# Ricker wavelet relations

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),

**(**)

**, being the peak frequency, has the Fourier transform [Figure 6.21a(ii)]**

**(**)

**where 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 is

**(**)

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

### Solution

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

**(**)

where . The transform of the first term is . To get the transform of the second term, we use Sheriff and Geldart, 1995, equation (15.142) which states that when , then,

that is, for ,

The transform of the second term now becomes

Adding the two transforms, we have

## Problem 6.21b

Show that is the peak of the frequency spectrum.

### Solution

To find the peak frequency, we set the derivative equal to zero. Thus, omitting the constant factor,

so for a maximum.

## Problem 6.21c

Show that (see Figure 6.21a) and that

### Solution

Since for , we have

**(**)

hence .

Moreover, is a minimum for , so is a root of

that is, of the equation

**(**)

Hence, and .

## Continue reading

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

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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