# Examples

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

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

Chapter | 8 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | 9781560801481 |

Store | SEG Online Store |

Let us now discuss a model consisting of a surface interface overlying two buried interfaces, with the three interfaces separated by arbitrary two-way layer traveltimes. Let the reflection coefficients be given by *a*, *b*, *c*. Let the two-way traveltime between the surface and the first buried interface be *S*, and let the two-way traveltime between the first buried interface and the second buried interface be *T*. In other words, the surface reflection coefficient is , the reflection coefficient for the first buried interface is , and the reflection coefficient for the second buried interface is . The *Z*-transform of the reflectivity is

**(**)

The right-hand side of the autocorrelation of the reflectivity is

**(**)

If we replace by 1, we obtain the feedback loop

**(**)

There are three reverberations. The reverberation between the surface and the first buried interface contributes , the reverberation between the second and third buried interfaces contributes , and the reverberation between the surface and third buried interface contributes . The synthetic trace now is given by the feedforward-feedback filter (Figure 11):

**(**)

Suppose that . Then the synthetic trace is given by

**(**)

which is

**(**)

The shot occurs at time 0, the first primary *b* occurs at time 2, the first-surface- first-interface multiple occurs at time 4, the second-surface–first-interface multiple occurs at time 6, the second primary occurs at time 7, the third-surface–first-interface multiple occurs at time 8, the peg-leg multiple occurs at time 9, and so on. Figure 11a shows the reflectivity for the case . The leading portion of the corresponding impulsive synthetic trace is shown in Figure 11b.

## Continue reading

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Synthetic seismogram with multiples | Small and white reflection coefficients |

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Wavelets | Wavelet Processing |

## Also in this chapter

- Introduction
- Polarity
- Reflection coefficients and transmission coefficients
- Ghost reflection
- Layer-cake model
- Synthetic seismogram without multiples
- Water reverberations
- Synthetic seismogram with multiples
- Small and white reflection coefficients
- Appendix H: Exercises