# Ondícula de Ricker

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(rik’ ∂r) Ondícula de fase cero, la segunda derivada de la función gaussiana o la tercera derivada de la función de densidad de probabilidad normal. La ondícula de Ricker se utiliza a menudo en modelado y fabricación de sismogramas sintéticos. Véase Figura R-14. Nombrado por Norman H. Ricker (1896–1980), geofísico estadounidense.

FIG. R-14. Ondícula de Ricker. (a) Representaciones en el dominio del tiempo y (b) en el dominio de las frecuencias.

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

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