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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

What is the energy of a wavelet? Wavelets with finite length die out completely (i.e., become zero) after a certain age. For example, the wavelet (4, 2, 1) dies out at time index 3. It is possible to have infinite-length wavelets, but for stability, we must require that they have finite energy. The energy of the infinite-length causal wavelet $\{b_{0},b_{\rm {l}},b_{2},\cdots \}$ is given by

${\begin{aligned}&b_{0}^{2}+b_{1}^{2}+b_{2}^{2}+\dots .\end{aligned}}$

(39)

If the coefficients of the wavelet are complex, the energy is given by

${\begin{aligned}&b_{0}\ b_{0}^{*}+b_{1}{\ }b_{1}^{*}+b_{2}b_{2}^{*}+\dots .\end{aligned}}$

(40)

(Note: The asterisk in the superscript position indicates the complex conjugate of the quantity to which it is attached. For example, if $b_{0}=u+iv$, then $b_{0}^{*}=u-iv$ and $b_{0}$ $b_{0}^{*}=u^{2}+v^{2}$). An example of an infinite-length causal wavelet is

${\begin{aligned}&b={\ }\left({1,\ }{\frac {1}{2}}{\ ,\ }{\frac {1}{4}}{\ ,\ }{\frac {1}{8}}{\ ,\ }{\frac {1}{16}}{,\ }\cdots \right)=\left({1,0.5,0.25,0.125,0.0625,\ }\dots \right),\end{aligned}}$

(41)

where the first coefficient, 1, is the coefficient for time index 0; the next coefficient, 1/2, is the coefficient for time index 1; and so on. Because of the stability property, the magnitudes of the coefficients asymptotically approach zero as time increases.

What is energy buildup (or partial energy)? The energy distribution of a wavelet is displayed by its energy buildup (or partial energy). Consider the (real) wavelet $\left\{b_{0}{,\ }b_{1}{,\ }b_{2},{\ }b_{3}\right\}.$. The energy buildup for time 0, denoted by $p_{0}$, is $b_{0}^{2}$. Because energy buildup is cumulative, the energy buildup $p_{1}$ at time 1 is $b_{0}^{2}+b_{\rm {l}}^{2}$, and so on. That is, $p_{0}=b_{0}^{2},p_{\rm {l}}=b_{0}^{2}+$ $b_{\rm {l}}^{2}=p_{0}+b_{\rm {l}}^{2},p_{2}=b_{0}^{2}+b_{\rm {l}}^{2}+b_{2}^{2}=p_{\rm {l}}+b_{2}^{2}$ and $p_{3}=b_{0}^{2}+b_{1}^{2}+b_{2}^{2}+b_{3}^{2}=p_{2}+b_{3}^{2}$. The last value of the energy buildup, in this case $p_{3}$, is the total energy of the wavelet. (If the wavelet is complex, we would use $b_{0}b_{0}^{*}$ instead of $b_{0}^{2}$, and so on.)

Let us now describe minimum delay in terms of energy buildup. Another way of describing the delay properties of wavelets is by means of the energy buildup. For example, the energy buildup of the minimum-delay wavelet (2, 1, 0.5, 0.25) is

$p_{0}=4$

$p_{1}=4+l=5$

${\begin{aligned}&p_{2}=4+1+0.25=5.25\end{aligned}}$

(42)

$p_{3}=4+1+0.25+0.0625=5.3125.$

On the other hand, the energy buildup of the maximum-delay four-length wavelet (0.25, 0.5, 1, 2) is

$p_{0}=0.0625$

$p_{1}=0.0625+0.25=0.3125$

${\begin{aligned}&p_{2}=0.0625+0.25+1=1.3125\end{aligned}}$

(43)

$p_{3}=0.0625+0.25+1+4=5.3125.$

By comparing the two energy-buildup curves, we see that the energy buildup of the maximum-delay wavelet never exceeds that of the minimum-delay wavelet. We would expect this behavior from the way we constructed the two wavelets. The minimum-delay wavelet is the one with the energy concentrated at the front, and the maximum-delay wavelet is the one with the energy concentrated at the end.

The energy-buildup curves of the mixed-delay (N + 1)-length wavelets in the suite lie between the energy-buildup curve of the minimum-delay wavelet and that of the maximum-delay (N + 1)-length wavelet of the suite. That is, the mixed-delay (N + 1)-length wavelets have their energy concentrated between the two extremes. Thus, the mixed-delay wavelet (1, 2, 0.25, 0.5) has energy buildup

$p_{0}=1$

$p_{1}=1+4=5$

${\begin{aligned}&p_{2}=1+4+0.0625={5.0625}\end{aligned}}$

(44)

$p_{3}=1+4+0.0625+0.25=5.3125,$

and this curve lies between the energy-buildup curve of the minimum-delay wavelet and that of the maximum-delay four-length wavelet of the suite, as illustrated by Table 4.

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Table of Contents (book)

Also in this chapter

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

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