${\displaystyle \alpha }$ represents the magnitude of the root and ${\displaystyle \theta }$ represents the angular frequency of the root.

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

 {\displaystyle {\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 ${\displaystyle \alpha }$. As a result, the coefficients ${\displaystyle \left(h_{0}{,\ }h_{\rm {l}}{,\ }h_{2}{,\ ..\ .}\right)}$ represent the inverse, which is a stable causal filter. In particular, H(Z) converges on the unit circle, so the Fourier transform ${\displaystyle H\left(e^{-i\omega }\right)}$ exists. For example, the stable causal inverse of (2, –1) is given by the coefficients in the expansion

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

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

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

 {\displaystyle {\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 (..., ${\displaystyle \left({\ }k_{-3}{,\ }k_{-2}{,\ }k_{-{\rm {l}}}\right)}$) 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.