# Calculation of inverse filters

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 |

## Problem 9.18

Assuming that the signature of an air-gun array is a unit impulse and that the recorded wavelet after transmission through the earth is , find the inverse filter that will remove the earth filtering. How many terms should the filter include?

### Solution

The inverse filter (see problem 9.7) is a filter that will restore the source impulse, i.e.,

or in the frequency domain where ,

Thus,

**(**)

where .

Since has the magnitude , the magnitude of for all values of , and we can expand equation (9.18a) using the binomial theorem [see equation (4.1b)] and Sheriff and Geldart, 1995, equation (15.43):

**(**)

We first find neglecting powers higher than :

We can verify the accuracy of by multiplying by . We have

We see that the inverse filter is exact as far as the term and terms for higher powers are small. The overall effect is to create a small tail whose energy is 0.00149 or 0.1%.

To determine how the accuracy depends on the number of terms used in , we observe the effect on the product as we successively drop high powers in . Dropping the term in yields the product and the energy of the tail is now 0.01063 or 1.1%. If we want accuracy of at least 1%, we must therefore retain the term.

If we also delete the term in [but not in ] the product becomes and the energy of the tail is 0.01850 or 1.8%. If we go one step further and drop the term in , we get and the energy of the tail is 0.13107 or 13.1%.

## Continue reading

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Deconvolution methods | Inverse filter to remove ghosting and recursive filtering |

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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