# Space-domain convolution and vibroseis acquisition

Series | Geophysical References Series |
---|---|

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

The techniques and concepts of convolution, aliasing, -transforms, and so on can be applied to other domains than the time-domain. Express the source and group patterns of Figure 9.2a as functions of (horizontal coordinate) and convolve the two to verify the effective pattern shown.

### Background

The *convolution* of two functions and , written , is defined as

**(**)

For digital functions (see problem 9.4), the integrals become sums:

**(**)

These equations state that convolution is equivalent to superposition in which each element of one function is replaced by the other function multiplied (weighted) by the element being replaced. The sum of all values at a given time is the value of at that instant.

Another explanation of convolution is the following: We note that is reflected in the -axis; in other words, the curve of is the same as that of except that it is reversed in direction [keeping fixed]. The function is displaced units to the right. Thus, in equation (9.2a) the value of at the time is obtained by moving to the right units, reflecting it in the -axis, then summing (integrating) the products of corresponding abscissas. The result is the same whichever function is displaced and reflected.

Arrays are discussed in problem 8.6, aliasing in problem 9.4, and -transforms in Sheriff and Geldart, 1995, section 15.5.3.

### Solution

Taking the spatial sampling interval as 16.5 m, the source pattern consists of [1,1,2,2,3,3,3, 3,2,2,1,1] for a total array length of 200 m. The geophone group is an array 100 m long; to convolve the source and receiver arrays, they should have the same spatial intervals, so we take six receivers spaced 16.5 m apart. To convolve the two arrays, we replace each element of the receiver array with the source array (taking the weights as unity). The result is that shown at the bottom of Figure 9.2a.

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Fourier series | Fourier transforms of the unit impulse and boxcar |

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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