# Difference between revisions of "Ricker wavelet relations"

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

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+ | [[file:Ch06_fig6-21a.png|thumb|center|{{figure number|6.21a.}} Ricker wavelet (i) in time domain and (ii) in frequency domain.]] | ||

=== Solution === | === Solution === |

## Latest revision as of 15:34, 8 November 2019

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 |

## Contents

## Problem 6.21a

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} g(t)=(1-2\pi^{2} f_{m}^{2} t^{2})e^{-(\pi f_{m} t)^{2}}, \end{align} }****(**)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{m}}**
, being the peak frequency, has the Fourier transform [Figure 6.21a(ii)]

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} G(f)=(2/\sqrt{\pi })(f/f_{m})^{2} e^{-(f/f_{m})^{2}},\; \gamma(f) = 0, \end{align} }****(**)

**where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{-at^{2}}}**
is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} e^{-at^{2}} \leftrightarrow (\pi /a)^{1/2} e^{-\omega^{2} /4a} \end{align} }****(**)

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

### Solution

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} g(t)=(1-2at^{2})e^{-at^{2}} = e^{-at^{2} } -2at^{2} e^{-at^{2}}, \end{align} }****(**)

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a=(\pi f_{M} )^{2}}**
. The transform of the first term is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(t)\leftrightarrow G(\omega)}**
, then,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} (-jt)^{n} g(t)\leftrightarrow \frac{{\rm d}^{n} G(\omega )}{{\rm d}\omega ^{n}}, \end{align} }**

that is, for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=2}**
,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} -t^{2} g(t)\leftrightarrow \frac{{\rm d}^{2} G(\omega )}{{\rm d}\omega ^{2}}. \end{align} }**

The transform of the second term now becomes

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align} }**

Adding the two transforms, we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} &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{align} }**

## Problem 6.21b

Show that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{M}}**
is the peak of the frequency spectrum.

### Solution

To find the peak frequency, we set the derivative **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm d}G(f)/{\rm d}f}**
equal to zero. Thus, omitting the constant factor,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align} }**

so **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f=f_{M}}**
for a maximum.

## Problem 6.21c

Show that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_{D} /T_{R} =\sqrt{3}}**
(see Figure 6.21a) and that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_{D} f_{M} =\sqrt{6} /\pi.}**

### Solution

Since **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(t)=0}**
for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t=+T_{R} /2}**
, we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} [1-2(\pi f_{M} T_{R} /2)^{2}]e^{-(\pi f_{M} T_{R} /2)^{2} } =0, \end{align} }****(**)

hence **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_{R} =\pm \sqrt{2} /(\pi f_{M})}**
.

Moreover, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(t)}**
is a minimum for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t=\pm T_{D} /2}**
, so **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_{D} /2}**
is a root of

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} ({\rm d}/{\rm d}t)[(1-2at^{2} )e^{-at^{2} } ]=0, \end{align} }**

that is, of the equation

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} e^{-at^{2} } [-4at+(1-2at^{2} )(-2at)]=0,\\ \mbox {so} \qquad\qquad\ t=T_{D} /2=[\sqrt{(3/2)} ]/\pi f_{M}. \end{align} }****(**)

Hence, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_{D} /T_{R} =\sqrt{3}}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_{D} f_{M} =\sqrt{6/\pi}}**
.

## Continue reading

Previous section | Next section |
---|---|

Causes of high-frequency losses | Improvement of signal/noise ratio by stacking |

Previous chapter | Next chapter |

Geometry of seismic waves | Characteristics of seismic events |

## 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