Energy - book
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| 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 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,b_{{\rm l}}, b_{2}, \cdots \} is given by
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} &b^{2}_0+b^{2}_{1}+b^{2}_{2}+\dots . \end{align} ()
If the coefficients of the wavelet are complex, the energy is given by
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} &b_0\ b^*_0+b_{1}{\ }b^*_{1}+b_{2}b^{*}_{2}+\dots. \end{align} ()
(Note: The asterisk in the superscript position indicates the complex conjugate of the quantity to which it is attached. For example, 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/":): b_0=u+iv , then 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=u-iv 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/":): b_0 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=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}} $ ()
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 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}{,\ }b_{2},{\ }b_{3}\right\}. . The energy buildup for time 0, denoted by 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/":): p_0 , 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/":): b^{2}_0 . Because energy buildup is cumulative, the energy buildup 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/":): p_{1} at time 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/":): b^{2}_0+b^{2}_{{\rm l}} , and so on. That is, $ p_{0}=b_{0}^{2},p_{\rm {l}}=b_{0}^{2}+ $ 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^{2}_{{\rm l}}=p_0+b^{2}_{{\rm l}},p_{2}=b^{2}_0+b^{2}_{{\rm l}}+b^{2}_{2}=p_{{\rm l}}+b^{2}_{2} 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/":): p_{3}=b^{2}_0+b^{2}_{1}+b^{2}_{2}+b^{2}_{3}=p_{2}+b^{2}_{3} . The last value of the energy buildup, in this case 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/":): p_3 , is the total energy of the wavelet. (If the wavelet is complex, we would use 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_0b^{*}_0 instead of 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^{2}_0 , 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
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/":): p_0=4
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/":): p_{1}=4+l=5
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} &p_{2}=4+1+0.25=5.25 \end{align} ()
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/":): 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
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/":): p_0=0.0625
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/":): p_{1}=0.0625+0.25=0.3125
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} &p_{2}=0.0625+0.25+1=1.3125 \end{align} ()
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/":): 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
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/":): p_0=1
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/":): p_{1}=1+4=5
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} &p_{2}=1+4+0.0625={5.0625} \end{align} ()
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/":): 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.
Continue reading
| Previous section | Next section |
|---|---|
| Delay in general | Autocorrelation |
| Previous chapter | Next chapter |
| Frequency | Synthetics |
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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