Difference between revisions of "Dictionary:Ricker wavelet"

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{{#category_index:R|Ricker wavelet}}
 
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(rik&#x2019; &#x2202;r) A zero-phase wavelet, the second derivative of the Gaussian function or the third derivative of the normal-probability density function. A Ricker wavelet is often used as a zero-phase embedded wavelet in modeling and synthetic seismogram manufacture. See Figure [[Special:MyLanguage/Dictionary:Fig_R-14|R-14]]. Named for Norman H. Ricker (1896&#x2013;1980), American geophysicist.
 
(rik&#x2019; &#x2202;r) A zero-phase wavelet, the second derivative of the Gaussian function or the third derivative of the normal-probability density function. A Ricker wavelet is often used as a zero-phase embedded wavelet in modeling and synthetic seismogram manufacture. See Figure [[Special:MyLanguage/Dictionary:Fig_R-14|R-14]]. Named for Norman H. Ricker (1896&#x2013;1980), American geophysicist.
  
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[[File:Segr14.jpg|center]]
 
[[File:Segr14.jpg|center]]
  
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FIG. R-14.  <b>Ricker wavelet</b>. (<b>a</b>) Time-domain and (<b>b</b>) frequency-domain representations.  
 
FIG. R-14.  <b>Ricker wavelet</b>. (<b>a</b>) Time-domain and (<b>b</b>) frequency-domain representations.  
  
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The amplitude ''f(t)'' of the Ricker wavelet with peak frequency ''f<sub>M'' at time ''t'' is given by,
 
The amplitude ''f(t)'' of the Ricker wavelet with peak frequency ''f<sub>M'' at time ''t'' is given by,
  
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:<math>f(t) = (1-2 \pi^2 f_{_{M}}^2 t^2) e^{-\pi^2 f_{_{M}}^2 t^2}</math>.
 
:<math>f(t) = (1-2 \pi^2 f_{_{M}}^2 t^2) e^{-\pi^2 f_{_{M}}^2 t^2}</math>.
  
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The frequency domain representation of the wavelet is given by,
 
The frequency domain representation of the wavelet is given by,
  
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<math>F(f) = \frac{2}{\sqrt{\pi}}\frac{f^{2}}{f_{M}^3}e^{-\frac{f^2}{f_{M}^2}}</math>  
 
<math>F(f) = \frac{2}{\sqrt{\pi}}\frac{f^{2}}{f_{M}^3}e^{-\frac{f^2}{f_{M}^2}}</math>  
  
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Where,  
 
Where,  
  
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:<math>T_{D}=\frac{\sqrt{6}}{\pi f_{M}} </math> and <math>T_{R}=T_{D}/\sqrt{3}</math>.
 
:<math>T_{D}=\frac{\sqrt{6}}{\pi f_{M}} </math> and <math>T_{R}=T_{D}/\sqrt{3}</math>.
  
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The mean frequency <math>f_{mean} =\frac{2}{\sqrt{\pi}}f_{M}</math> and the median frequency <math>f_{median}=1.08f_{M}</math>.  
 
The mean frequency <math>f_{mean} =\frac{2}{\sqrt{\pi}}f_{M}</math> and the median frequency <math>f_{median}=1.08f_{M}</math>.  
  
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Sometimes the period (somewhat erroneously referred to occasionally as the wavelength) is given as 1/''f'', but since it has mixed frequencies, this is not quite correct, and for some wavelets is not even a good approximation. In fact, the Ricker wavelet has its sidelobe minima at  
 
Sometimes the period (somewhat erroneously referred to occasionally as the wavelength) is given as 1/''f'', but since it has mixed frequencies, this is not quite correct, and for some wavelets is not even a good approximation. In fact, the Ricker wavelet has its sidelobe minima at  
  
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:<math>\pm \frac{\sqrt{3/2}}{f_{_{M}}\pi} </math>
 
:<math>\pm \frac{\sqrt{3/2}}{f_{_{M}}\pi} </math>
  
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These minima have the value
 
These minima have the value
  
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:<math>A_\mathrm{min} = -\frac{2}{e^{3/2}} </math>
 
:<math>A_\mathrm{min} = -\frac{2}{e^{3/2}} </math>
 
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Revision as of 16:05, 1 September 2017

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(rik’ ∂r) A zero-phase wavelet, the second derivative of the Gaussian function or the third derivative of the normal-probability density function. A Ricker wavelet is often used as a zero-phase embedded wavelet in modeling and synthetic seismogram manufacture. See Figure R-14. Named for Norman H. Ricker (1896–1980), American geophysicist.

Segr14.jpg

FIG. R-14. Ricker wavelet. (a) Time-domain and (b) frequency-domain representations.


The amplitude f(t) of the Ricker wavelet with peak frequency fM at time t is given by,

.

The frequency domain representation of the wavelet is given by,

Where,

and .

The mean frequency and the median frequency .

Sometimes the period (somewhat erroneously referred to occasionally as the wavelength) is given as 1/f, but since it has mixed frequencies, this is not quite correct, and for some wavelets is not even a good approximation. In fact, the Ricker wavelet has its sidelobe minima at

These minima have the value