# The convolutional model – book

Series | Geophysical References Series |
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Title | Problems in Exploration Seismology and their Solutions |

Author | Lloyd P. Geldart and Robert E. Sheriff |

Chapter | 9 |

Pages | 295 - 366 |

DOI | http://dx.doi.org/10.1190/1.9781560801733 |

ISBN | ISBN 9781560801153 |

Store | SEG Online Store |

## Contents

## Problem 9.6a

The earth acts as a filter to seismic energy to change the waveshape so that what we record is a distorted version of the waveshape we send into the earth. Assume that the signal generated by a seismic source is so short that it can be represented adequately by an impulse; what aspects of passage through the earth will distort the impulse and what will be the effects?

### Solution

The signal from a seismic source will be distorted: (a) by the region near the source where the stresses are so high as to produce nonlinear effects , (b) by the near-surface region where absorption is especially large , (c) by absorption in passage through the main body of the earth , and (d) perhaps by other factors. The first of these will change an impulsive signal to a waveform , which we can think of as a series of impulses with amplitudes that give the shape . Passage through the near surface will replace each of these impulses by multiplied by the scaled value , that is, . Absorption will change this series to (where the absorption is frequency dependent). Reflection from interfaces in the earth can also be thought of as additional convolutions changing the waveform to , etc. Thus the effect on the waveform can be thought of as a sequence of convolutions. In ordinary reflection seismic work the objective is to determine the reflectivity so we combine all the other waveshape-changing factors into an equivalent wavelet, generally called the *embedded wavelet* :

where . This concept is called the *convolutional model*.

## Problem 9.6b

What is the effect if the source is not impulsive?

### Solution

A nonimpulsive source , such as the vibroseis, has the effect of replacing with , but otherwise the reasoning in part (a) is unchanged.

## Continue reading

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Alias filters | Water reverberation filter |

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Reflection field methods | Geologic interpretation of reflection data |

## Also in this chapter

- Fourier series
- Space-domain convolution and vibroseis acquisition
- Fourier transforms of the unit impulse and boxcar
- Extension of the sampling theorem
- Alias filters
- The convolutional model
- Water reverberation filter
- Calculating crosscorrelation and autocorrelation
- Digital calculations
- Semblance
- Convolution and correlation calculations
- Properties of minimum-phase wavelets
- Phase of composite wavelets
- Tuning and waveshape
- Making a wavelet minimum-phase
- Zero-phase filtering of a minimum-phase wavelet
- Deconvolution methods
- Calculation of inverse filters
- Inverse filter to remove ghosting and recursive filtering
- Ghosting as a notch filter
- Autocorrelation
- Wiener (least-squares) inverse filters
- Interpreting stacking velocity
- Effect of local high-velocity body
- Apparent-velocity filtering
- Complex-trace analysis
- Kirchhoff migration
- Using an upward-traveling coordinate system
- Finite-difference migration
- Effect of migration on fault interpretation
- Derivative and integral operators
- Effects of normal-moveout (NMO) removal
- Weighted least-squares