Ondícula de Ricker

(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.

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La amplitud f(t) de la ondícula de Ricker con una frecuencia pico f_M en tiempo t está dada por,

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

The frequency domain representation of the wavelet is given by,

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

Where,

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

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

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

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

These minima have the value

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