# Dictionary:Ricker wavelet

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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.

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,

${\displaystyle f(t)=(1-2\pi ^{2}f_{_{M}}^{2}t^{2})e^{-\pi ^{2}f_{_{M}}^{2}t^{2}}}$.

The frequency domain representation of the wavelet is given by,

${\displaystyle F(f)={\frac {2}{\sqrt {\pi }}}{\frac {f^{2}}{f_{M}^{3}}}e^{-{\frac {f^{2}}{f_{M}^{2}}}}}$

Where,

${\displaystyle T_{D}={\frac {\sqrt {6}}{\pi f_{M}}}}$ and ${\displaystyle T_{R}=T_{D}/{\sqrt {3}}}$.

The mean frequency ${\displaystyle f_{mean}={\frac {2}{\sqrt {\pi }}}f_{M}}$ and the median frequency ${\displaystyle f_{median}=1.08f_{M}}$.

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

${\displaystyle \pm {\frac {\sqrt {3/2}}{f_{_{M}}\pi }}}$

These minima have the value

${\displaystyle A_{\mathrm {min} }=-{\frac {2}{e^{3/2}}}}$