Canonical representation
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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 canonical representation of a causal wavelet? 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/":): w_M be the minimum-phase wavelet with the same amplitude spectrum as that of the causal wavelet w. 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/":): w_M is called the minimum-phase counterpart of w. The canonical representation' states that any causal wavelet w can be represented as the convolution of its minimum-phase counterpart 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/":): w_M and a causal all-pass wavelet p; that 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/":): \begin{align} &w=w_M*p. \end{align} ()
Because the inverse of a minimum-phase wavelet is minimum phase, 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/":): W_M ^{ - 1} is minimum phase and hence is causal. From the canonical representation, we see that the inverse wavelet 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} &w^{-1}=w^{-1}_M*p^{-{ l}} . \end{align} ()
Two cases can occur. In the first case, the causal wavelet w is itself minimum phase. Then the causal all-pass wavelet is trivial, so the inverse wavelet is simply 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/":): w^{-{ 1}}=w^{-{ 1}}_M . In this case, the inverse wavelet is minimum phase and causal.
In the second case, the causal wavelet w is not minimum phase. Then the causal all-pass wavelet p is not trivial and therefore its 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/":): p^{-1}=p^R is a one-sided noncausal wavelet. In this case, the inverse 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/":): w^{-1} is a two-sided noncausal wavelet.
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| Autocorrelation | Zero-phase wavelets |
| 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
- Energy
- Autocorrelation
- Zero-phase wavelets
- Symmetric wavelets
- Ricker wavelet
- Appendix G: Exercises
External links
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