# Derivative and integral operators

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 9.31a

The operator is called the “derivative operator”; explain why.

### Solution

In terms of finite differences (see problem 9.29), the derivative of a function is approximately equal to the change in values between two adjacent points divided by the distance between the two points. Convolving with gives values of the finite-difference derivative of for different values of . To show this, we find the term of . From equation (9.2b), we have

Each term (except the first) equals at . Taking , we see that is minus the finite difference derivative at .

## Problem 9.31b

What is the integral operator?

### Solution

The integral, , for continuous functions becomes a summation for digital functions, the equivalent of the above integral being . If we take the operator where both and have terms, we obtain the following for the terms of (omitting ):

The terms are equivalent to the integrals between 0 and for . After the value is reached, the upper limit remains while the lower limit is successively 1, 2, . . . , . Thus we conclude that the operator [1, 1, 1, . . . , 1] is an integral operator. By taking with fewer terms than , we can get the equivalent of an integral over selected parts of .

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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