Ricker wavelet relations

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	6
Pages	181 - 220
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 6.21a

Verify that the Ricker wavelet in Figure 6.21a(i),

${\begin{aligned}g(t)=(1-2\pi ^{2}f_{m}^{2}t^{2})e^{-(\pi f_{m}t)^{2}},\end{aligned}}$

(6.21a)

$f_{m}$, being the peak frequency, has the Fourier transform [Figure 6.21a(ii)]

${\begin{aligned}G(f)=(2/{\sqrt {\pi }})(f/f_{m})^{2}e^{-(f/f_{m})^{2}},\;\gamma (f)=0,\end{aligned}}$

(6.21b)

where $\gamma (f)$ is the phase.

Background

Fourier transforms are discussed in problem 9.3 and theorems on Fourier transforms in Sheriff and Geldart, 1995, section 15.2.6.

The transform of $e^{-at^{2}}$ is

${\begin{aligned}e^{-at^{2}}\leftrightarrow (\pi /a)^{1/2}e^{-\omega ^{2}/4a}\end{aligned}}$

(6.21c)

[Papoulis, 1962: p. 25, equation (2-68)].

Figure 6.21a.  Ricker wavelet (i) in time domain and (ii) in frequency domain.

Solution

The time-domain expression for the Ricker wavelet can be written in the form

${\begin{aligned}g(t)=(1-2at^{2})e^{-at^{2}}=e^{-at^{2}}-2at^{2}e^{-at^{2}},\end{aligned}}$

(6.21d)

where $a=(\pi f_{M})^{2}$. The transform of the first term is $(\pi /a)^{1/2}e^{-\omega ^{2}}/4a$. To get the transform of the second term, we use Sheriff and Geldart, 1995, equation (15.142) which states that when $g(t)\leftrightarrow G(\omega )$, then,

${\begin{aligned}(-jt)^{n}g(t)\leftrightarrow {\frac {{\rm {d}}^{n}G(\omega )}{{\rm {d}}\omega ^{n}}},\end{aligned}}$

that is, for $n=2$,

${\begin{aligned}-t^{2}g(t)\leftrightarrow {\frac {{\rm {d}}^{2}G(\omega )}{{\rm {d}}\omega ^{2}}}.\end{aligned}}$

The transform of the second term now becomes

${\begin{aligned}2a{\frac {{\rm {d}}^{2}}{{\rm {d}}\omega ^{2}}}\left[\left({\frac {\pi }{a}}\right)^{1/2}e^{-(\omega ^{2}/4a)}\right]=2(\pi a)^{1/2}{\frac {\rm {d}}{{\rm {d}}\omega }}\left[-\left({\frac {\omega }{2a}}\right)e^{-(\omega ^{2}/4a)}\right]\\=-(\pi /a)^{1/2}{\frac {\rm {d}}{{\rm {d}}\omega }}\left[\omega e^{-(\omega ^{2}/4a)}\right]=-(\pi /a)^{1/2}\left[1-\left({\frac {\omega ^{2}}{2a}}\right)\right]e^{-(\omega ^{2}/4a)}.\end{aligned}}$

Adding the two transforms, we have

${\begin{aligned}&g(t)\leftrightarrow (2/{\sqrt {\pi }})(\omega /\omega _{M})^{2}e^{-(\omega /\omega _{M})^{2}}=(\pi /a^{3})^{1/2}(\omega ^{2}/2)e^{-(\omega /\omega _{M})^{2}}\\&\leftrightarrow (2/{\sqrt {\pi }})(f^{2}/f_{M}^{3})e^{-(f/f_{M})^{2}}.\end{aligned}}$

Problem 6.21b

Show that $f_{M}$ is the peak of the frequency spectrum.

Solution

To find the peak frequency, we set the derivative ${\rm {d}}G(f)/{\rm {d}}f$ equal to zero. Thus, omitting the constant factor,

${\begin{aligned}\left({\frac {\rm {d}}{{\rm {d}}f}}\right)\left[f^{2}e^{-(f/f_{M})^{2}}\right]=e^{-(f/f_{M})^{2}}\left[2f-f^{2}(2f/f_{M}^{2})\right]=0,\end{aligned}}$

so $f=f_{M}$ for a maximum.

Problem 6.21c

Show that $T_{D}/T_{R}={\sqrt {3}}$ (see Figure 6.21a) and that $T_{D}f_{M}={\sqrt {6}}/\pi .$

Solution

Since $g(t)=0$ for $t=+T_{R}/2$, we have

${\begin{aligned}[1-2(\pi f_{M}T_{R}/2)^{2}]e^{-(\pi f_{M}T_{R}/2)^{2}}=0,\end{aligned}}$

(6.21e)

hence $T_{R}=\pm {\sqrt {2}}/(\pi f_{M})$.

Moreover, $g(t)$ is a minimum for $t=\pm T_{D}/2$, so $T_{D}/2$ is a root of

${\begin{aligned}({\rm {d}}/{\rm {d}}t)[(1-2at^{2})e^{-at^{2}}]=0,\end{aligned}}$

that is, of the equation

${\begin{aligned}e^{-at^{2}}[-4at+(1-2at^{2})(-2at)]=0,\\{\mbox{so}}\qquad \qquad \ t=T_{D}/2=[{\sqrt {(3/2)}}]/\pi f_{M}.\end{aligned}}$

(6.21f)

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

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Table of Contents (book)

Also in this chapter

• Characteristics of different types of events and noise
• Horizontal resolution
• Reflection and refraction laws and Fermat’s principle
• Effect of reflector curvature on a plane wave
• Diffraction traveltime curves
• Amplitude variation with offset for seafloor multiples
• Ghost amplitude and energy
• Directivity of a source plus its ghost
• Directivity of a harmonic source plus ghost
• Differential moveout between primary and multiple
• Suppressing multiples by NMO differences
• Distinguishing horizontal/vertical discontinuities
• Identification of events
• Traveltime curves for various events
• Reflections/diffractions from refractor terminations
• Refractions and refraction multiples
• Destructive and constructive interference for a wedge
• Dependence of resolvable limit on frequency
• Vertical resolution
• Causes of high-frequency losses
• Ricker wavelet relations
• Improvement of signal/noise ratio by stacking

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